Existence, sign and asymptotic behaviour for a class of integro-differential elliptic type problems

Abstract

In this work we study existence, sign and asymptotic behaviour of solutions for a class of elliptic problems of the integral-differential type under the presence of a parameter. A careful analysis of the influence of the referred parameter on the structure of the set of solutions is made, by considering different reaction terms. Among our main contributions are: (1) a positive answer to Remark 2.4 in Allegretto and Barabanova (Proc R Soc Edinb A 126(3):643–663, 1996); (2) a detailed treatment of the associated eigenvalue problem; (3) The first result involving the existence of a ground-state solution for this class of problems.

1 Introduction

In this paper we are concerned with the following class of problems with a nonlocal term coupled:

where $a\in \mathbb {R}$, $b, c, d:\Omega \rightarrow \mathbb {R}$ and $g:\mathbb {R}\rightarrow \mathbb {R}$ are continuous functions and $\Omega$ is bounded smooth domain in $\mathbb {R}^N$.

Problems like (PP) are the steady-state version of quite relevant models in applications and appear in a great range of real-life situations. For instance, when $b(x)=a=1$, problem (PP) is the stationary version of the parabolic equation

$$\begin{aligned} \left\{ \begin{array}{ll} u_t-\Delta u +c(x)\int _{\Omega }d(x)u dx=g(u) &{} \text{ in } \Omega \times (0,\infty ),\\ u(x, t)=0 &{} \text{ on } \partial \Omega \times (0, \infty ). \end{array}\right. \end{aligned}$$

(1.1)

A biological interpretation of problem (1.1) is provided in [11], in which, it describes the poisoning of a population by metabolic products. In such a context, u(x) is the population density at x, g(u) is a birth/death rate, $\int _{\Omega }d(x)u dx$ is a coupling that represents a weighted average of the population, c(x) is a positive local rate that determines how it individual is affected by the average and the Dirichlet boundary condition means that the population is surrounded by a hostile exterior region, where the individuals cannot survive. In [11], among other things, the authors highlight the particular situation in which c(x) is a constant and $d(x)=1$, where the weighted average is replaced by the total population. Another biological model, considering the influence of the total population in the diffusion of bacterias, can be found in [9]. We also recommend reference [12] for an interesting discussion about the use of averaged density or its $L^1$ norm, as a form to incorporate the so-called “crowding effects”, in dispersion models of ecological systems.

In the general case, as demonstrated in [2], problem (PP) is also significant from a physical point of a view. In fact, the study of (PP) is motivated by a nonlinear system of equations proposed in [2] as model micro-sensor thermistor behaviour in the presence of a gas. In [1], a particular case of the nonlinear system in [2] is considered ($a>0, c(x)=d(x)=1$). In such model, u describes the distribution of temperature, the integral term corresponds to the heat loss from the surface of the thermistor to the surrounding gas and a is a parameter involving the gas pressure and other factors. Still considering physical applications, a variant of the Eq. (1.1) with Neumann boundary condition is studied in [7] as a model for an electric wire.

The authors in [1] investigated the existence of solution for the general problem (PP) when $g=g(x)$ and generalized a result previously obtained in [6], for a one-dimensional initial value problem with $c=d$, describing in which cases the solution is unique. Moreover, the main result in [1] is the proof of a sort of local maximum principle for (PP), when the modulus of the parameter a is small enough (see for instance Theorem 2.3 and Corollary 2.5 in [1]). Since the approach used in [1], which is based in the Fredholm Theory, is not so constructive, the authors did not get many informations about qualitative/asymptotic behaviour of the solutions with respect to a. Actually, the authors only conjecture that when a is large, then the solution may change sign (see Remark 2.4 in [1]).

To the best of our knowledge, the eigenvalue problem, that is, $g(u)=\lambda u$ in (PP), has been mainly treated in the unidimensional case (see for example [11] and [13]). In [1], the N-dimensional eigenvalue problem is also considered (with $b=1$ and $c=d$), but the authors are mainly concentrated in the study of the first eigenvalue and its associated eigenfunctions. More recent works involving the integro-differential operator in (PP) and other kinds of nonlinearities can be found in the literature. We refer the interested reader to [3], [8] and [10].

In this work, we are focused in studying problem (PP) when $b=c=d=1$ and N is a general dimension. To be more precise, we are going to investigate the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u +a\int _{\Omega }u dx=g(x, u) &{} \text{ in } \Omega ,\\ u=0 &{} \text{ on } \partial \Omega , \end{array}\right. \end{aligned}$$

(1.2)

where $\Omega \subset \mathbb {R}^{N}$ is a smooth bounded domain, a is a real parameter and $g=g(x)$, $g(u)=\lambda u$ or g is an asymptotically linear nonlinearity at the origin. Our main contribution regarding case $g=g(x)$ has been to get more information about how the solution of (1.2) varies with the parameter a (see Theorem 2.2). This fact allowed us to provide a complete description about the influence of the parameter a on the sign of the solution. As a consequence, we give a positive answer to an Allegretto-Barabanova conjecture involving the parameter a (see Theorem 2.5).

We also give contributions in the study of the eigenvalue problem ($g(u)=\lambda u$): 1) It is observed an interesting relation between the existence of eigenvalues for problem (1.2) and a particular convergente real series involving the eigenvalues and the integral of eigenfunctions of the laplacian operator with Dirichlet Boundary condition (see Theorems 3.3 and 3.4); 2) We completely describe the spectrum of the integro-differential operator in (1.2) and proved that, under suitable assumptions, the eigenvalues have a variational characterization. In particular, we provide a sort of generalization of Proposition 4.1 in [1] (see Corollary 3.5); 3) We study the behaviour of the curve of eigenvalues with respect to a and the sign of the associated. eigenfunctions (see Corollaries 3.6 and 3.7).

Finally, we think that this is the first work to investigate existence and asymptotic behaviour of ground-state solutions for (1.2) involving an asymptotically linear nonlinearity at the origin. Our main result in this direction is enunciated in Theorem 4.4.

The paper is divided as follows: In Sects. 2 and 3 we study the linear and the eigenvalue problems. In Sect. 4 it is studied the asymptotically linear problem at the origin. Finally, Sect. 5 presents some related open questions and provide a biological understanding about some of the results obtained in previous sections.

2 The linear problem

In the present section we are interested in investigating the following class of linear problems

where $\Omega \subset \mathbb {R}^{N}$ is a smooth bounded domain, a is a real parameter and $f\in L^{2}(\Omega )$ is a given function.

As will be clear, the existence and qualitative properties of solution for ($P_a$), with $a\ne 0$, is closely related with the behaviour of the unique solution of ($P_a$), with $a=0$. In this sense, to highlight this link, next result is crucial. Before stating it, we point out that, throughout this paper, e denotes the unique solution of

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta w =1 &{} \text{ in } \Omega ,\\ w=0 &{} \text{ on } \partial \Omega . \end{array}\right. \end{aligned}$$

(2.1)

Moreover, in the sequel, c is a generic constant and $\{\varphi _j\}$ denotes an orthonormal basis of $H_{0}^{1}(\Omega )$ composed by eigenfunctions associated to eigenvalues of the laplacian operator with Dirichlet boundary condition.

Lemma 2.1

Let $e^{\perp }$ be the orthogonal complement of e in $H_{0}^{1}(\Omega )$. Then

$$\begin{aligned} e=\sum _{j=1}^{\infty }\left( \int _{\Omega }\varphi _j dx\right) \varphi _j \end{aligned}$$

(2.2)

and

$$\begin{aligned} e^{\perp }=\left\{ u\in H_{0}^{1}(\Omega ): \int _{\Omega }u dx=0\right\} . \end{aligned}$$

(2.3)

Proof

In order to show (2.3), it is enough to note that

$$\begin{aligned} \int _{\Omega }\nabla e\nabla v dx=\int _{\Omega }v dx, \ \forall \ v\in H_{0}^{1}(\Omega ). \end{aligned}$$

(2.4)

To prove (2.2), let

$$\begin{aligned} e=\sum _{j=1}^{\infty }\alpha _j\varphi _j \end{aligned}$$

be the decomposition of e in terms of the orthonormal basis $\{\varphi _j\}$ of $H_{0}^{1}(\Omega )$. By choosing $v=\varphi _j$ in (2.4), we get

$$\begin{aligned} \alpha _j=\int _{\Omega }\nabla e\nabla \varphi _j dx=\int _{\Omega }\varphi _j dx, \end{aligned}$$

and the result follows. $\square$

An immediate consequence of previous lemma is the following orthogonal decomposition $H_{0}^{1}(\Omega )=\mathbb {R}e\oplus X$, where

$$\begin{aligned} X=\left\{ u\in H_{0}^{1}(\Omega ): \int _{\Omega }u dx=0\right\} \end{aligned}$$

is the linear space of zero mean functions of $H_{0}^{1}(\Omega )$. From now on, we are going to denote by

$$\begin{aligned} u_{f}=\alpha _{f}e+P_{X}u_{f}, \end{aligned}$$

(2.5)

the unique weak solution of the problem

where $\alpha _{f}\in \mathbb {R}$ and $P_{X} u_{f}$ is the orthogonal projection of $u_f$ onto X. Note that

$$\begin{aligned} \alpha _{f}\int _{\Omega }e dx=\alpha _{f}\Vert e\Vert ^{2}=\int _{\Omega }\nabla u_{f}\nabla e dx=\int _{\Omega }fe dx, \end{aligned}$$

showing that

$$\begin{aligned} \alpha _{f}=\frac{\int _{\Omega }fe dx}{\int _{\Omega }e dx}. \end{aligned}$$

(2.6)

Remark 1

It is important to point out that $P_Xu_f$ is the unique solution of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u =f(x)-\frac{\int _{\Omega }fe dx}{\int _{\Omega }e dx} &{} \text{ in } \Omega ,\\ u=0 &{} \text{ on } \partial \Omega . \end{array}\right. \end{aligned}$$

(2.7)

Consequently, it follows by (2.5), (2.6) and (2.7), that $f\equiv c$ is constant if, and only if,

$$\begin{aligned} P_{X}u_f= 0 \ \text{ and } \ \alpha _f=c. \end{aligned}$$

Thus, when f is not constant, $P_{X}u_f$ is a nontrivial solution of (2.7) which changes sign (since $P_{X}u_f\in X$).

Theorem 2.2

Let $f\in L^{2}(\Omega )$, $\alpha _{f}\in \mathbb {R}$ and $P_{X} u_{f}\in X$ be defined as in (2.5), (2.6) and (2.7). The following assertions hold:

• (i) If $a\ne -1/\int _{\Omega }e dx$, then the problem ($P_a$) has a unique solution $u_{a}$ given by

  $$\begin{aligned} u_{a}=\frac{\alpha _{f}}{\left( 1+a\int _{\Omega }e dx\right) }e+P_{X} u_{f}. \end{aligned}$$

  (2.8)

• (ii) If $a= -1/\int _{\Omega }e dx$, then the problem ($P_a$) has solution if, and only if, $\int _{\Omega }fe dx=0$. Furthermore, in this case, a function u is a weak solution of ($P_a$) if, and only if,

  $$\begin{aligned} u=u_{r}=\frac{r}{\int _{\Omega }e dx}e+P_{X}u_{f}, \ r\in \mathbb {R}. \end{aligned}$$

  (2.9)

Proof

Let us consider a suitable auxiliary problem given by

where $r\in \mathbb {R}$ is a fixed real number. Since, for each $a\ne 0$, the function $f-ar$ belongs to $L^{2}(\Omega )$, there exists a unique $u(r)\in H_{0}^{1}(\Omega )$, such that

$$\begin{aligned} \int _{\Omega }\nabla u(r)\nabla v dx+ar\int _{\Omega }v dx=\int _{\Omega }fv dx, \end{aligned}$$

for all $v\in H_{0}^{1}(\Omega )$. Moreover, let e as defined in (2.1). Thus, there exist $\alpha (r)\in \mathbb {R}$ and $P_{X}u(r)\in X$ such that

$$\begin{aligned} u(r)=\alpha (r)e+P_{X}u(r). \end{aligned}$$

We are going to prove that

$$\begin{aligned} \alpha (r)=\alpha _{f}-ar \ \text{ and } \ P_{X}u(r)=P_{X}u_{f}, \end{aligned}$$

where $\alpha _{f}$ and $P_{X}u_{f}$ are given in (2.5), (2.6) and (2.7). That is,

$$\begin{aligned} u(r)=(\alpha _{f}-ar)e+P_{X}u_{f}. \end{aligned}$$

(2.10)

In fact, a straightforward calculation shows that

$$\begin{aligned}{} & {} \int _{\Omega }\nabla \left( (\alpha _{f}-ar) e+P_{X}u_{f}\right) \nabla v dx+ar\int _{\Omega }v dx=(\alpha _{f}-ar)\int _{\Omega }\nabla e\nabla v dx\\{} & {} \quad + \int _{\Omega }\nabla P_{X}u_{f} \nabla v dx +ar\int _{\Omega }v dx. \end{aligned}$$

By using the definition of e and standing for $P_{X}v$ the orthogonal projection of v onto X, we obtain

$$\begin{aligned}{} & {} \int _{\Omega }\nabla \left( (\alpha _{f}-ar) e+P_{X}u_{f}\right) \nabla v dx+ar\int _{\Omega }v dx=(\alpha _{f}-ar)\int _{\Omega }\nabla e\nabla vdx\\{} & {} \quad + \int _{\Omega }\nabla P_{X}u_{f} \nabla P_{X}v dx +ar\int _{\Omega }\nabla e\nabla v dx. \end{aligned}$$

Consequently, by (2.5),

$$\begin{aligned} \int _{\Omega }\nabla \left( (\alpha _{f}-ar) e+P_{X}u_{f}\right) \nabla v dx+ar\int _{\Omega }v dx=\int _{\Omega } \nabla (\alpha _{f}e)\nabla vdx+ \int _{\Omega }f P_{X}v dx. \end{aligned}$$

Since $v=\gamma e+P_{X}v$, for some $\gamma \in \mathbb {R}$, we obtain

$$\begin{aligned} \int _{\Omega }\nabla \left( (\alpha _{f}-ar) e+P_{X}u_{f}\right) \nabla v dx+ar\int _{\Omega }v dx=\int _{\Omega }\nabla u_{f}\nabla (\gamma e) dx+ \int _{\Omega }f P_{X}v dx. \end{aligned}$$

Thus,

$$\begin{aligned} \int _{\Omega }\nabla \left( (\alpha _{f}-ar) e+P_{X}u_{f}\right) \nabla v dx+ar\int _{\Omega }v dx=\int _{\Omega } f(\gamma e) dx+ \int _{\Omega }f P_{X}v dx. \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{\Omega }\nabla ((\alpha _{f}-ar) e+P_{X}u_{f})\nabla v dx+ar\int _{\Omega }v dx=\int _{\Omega } fv dx, \end{aligned}$$

for all $v\in H_{0}^{1}(\Omega )$. Showing that $(\alpha _{f}-ar) e+P_{X}u_{f}$ is a solution of ($P_{r}$) and, by uniqueness,

$$\begin{aligned} u(r)=(\alpha _{f}-ar) e+P_{X}u_{f}. \end{aligned}$$

(2.11)

Now, observe that u(r) in (2.11) is a solution of the problem ($P_a$) if

$$\begin{aligned} r=\int _{\Omega }u(r) dx, \end{aligned}$$

(2.12)

or yet

$$\begin{aligned} r=(\alpha _{f}-ar)\int _{\Omega }e dx. \end{aligned}$$

Last equality is equivalent to

$$\begin{aligned} \left( 1+a\int _{\Omega }e dx\right) r=\alpha _{f}\int _{\Omega }e dx. \end{aligned}$$

It follows from (2.6) that

$$\begin{aligned} \left( 1+a\int _{\Omega }e dx\right) r=\int _{\Omega }fe dx. \end{aligned}$$

(2.13)

Now we have to consider two cases:

Case 1: $a\ne -1/\int _{\Omega }e dx$

In this case, it follows from (2.13) that r is uniquely determined by

$$\begin{aligned} r=\frac{\int _{\Omega }fe dx}{1+a\int _{\Omega }e dx}. \end{aligned}$$

(2.14)

To prove that $u_a$ in (2.8) is a solution of ($P_a$), it is enough to replace (2.6) and (2.14) in (2.11). Finally to prove that $u_a$ is unique, suppose that

$$\begin{aligned} u=\beta e+P_{X}u \end{aligned}$$

(2.15)

is a solution of ($P_a$) with $a\ne -1/\int _{\Omega }e dx$, then taking $v=e$ as a test function, we have

$$\begin{aligned} \left( 1+a\int _{\Omega }e dx\right) \int _{\Omega }u dx=\int _{\Omega }u dx+a\int _{\Omega } u dx\int _{\Omega }e dx=\int _{\Omega } fe dx. \end{aligned}$$

Thus,

$$\begin{aligned} \int _{\Omega }u dx=\frac{\int _{\Omega } fe dx}{\left( 1+a\int _{\Omega }e dx\right) }=\frac{\alpha _f\int _{\Omega }e dx}{\left( 1+a\int _{\Omega }e dx\right) } \end{aligned}$$

(2.16)

On the other hand, by (2.15),

$$\begin{aligned} \int _{\Omega }u dx=\beta \int _{\Omega }e dx. \end{aligned}$$

(2.17)

By comparing (2.16) and (2.17), we conclude that

$$\begin{aligned} \beta =\frac{\alpha _f}{\left( 1+a\int _{\Omega }e dx\right) }. \end{aligned}$$

(2.18)

Again, since u is a solution of ($P_a$), taking any $w\in X$ as a test function, we get

$$\begin{aligned} \int _{\Omega }\nabla P_{X}u\nabla w dx=\int _{\Omega }\nabla u\nabla w dx=\int _{\Omega }f w dx=\int _{\Omega }\nabla P_{X}u_{f} \nabla w dx, \end{aligned}$$

where last equality follows from the fact that $u_{f}$ is the unique solution of problem (LP). Therefore,

$$\begin{aligned} P_{X}u=P_{X}u_{f}. \end{aligned}$$

(2.19)

The proof of item (i) follows now from (2.15), (2.18) and (2.19).

Case 2: $a= -1/\int _{\Omega }e dx$

In this case, if $\int _{\Omega } fe dx=0$ then, it follows from (2.13) that (2.12) holds for all $r\in \mathbb {R}$. Thus, replacing $a= -1/\int _{\Omega }e dx$ and $\alpha _f=\int _{\Omega } fe dx/\int _{\Omega }e dx=0$ in (2.11), we derive that $u_r$ in (2.9) is a solution of ($P_a$), for all $r\in \mathbb {R}$. To prove the second part of (ii), suppose that ($P_a$) has a weak solution u with $a= -1/\int _{\Omega }e dx$. It follows from (2.11), that

$$\begin{aligned} u=\left( \alpha _{f}-a\int _{\Omega }u dx\right) e+P_{X}u_{f}. \end{aligned}$$

(2.20)

Taking $v=e$ as a test function, we have

$$\begin{aligned} \int _{\Omega }\nabla u\nabla e dx+a\int _{\Omega }u dx\int _{\Omega }e dx=\int _{\Omega }fe dx. \end{aligned}$$

By definition of e and the choice of a, we obtain

$$\begin{aligned} 0=\left( 1+a\int _{\Omega } e dx\right) \int _{\Omega }u dx=\int _{\Omega }fe dx. \end{aligned}$$

By replacing, $\alpha _f=\int _{\Omega }fe/\int _{\Omega }e dx=0$ and $a= -1/\int _{\Omega }e dx$ in (2.20) we conclude that u is in the form $u_r$, with $r=\int _{\Omega }u dx$. $\square$

Corollary 2.3

Let $f\in L^{2}(\Omega )$, $P_{X} u_{f}\in X$ be defined as in (2.5) and $u_{a}$ the unique solution of problem ($P_a$) as $a\ne -1/\int _{\Omega }e dx$. Then, the following statements hold:

• (i) The mapping $\mathbb {R}\backslash \{-1/\int _{\Omega }e dx\}\ni a\mapsto u_a\in H_0^1(\Omega )$ is $C^{2}$, where

  $$\begin{aligned} \frac{du_{a}}{da}=-\frac{\int _{\Omega }fe dx}{\left( 1+a\int _{\Omega }e dx\right) ^{2}}e, \end{aligned}$$

  (2.21)

  is the unique solution of

  $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta w +a \int _{\Omega }w dx= -\frac{\int _{\Omega }fe dx}{1+a\int _{\Omega }e dx} &{} \text{ in } \Omega ,\\ w=0 &{} \text{ on } \partial \Omega , \end{array}\right. \end{aligned}$$

  (2.22)

  and

  $$\begin{aligned} \frac{d^{2}u_{a}}{da^{2}}=-\frac{2\int _{\Omega }fe dx\int _{\Omega }e dx}{\left( 1+a\int _{\Omega }e dx\right) ^{3}}e, \end{aligned}$$

  (2.23)

  is the unique solution of

  $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta w +a\int _{\Omega }w dx = \frac{2\int _{\Omega }fe dx\int _{\Omega }e dx}{\left( 1+a\int _{\Omega }e dx\right) ^{2}} &{} \text{ in } \Omega ,\\ u=0 &{} \text{ on } \partial \Omega . \end{array}\right. \end{aligned}$$

  (2.24)

• (ii) The map $\mathbb {R}\backslash \{-1/\int _{\Omega }e dx\}\ni a\mapsto \int _{\Omega }u_a dx$ is $C^{2}$, decreasing in the intervals $(-\infty , -1/\int _{\Omega }e dx)$ and $(-1/\int _{\Omega }e dx, +\infty )$, concave in $(-\infty , -1/\int _{\Omega }e dx)$ and convex in $(-1/\int _{\Omega }e dx, \infty )$, if $\int _{\Omega }fe dx>0$. Moreover, it is increasing in the intervals $(-\infty , -1/\int _{\Omega }e dx)$ and $(-1/\int _{\Omega }e dx, +\infty )$, convex in $(-\infty , -1/\int _{\Omega }e dx)$ and concave in $(-1/\int _{\Omega }e dx, \infty )$, if $\int _{\Omega }fe dx<0$.

• (iii) The following asymptotic behaviours hold:

  $$\begin{aligned} u_a\rightarrow & {} P_{X}u_f \ \hbox {in} H_{0}^{1}(\Omega ) \hbox {, as} a\rightarrow \pm \infty ,\\ u_a\rightarrow & {} u_f \ \hbox {in} H_{0}^{1}(\Omega ) \hbox {, as} a\rightarrow 0,\\ \int _{\Omega }u_a dx\rightarrow & {} +\infty \ \hbox { if} \ \int _{\Omega }fe dx>0 \ \text{ and } \ \int _{\Omega }u_a dx\\\rightarrow & {} -\infty \ \text{ if } \ \int _{\Omega }fe dx<0, \ \text{ as }\ a\rightarrow \left( -1/\int _{\Omega }e dx\right) ^+\\ \end{aligned}$$

  and

  $$\begin{aligned} \int _{\Omega }u_a dx\rightarrow & {} -\infty \ \hbox { if} \ \int _{\Omega }fe dx>0 \ \text{ and } \ \int _{\Omega }u_a dx\\\rightarrow & {} +\infty \ \text{ if } \ \int _{\Omega }fe dx<0, \ \text{ as }\ a\rightarrow \left( -1/\int _{\Omega }e dx\right) ^-. \end{aligned}$$

Proof

(i) Since $a\ne -1/\int _{\Omega }e dx$, it is an immediate consequence of item (i) of Theorem 2.2 that $\mathbb {R}\backslash \{-1/\int _{\Omega }e dx\}\ni a\mapsto u_a\in H_0^1(\Omega )$ is $C^{2}$ with derivatives as in (2.21) and (2.23), and problems (2.22) and (2.24) have unique solutions. Now, a straightforward calculation shows that (2.21) is the solution of (2.22) and (2.23) is the solution of (2.24).

(ii) It is sufficient to note that, by item (i), we have

$$\begin{aligned} \frac{d}{da}\int _{\Omega }u_{a} dx=\int _{\Omega }\frac{du_{a}}{da} dx=-\frac{\int _{\Omega }fe dx\int _{\Omega }e dx}{\left( 1+a\int _{\Omega }e dx\right) ^{2}} \end{aligned}$$

and

$$\begin{aligned} \frac{d^{2}}{da^{2}}\int _{\Omega }u_{a} dx=\int _{\Omega }\frac{d^{2}u_{a}}{da^{2}} dx=\frac{2\int _{\Omega }fe dx\left( \int _{\Omega }e dx\right) ^{2}}{\left( 1+a\int _{\Omega }e dx\right) ^{3}}. \end{aligned}$$

(iii) A direct consequence of item (i) of Theorem 2.2. See Figs. 1 and 2 for an illustration of the bifurcation diagram of $\int _{\Omega } u_{a} dx$ with respect to a. $\square$

Fig. 1

Behaviour of $\int _{\Omega } u_{a} dx$ regarding a, as $\int _{\Omega }fe dx>0$

Fig. 2

Behaviour of $\int _{\Omega } u_{a} dx$ regarding a, as $\int _{\Omega }fe dx<0$

Next result is an immediate consequence of Theorem 2.2 and its relevance lies on the fact that it provides a kind of maximum principle to the integral of the solution, at the same time that it gives us conditions to the nonexistence of nonnegative and nonpositive solutions of ($P_a$), depending on the size of the parameter a.

Corollary 2.4

Let $f\in L^{2}(\Omega )$ be such that $f\ge 0$ and $f\ne 0$. If $a<-1/\int _{\Omega }e dx$ then $\int _{\Omega } u_{a} dx<0$. On the other hand, if $a>-1/\int _{\Omega }e dx$ then $\int _{\Omega } u_{a} dx>0$.

The following theorem generalizes for higher dimensions and any nonnegative and non-constant f in $L^{\infty }(\Omega )$, a fact observed by Allegretto and Barabanova [1] (see introduction in [1]) for the particular case in which $f(x)=\sin (\pi x)$ and $\Omega =(0, 1)$ in the problem ($P_a$). Furthermore, it highlights the role played by the size of the parameter a in the rupture of a maximum principle for the problem ($P_a$).

Theorem 2.5

Let $f\in L^{\infty }(\Omega )$ be such that $f\ge 0$ and f is not constant. Then, there exists $a_{*}>0$ such that:

• (i) $u_{a}>0$ in $\Omega$ and $\partial u_{a}/\partial \eta <0$ on $\partial \Omega$, if $0\le a<a_{*}$;

• (ii) $u_{a_{*}}\ge 0$ and, there exists a set $\Omega _{*}\subset \Omega$ such that $u_{a_*}=0$ on $\Omega _*$ or there exists a set $\Gamma _*\subset \partial \Omega$ such that the outward normal derivative $\partial u_a/\partial \eta$ vanishes on $\Gamma _*$;

• (iii) $u_{a}$ changes sign in $\Omega$, if $a>a_{*}$.

Proof

It is well known that the positive cone $\mathcal {P}=\{u\in C^{1, \beta }_{0}(\overline{\Omega }): u\ge 0\}$ of

$$\begin{aligned} C^{1, \beta }_{0}(\overline{\Omega })=\{u\in C^{1, \beta }(\overline{\Omega }): u=0 \ \hbox { on}\ \partial \Omega \} \end{aligned}$$

has nonempty interior, with

$$\begin{aligned} int(\mathcal {P})=\{u\in \mathcal {P}: u>0 \ \hbox {in} \Omega \hbox {and} \partial u/\partial \eta <0\hbox { on }\partial \Omega \}, \end{aligned}$$

see, for instance, [4]. Let us consider the set

$$\begin{aligned} A=\{a\in [0, \infty ): u_{a}\in int(\mathcal {P})\}. \end{aligned}$$

Since $f\in L^{\infty }(\Omega )$ and, for each $a\in [0, \infty )$, $\int _{\Omega }u_a dx$ is well defined. From elliptic regularity, it is known that $u_a\in C^{1, \beta }_{0}(\overline{\Omega })$. In what follows, we divide the proof into several steps.

Step 1: $A=[0, a_{*})$, where $a_{*}:=\sup A>0$.

First, note that $0\in A$. In fact, it follows from Theorem 2.2 that $u_0=u_{f}\in int(\mathcal {P})$, since $u_{f}$ is the unique solution of problem (LP).

Now, we are going to prove that A is a relative open set in $[0,\infty )$. For that, we first point out that $int(\mathcal {P})$ is open and the map $[0, \infty )\ni a\mapsto u_{a}\in C^{1, \beta }_{0}(\overline{\Omega })$ is a continuous function (see Theorem 2.2). Consequently, for $\varepsilon >0$ small enough, we have $[0, \varepsilon )\subset A$.

Let us define $a_{*}:=\sup A$. It is clear that $a_*>0$ because $[0, \varepsilon )\subset A$. We claim that $a_{*}<\infty$. In effect, again by Theorem 2.2, $u_{a}\rightarrow P_{X}u_{f}\in X$ in $C^{1, \beta }_{0}(\overline{\Omega })$ as $a\rightarrow +\infty$. Thus, since f is not constant, it follows from Remark 1 that for $a^{*}$ large enough, $u_{a^*}$ changes sign. On the other hand, by Theorem 2.2,

$$\begin{aligned} a_1>a_2>0 \ \Longrightarrow \ u_{a_2}>u_{a_1}. \end{aligned}$$

(2.25)

Thus, it must necessarily occur $a_{*}\le a^{*}$. Moreover, (2.25) also implies that any positive number less than $a_*$ belongs to A, that is, $[0, a_*)\subset A$. To conclude that A is open in $[0, \infty )$, it remains us to prove that $a_*\not \in A$. Otherwise, $u_{a_*}\in int(\mathcal {P})$ and, since $a\mapsto u_{a}\in C^{1, \beta }_{0}(\overline{\Omega })$ is continuous and $int(\mathcal {P})$ is open in $C^{1, \beta }_{0}(\overline{\Omega })$, it follows that there exists $\delta >0$ such that $[0, a_{*}+\delta )\subset A$. Since $a_*=\sup A$, last inclusion cannot occur. Therefore, $A=[0, a_{*})$.

Step 2: $u_{a_*}\in \partial \mathcal {P}$.

From previous step, $u_{a_{*}}\not \in int(\mathcal {P})$. On the other hand, let $\{a_{n}\}\subset A$ be such that $a_{n}\rightarrow a_{*}^{-}$. Then $u_{a_{n}}\rightarrow u_{a_{*}}$ in $C^{1, \beta }_{0}(\overline{\Omega })$ with $\{u_{a_n}\}\subset int(\mathcal {P})$. This shows that $u_{a_*}\in \partial \mathcal {P}$ and $u_{a_{*}}\ge 0$.

Step 3: $u_{a}$ changes sign if $a>a_{*}$.

Since $u_{a_*}\in \partial \mathcal {P}$, there exists a set $\Omega _{*}\subset \Omega$ such that $u_{a_{*}}=0$ in $\Omega _{*}$, or there exists $\Gamma _*\subset \partial \Omega$ such that $\partial u_{a_{*}}/\partial \eta =0$ on $\Gamma _*$. On the other hand, it follows from Theorem 2.2, that $a>a_{*}$ implies

$$\begin{aligned} 0=u_{a_{*}}>u_{a} \ \hbox { in }\ \Omega _{*} \end{aligned}$$

(2.26)

or

$$\begin{aligned} \partial u_{a}/\partial \eta >\partial u_{a_{*}}/\partial \eta =0 \ \hbox { on}\ \Gamma _*. \end{aligned}$$

(2.27)

It follows from (2.26) and (2.27) that $u_a$ assumes negative values if $a>a_*$. Finally, Corollary 2.4 implies that, if $a>a_{*}$, then $u_{a}$ also assumes positive values.

The result now follows from Steps 1, 2 and 3. $\square$

3 The eigenvalue problem

In the present section we are going to investigate the problem

where $\Omega \subset \mathbb {R}^{N}$ is a smooth bounded domain, $a\ne 0$ and $\lambda$ are real parameters.

A weak solution of (EP) is a function $u\in H_{0}^{1}(\Omega )$ satisfying

$$\begin{aligned} \int _{\Omega }\nabla u\nabla v dx+a\int _{\Omega }u dx\int _{\Omega }v dx=\lambda \int _{\Omega }uv dx, \ \forall \ v\in H_{0}^{1}(\Omega ). \end{aligned}$$

(3.1)

If u is nontrivial, we say that u is an eigenfunction associated to the eigenvalue $\lambda$. Since $\int _{\Omega }u dx$ is a real number, it is standard to show that any eigenfunction of (EP) is also a classical solution of (EP).

As in the previous section, $\{\varphi _k\}_{k\in \mathbb {N}}$ denotes an orthonormal basis of $H_{0}^{1}(\Omega )$ composed by eigenvalues of the laplacian operator and $V_{\lambda _k}$ is the finite-dimensional eigenspace associated to the eigenvalue $\lambda _k$, where

$$\begin{aligned} 0<\lambda _1<\lambda _2\le \lambda _3\le \cdots \le \lambda _k\le \cdots . \end{aligned}$$

(3.2)

Thus, any function $u\in H_{0}^{1}(\Omega )$ is uniquely determined by a series

$$\begin{aligned} u=\sum _{k=1}^{\infty }\alpha _k\varphi _k, \end{aligned}$$

(3.3)

where $\{\alpha _k\}_{k\in \mathbb {N}}\subset \mathbb {R}$.

Again, our approach will be concentrated in the study of a suitable auxiliary problem associated with (EP). In this way, we are going to consider the problem

where $a\ne 0$ and $r, \lambda \in \mathbb {R}$. As we will see, the existence of solutions to problem (AEP) is closely related with the existence of solution for the problem (EP).

Before stating our first result in this section, let us fix some notation. From now on, if $\lambda \ne \lambda _k$ for all $k\in \mathbb {N}$, we define

$$\begin{aligned} \mu (\lambda ):=\sum _{k=1}^{\infty }\frac{\lambda _k}{\lambda -\lambda _k}\left( \int _{\Omega }\varphi _k dx\right) ^{2}. \end{aligned}$$

(3.4)

On the other hand, if $\lambda =\lambda _k$ for some $k\in \mathbb {N}$, we define

$$\begin{aligned} \mu _k:=\sum _{j\in \Sigma _k}\frac{\lambda _j}{\lambda _k-\lambda _j}\left( \int _{\Omega }\varphi _{j} dx\right) ^{2}, \end{aligned}$$

(3.5)

where

$$\begin{aligned} \Sigma _k:=\{j\in \mathbb {N}: \lambda _j\ne \lambda _k\}. \end{aligned}$$

(3.6)

Among other facts, next result tells us that (3.4) and (3.5) are real numbers.

Proposition 3.1

The following assertions about problem (AEP) hold.

• (i) Suppose that $\lambda \ne \lambda _k$, for all $k\in \mathbb {N}$. Then, $u_{a, \lambda , r}:=ar u_{\lambda }$ is the unique weak solution of problem (AEP), where

  $$\begin{aligned} u_{\lambda }=\sum _{k=1}^{\infty }\frac{\lambda _{k}}{\lambda -\lambda _k}\left( \int _{\Omega }\varphi _k dx\right) \varphi _k \end{aligned}$$

  (3.7)

  is the unique solution of

  $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u +1=\lambda u &{} \text{ in } \Omega ,\\ u=0 &{} \text{ on } \partial \Omega . \end{array}\right. \end{aligned}$$

  (3.8)

  Moreover, the following identity holds

  $$\begin{aligned} \mu (\lambda )=\int _{\Omega }u_{\lambda }dx. \end{aligned}$$

  (3.9)

• (ii) Suppose that $\lambda = \lambda _k$, for some $k\in \mathbb {N}$. Then, problem (AEP) has a weak solution if, and only if, $V_{\lambda _{k}}\subset X$. In this case, any solution of (AEP) is in the form $aru_{\lambda _k}+\varphi$, where

  $$\begin{aligned} u_{\lambda _{k}}=\sum _{j\in \Sigma _k}\frac{\lambda _{j}}{\lambda _k-\lambda _j}\left( \int _{\Omega }\varphi _j dx\right) \varphi _j \end{aligned}$$

  (3.10)

  is a solution of

  $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u +1=\lambda _k u &{} \text{ in } \Omega ,\\ u=0 &{} \text{ on } \partial \Omega , \end{array}\right. \end{aligned}$$

  (3.11)

  $\varphi \in V_{\lambda _{k}}$ and $\Sigma _k$ is given in (3.6). Moreover, the following identity holds

  $$\begin{aligned} \mu _k=\int _{\Omega }u_{\lambda _{k}}dx. \end{aligned}$$

  (3.12)

Proof

(i) It follows from Theorem 6.6 in [5] that, if $\lambda \ne \lambda _{k}$ for all $k\in \mathbb {N}$, then problem (AEP) has a unique solution $u_{a, \lambda , r}$. Thus,

$$\begin{aligned} \int _{\Omega }\nabla u_{a, \lambda , r}\nabla v dx+ar\int _{\Omega }v dx=\lambda \int _{\Omega }u_{a, \lambda , r}v dx, \ \forall \ v\in H_{0}^{1}(\Omega ). \end{aligned}$$

(3.13)

Choosing $v=\varphi _k$ as a test function in (3.13) and writing $u_{a, \lambda , r}$ as in (3.3), that is,

$$\begin{aligned} u_{a, \lambda , r}=\sum _{k=1}^{\infty }\alpha _k\varphi _k, \end{aligned}$$

(3.14)

we get

$$\begin{aligned} \alpha _k+ar \int _{\Omega }\varphi _k dx=(\lambda /\lambda _k)\alpha _k. \end{aligned}$$

Thus,

$$\begin{aligned} ar \int _{\Omega }\varphi _k dx=[(\lambda /\lambda _k) - 1] \alpha _k. \end{aligned}$$

(3.15)

It follows from (3.15) that

$$\begin{aligned} \alpha _k=\frac{ar\lambda _k}{\lambda -\lambda _k}\int _{\Omega }\varphi _k dx, \ \hbox { for all}\ k\in \mathbb {N}. \end{aligned}$$

From (3.14), we have

$$\begin{aligned} u_{a, \lambda , r}=ar\sum _{k=1}^{\infty }\frac{\lambda _k}{\lambda -\lambda _k} \left( \int _{\Omega }\varphi _k dx\right) \varphi _k. \end{aligned}$$

(3.16)

Consequently, by defining

$$\begin{aligned} u_{\lambda }:=\sum _{k=1}^{\infty }\frac{\lambda _{k}}{\lambda -\lambda _k}\left( \int _{\Omega }\varphi _k dx\right) \varphi _k, \end{aligned}$$

we have that

$$\begin{aligned} u_{a, \lambda , r}=aru_{\lambda }, \end{aligned}$$

(3.17)

and $u_{\lambda }\in H_{0}^{1}(\Omega )$. It follows from $\lambda \ne \lambda _k$ for all $k\in \mathbb {N}$ and Theorem 6.6 in [5] that the problem (3.8) has a unique solution. Moreover, the same class of approach (with ar replaced by 1) shows that $u_\lambda$ is the solution of (3.8). By integrating both sides of (3.16), we have

$$\begin{aligned} \int _{\Omega }u_{a, \lambda , r} dx=ar\mu (\lambda ). \end{aligned}$$

(3.18)

It follows from (3.18) that $\mu (\lambda )$ is finite, and by comparing (3.17) and (3.18), we conclude (3.9).

(ii) Again, from Theorem 6.6 in [5], problem (AEP) has a weak solution u if, and only if, $V_{\lambda _{k}}\subset X$. If this is the case, by reasoning as in the proof of item (i) with $v=\varphi _j$ and $j\in \Sigma _k$, we conclude that

$$\begin{aligned} u=aru_{\lambda _k}+\varphi _{*}=ar\sum _{j\in \Sigma _k}\frac{\lambda _j}{\lambda _k-\lambda _j} \left( \int _{\Omega }\varphi _j dx\right) \varphi _j+\varphi _{*}, \end{aligned}$$

(3.19)

for some $\varphi _{*}\in V_{\lambda _{k}}$. By taking the integral in both sides of (3.19) and using (3.5), we get

$$\begin{aligned} \int _{\Omega }u_{\lambda _{k}} dx=\mu _k. \end{aligned}$$

To finish the proof, it is enough to show that, for each $\varphi \in V_{\lambda _{k}}$, the function $aru_{\lambda _k}+\varphi$ is a solution of (AEP), but this follows from a straightforward calculation combined with the fact that $V_{\lambda _{k}}\subset X$. Thus, taking $ar=1$, we conclude that problem (3.11) has solution if, and only if, $V_{\lambda _{k}}\subset X$ and in this case the set of solution is given by $u_{\lambda _k}+\varphi$, for all $\varphi \in V_{\lambda _{k}}$. The result now is proved. $\square$

It follows from Proposition 3.1 that the mapping $\mu :\mathbb {R}\backslash \{\lambda _k\}_{k\in \mathbb {N}}\rightarrow \mathbb {R}$, given by

$$\begin{aligned} \mu (\lambda )=\sum _{k=1}^{\infty }\frac{\lambda _k}{\lambda -\lambda _k}\left( \int _{\Omega }\varphi _k dx\right) ^{2}=\int _{\Omega } u_{\lambda } dx, \end{aligned}$$

(3.20)

is well defined, where $u_{\lambda }$ is as in (3.7). In the next result we study the graph of function $\mu$ and, consequently, the asymptotic behaviour of $\int _{\Omega }u_\lambda dx$ with respect to $\lambda$.

Proposition 3.2

The following assertions hold:

• (i) The mapping $\mathbb {R}\backslash \{\lambda _k\}_{k\ge 1}\ni \lambda \mapsto u_{\lambda }\in H_{0}^{1}(\Omega )$ is $C^{2}$, where

  $$\begin{aligned} \frac{du_{\lambda }}{d\lambda }=-\sum _{k=1}^{\infty }\frac{\lambda _k}{(\lambda -\lambda _k)^{2}}\left( \int _{\Omega }\varphi _k dx\right) \varphi _k \end{aligned}$$

  (3.21)

  is the unique solution of

  $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u =\lambda u +u_\lambda &{} \text{ in } \Omega ,\\ u=0 &{} \text{ on } \partial \Omega , \end{array}\right. \end{aligned}$$

  (3.22)

  and

  $$\begin{aligned} \frac{d^{2}u_{\lambda }}{d\lambda ^{2}}=\sum _{k=1}^{\infty }\frac{\lambda _k}{(\lambda -\lambda _k)^{3}}\left( \int _{\Omega }\varphi _k dx\right) \varphi _k \end{aligned}$$

  (3.23)

  is the unique solution of

  $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u =\lambda u +2u_\lambda &{} \text{ in } \Omega ,\\ u=0 &{} \text{ on } \partial \Omega . \end{array}\right. \end{aligned}$$

  (3.24)

• (ii) The map $\mu$ is $C^2$, decreases and is concave in $(-\infty , \lambda _1)$. Furthermore, for all integer $k\ge 1$ such that $\lambda _k<\lambda _{k+1}$, $\mu$ decreases and changes concavity in $(\lambda _k, \lambda _{k+1})$, and satisfies

  $$\begin{aligned} \mu (0)=-\int _{\Omega }e dx \ , \ \mu (\lambda )<0 \ \hbox { if }\ \lambda \in (-\infty , \lambda _1] \ , \ \lim _{\lambda \rightarrow -\infty }\mu (\lambda )=0 \ \text{ and } \ \lim _{\lambda \rightarrow \lambda _k^{\pm }}\mu (\lambda )=\pm \infty . \end{aligned}$$

  (3.25)

Proof

(i) The regularity of the mapping $\lambda \mapsto u_\lambda$ and the identities (3.21) and (3.23) are straightforward consequences of (3.7). Moreover, since $\lambda \ne \lambda _k$ for all $k\in \mathbb {N}$, it follows from Theorem 6.6 in [5] that the problems (3.22) and (3.24) have a unique solution. Now, by using the identities (3.21) and (3.23), a direct calculation proves that $du_\lambda /d\lambda$ is the solution of (3.22) and $d^{2}u_\lambda /d\lambda ^{2}$ is the solution of (3.24).

(ii) It follows from definition of $\mu$ and item (i), that $\mu \in C^{2}(\mathbb {R}\backslash \{\lambda _k\}_{k\in \mathbb {N}}, \mathbb {R})$ and

$$\begin{aligned} \mu '(\lambda )=\int _{\Omega }\frac{d u_\lambda }{d\lambda } dx=-\sum _{k=1}^{\infty }\frac{\lambda _k}{(\lambda -\lambda _k)^{2}}\left( \int _{\Omega }\varphi _k dx\right) ^{2}=-\int _{\Omega }u_{\lambda }^{2}dx \end{aligned}$$

(3.26)

and

$$\begin{aligned} \mu ''(\lambda )=\int _{\Omega }\frac{d^{2} u_\lambda }{d\lambda ^{2}} dx=\sum _{k=1}^{\infty }\frac{\lambda _k}{(\lambda -\lambda _k)^{3}}\left( \int _{\Omega }\varphi _k dx\right) ^{2}=-2\int _{\Omega }u_{\lambda } (du_{\lambda }/d\lambda )dx. \end{aligned}$$

(3.27)

The monotonicity of $\mu$ is a consequence of (3.26). To show that $\mu$ changes concavity, observe that, from (3.27), it results that, in each interval $(\lambda _k, \lambda _{k+1})$, $\mu ''(\lambda )$ is positive for $\lambda$ close to $\lambda _k$, and negative, for $\lambda$ close from $\lambda _{k+1}$.

The conclusion of the proof follows from $e=\sum _{k=1}^{\infty }(\int _{\Omega }\varphi _k dx)\varphi _k$ and (3.20). See Fig. 3. $\square$

Remark 2

Let $\{\lambda ^{k}\}_{k\in \mathbb {N}}$ be the increasing subsequence of $\{\lambda _k\}_{k\in \mathbb {N}}$, defined by

$$\begin{aligned} \lambda ^{1}=\lambda _{1} \ \text{ and } \ \lambda ^{k+1}=\min \{\lambda _j: \lambda _j>\lambda _k\}. \end{aligned}$$

(3.28)

Observe that, by (3.2), we get $\lambda ^2=\lambda _2$. An interesting consequence of Propositions 3.1 and 3.2 is that, for each $k\in \mathbb {N}$, there exists $\lambda _{*}(k)\in (\lambda ^k, \lambda ^{k+1})$, such that the nontrivial solution $u_{\lambda _{*}(k)}$ of problem (3.8) (with $\lambda =\lambda _{*}(k)$) satisfies $\int _{\Omega }u_{\lambda _{*}(k)} dx=0$. Moreover, since $\mu$ is decreasing in any interval where it is defined, $\{\lambda _{*}(k)\}$ are the unique numbers $\lambda$ for which problem (3.8) admits solutions with null integral. See Fig. 3.

Fig. 3

Behaviour of $\int _{\Omega }u_{\lambda } dx$ regarding $\lambda$

Theorem 3.3

Problem (EP) admits only the trivial solution in the following cases:

• (i) $\lambda \ne \lambda _{k}$, for all $k\in \mathbb {N}$, and $a \mu (\lambda )\ne 1$;

• (ii) $\lambda = \lambda _{k}$, for some $k\in \mathbb {N}$, and $a \mu _k\ne 1$;

• (iii) $\lambda = \lambda _{k}$, for some $k\in \mathbb {N}$, $a \mu _k= 1$ and $V_{\lambda _k}\not \subset X$;

Proof

(i) Suppose that (EP) has a solution u. By choosing $v=\varphi _k$ as a test function in (3.1), writing u as in (3.3) and reasoning as in the proof of item (i) of Proposition 3.1 with $r=\int _{\Omega }u dx$, we get

$$\begin{aligned} u=a\left( \int _{\Omega }u dx\right) \sum _{k=1}^{\infty }\frac{\lambda _k}{\lambda -\lambda _k} \left( \int _{\Omega }\varphi _k dx\right) \varphi _k \end{aligned}$$

(3.29)

If $\int _{\Omega }u dx\ne 0$, by integrating both sides of (3.29) and dividing by $\int _{\Omega }u dx$, we obtain $1=a\mu (\lambda )$. Since this equality does not occur, we get a contradiction. Therefore $\int _{\Omega }u dx=0$. In this case, we have two possibilities: either $u=0$, or u is an eigenfunction of the Laplacian operator. The second one is not possible because $\lambda \ne \lambda _{k}$, for all $k\in \mathbb {N}$.

(ii) An analogous reasoning for $v=\varphi _j$, $j\in \Sigma _k$, shows the same result with $\mu (\lambda )$ replaced by $\mu _k$.

(iii) Suppose that (EP) has a solution u and let $\varphi _{k}\in V_{\lambda _k}\backslash X$. By choosing $v=\varphi _k$ as a test function in (3.1), writing u as in (3.3) and reasoning as in the proof of item (i) of Proposition 3.1 with $r=\int _{\Omega }u dx$, we get

$$\begin{aligned} a\int _{\Omega }u dx \int _{\Omega }\varphi _k dx=[(\lambda /\lambda _k) - 1] \alpha _k. \end{aligned}$$

(3.30)

Since $\lambda =\lambda _k$, $a\ne 0$ and $\varphi _{k}\in V_{\lambda _k}\backslash X$, it follows from (3.30), that

$$\begin{aligned} \int _{\Omega }u dx=0. \end{aligned}$$

(3.31)

On the other hand, by reasoning as in the proof of (ii), we know that

$$\begin{aligned} u=a\left( \int _{\Omega }u dx\right) \sum _{j\in \Sigma _k}^{\infty }\frac{\lambda _j}{\lambda _k-\lambda _j} \left( \int _{\Omega }\varphi _j dx\right) .\varphi _j \end{aligned}$$

(3.32)

The result follows from (3.31) and (3.32). $\square$

Remark 3

An immediate consequence of items (ii) and (iii) of Theorem 3.3 is that the problem (EP) has only the trivial solution when $\lambda =\lambda _1$.

Theorem 3.4

The following statements about problem (EP) hold.

• (i) If $\lambda \ne \lambda _{k}$, for all $k\in \mathbb {N}$, and $a \mu (\lambda )= 1$, then the multiples of

  $$\begin{aligned} u_{\lambda }=\sum _{k=1}^{\infty }\frac{\lambda _k}{\lambda -\lambda _k} \left( \int _{\Omega }\varphi _k dx\right) \varphi _k \end{aligned}$$

  are the only solutions of (EP).

• (ii) If $\lambda = \lambda _{k}$, $V_{\lambda _k}\subset X$ and $a \mu _k= 1$, for some $k\in \mathbb {N}$, then, the functions $\gamma u_{\lambda _k}+\varphi$, where $\gamma \in \mathbb {R}$, $\varphi \in V_{\lambda _k}$,

  $$\begin{aligned} u_{\lambda _k}=\sum _{j\in \Sigma _k}\frac{\lambda _j}{\lambda _k-\lambda _j} \left( \int _{\Omega }\varphi _j dx\right) \varphi _j, \end{aligned}$$

  and $\Sigma _k$ is as in (3.6) are the only solutions of (EP).

Proof

It is enough to prove the existence of $r>0$ for which the solutions u obtained for problem (AEP) satisfy $r=\int _{\Omega }u dx$. Fortunately, this is a straightforward consequence of $a \mu (\lambda )= 1$ (in case of item (i)), $a \mu _k= 1$ (in case of item (ii)), and equalities (3.7), (3.9), (3.10) and (3.12) in Proposition 3.1. $\square$

Item (i) of Theorem 3.4 can be understood in two ways: 1) Given $\lambda \in \mathbb {R}$, with $\lambda \ne \lambda _{k}$ and $\lambda \ne \lambda _{*}(k)$ for all $k\in \mathbb {N}$, where $\lambda _{*}(k)$ is as Remark 2, there exists a unique $a=1/\mu (\lambda )$ for which problem (EP) has nontrivial solution. 2) Given $a\ne 0$, problem (EP) admits nontrivial solutions only in the numbers $\lambda$ for which $\mu (\lambda )=1/a$. In next results we elucidate the second possibility.

Corollary 3.5

For each integer $k\ge 1$, let $E_{k+1}$ be the orthogonal complement of $V_{\lambda ^1}\oplus \cdots \oplus V_{\lambda ^{k}}$ in $H_{0}^{1}(\Omega )$. The following statements hold:

• (i) If $a\ne 0$, then the eigenvalues of problem (EP), different from $\{\lambda _k\}_{k\in \mathbb {N}}$, constitutes an increasing real sequence

  $$\begin{aligned} \lambda _{1}(a)<\lambda _{2}(a)<\cdots< \lambda _{k}(a)< \cdots , \end{aligned}$$

  with unidimensional associated eigenspaces, and satisfy $\lambda _{k}(a)\in (\lambda ^k, \lambda ^{k+1})$, where $\{\lambda ^k\}_{k\ge 1}$ is given in Remark 2. Additionally, when $a<0$, there exists one more eigenvalue $\lambda _{0}(a)$ for problem (EP), with $0<\lambda _{0}(a)<\lambda _1$, if $-1/\int _{\Omega }e dx<a<0$, and $\lambda _{0}(a)<0$, if $a<-1/\int _{\Omega }e dx$. The eigenspace associated to $\lambda _{0}(a)$ is also unidimensional.

• (ii) If $0<a<(\lambda _{2}-\lambda _{1})/|\Omega |$, then

  $$\begin{aligned} \lambda _{1}(a)=\min _{u\in H_{0}^{1}(\Omega )\backslash \{0\}}\left[ \frac{\int _{\Omega }|\nabla u|^{2}dx+a\left( \int _{\Omega }udx\right) ^{2}}{\int _{\Omega }u^{2}dx}\right] . \end{aligned}$$

  (3.33)

  Moreover, if $0<a<(\lambda ^{k+2}-\lambda ^{k+1})/|\Omega |$ and $V_{\lambda ^{k+1}}\cap X=\emptyset$, for some integer $k\ge 1$, then

  $$\begin{aligned} \lambda _{k+1}(a)=\min _{u\in E_{k+1}\backslash \{0\}}\left[ \frac{\int _{\Omega }|\nabla u|^{2}dx+a\left( \int _{\Omega }u dx\right) ^{2}}{\int _{\Omega }u^{2}dx}\right] . \end{aligned}$$

  (3.34)

Proof

(i) For all $a\ne 0$, it is an immediate consequence of item (ii) in Proposition 3.2 that, for each $k\ge 1$, the equation $1/a=\mu (\lambda )$ has a unique root in $(\lambda ^k, \lambda ^{k+1})$. Moreover, if $a<0$, there exists one more root of $1/a=\mu (\lambda )$ in $(-\infty , \lambda _1)$. The result follows now from item (i) in Theorem 3.4 combined with the behaviour of the function $\mu$ described in Proposition 3.2. See Fig. 4 for a descriptive illustration.

(ii) For each integer $k\ge 0$, consider the set

$$\begin{aligned} M_{k+1}=\{u\in E_{k+1}: \int _{\Omega }u^{2}dx=1\}, \end{aligned}$$

and the functional

$$\begin{aligned} \Psi (u)=\int _{\Omega }|\nabla u|^{2}dx+a\left( \int _{\Omega }udx\right) ^{2}, \end{aligned}$$

where $E_{k+1}:=H_{0}^{1}(\Omega )$, if $k=0$, and $E_{k+1}$ is the orthogonal complement of $V_{\lambda ^1}\oplus \cdots \oplus V_{\lambda ^{k}}$ in $H_{0}^{1}(\Omega )$, if $k\ge 1$. Since $\Psi$ is bounded from below, weakly lower semicontinuous and $M_{k+1}$ is weakly compact, it follows that $\Psi$ assumes a minimum $\Psi (u)=\gamma _{k+1}>0$ on $M_{k+1}$. Consequently, from Lagrange Multipliers Theorem, we conclude that u is an eigenfunction associated to the eigenvalue

$$\begin{aligned} \gamma _{k+1}=\min _{u\in E_{k+1}\backslash \{0\}}\left[ \frac{\int _{\Omega }|\nabla u|^{2}dx+a\left( \int _{\Omega }u dx\right) ^{2}}{\int _{\Omega }u^{2}dx}\right] \end{aligned}$$

of problem (EP), for each $k\ge 0$. Since $a>0$, we have

$$\begin{aligned} \lambda ^{k+1}=\min _{u\in E_{k+1}\backslash \{0\}}\left[ \frac{\int _{\Omega }|\nabla u|^{2}dx}{\int _{\Omega }u^{2}dx}\right] \le \gamma _{k+1}\le \lambda ^{k+1}+a|\Omega |, \ \text{ for } \text{ all } \text{ integer } k\ge 0, \end{aligned}$$

(3.35)

where the last inequality follows from Jensen’s inequality.

Claim 1: $\lambda _1<\gamma _1$

Suppose by contradiction that $\lambda _1=\gamma _1$. Let $u\in H_{0}^{1}(\Omega )\backslash \{0\}$ be such that

$$\begin{aligned} \gamma _{1}=\frac{\int _{\Omega }|\nabla u|^{2}dx+a\left( \int _{\Omega }u dx\right) ^{2}}{\int _{\Omega }u^{2}dx}. \end{aligned}$$

(3.36)

Observe that $u\not \in X$, because, in this case, u would be a changing sign eigenfunction of the Laplacian operator with Dirichlet boundary condition associated to $\lambda _1$, which is not possible. Therefore $\int _{\Omega } u dx\ne 0$ and, by (3.36), we get

$$\begin{aligned} \frac{\int _{\Omega }|\nabla u|^{2}dx}{\int _{\Omega }u^{2}dx}<\frac{\int _{\Omega }|\nabla u|^{2}dx+a\left( \int _{\Omega }udx\right) ^{2}}{\int _{\Omega }u^{2}dx}=\gamma _1=\lambda _1, \end{aligned}$$

contradicting the variational characterization of $\lambda _1$. Claim 1 is now proved.

Claim 2: $\lambda ^{k+1}<\gamma _{k+1}$ for all $k\ge 1$.

Let $u\in E_{k+1}\backslash \{0\}$ be such that

$$\begin{aligned} \gamma ^{k+1}=\frac{\int _{\Omega }|\nabla u|^{2}dx+a\left( \int _{\Omega }udx\right) ^{2}}{\int _{\Omega }u^{2}dx}. \end{aligned}$$

(3.37)

Suppose by contradiction that $\lambda ^{k+1}=\gamma _{k+1}$. As in Claim 1, we are going to prove that $u\not \in X$. In fact, in the contrary, $u\in X$ would be an eigenfunction associated to $\lambda ^{k+1}$, contradicting the hypothesis that $V_{\lambda ^{k+1}}\cap X=\emptyset$. Therefore, $\int _{\Omega } u dx\ne 0$. The conclusion of the proof follows exactly as in Claim 1, by using the fact that

$$\begin{aligned} \lambda ^{k+1}=\min _{u\in E_{k+1}\backslash \{0\}}\left[ \frac{\int _{\Omega }|\nabla u|^{2}dx}{\int _{\Omega }u^{2}dx}\right] . \end{aligned}$$

Finally, it follows from $0<a<(\lambda _{2}-\lambda _{1})/|\Omega |$, (3.35) with $k=0$ and Claim 1, that $\gamma _1$ is an eigenvalue of (EP) with $\lambda _1<\gamma _1<\lambda _2$. From item (i), we conclude that $\gamma _1=\lambda _1(a)$, proving that (3.33) holds. In an analogous way, (3.34) follows from $0<a<(\lambda ^{k+2}-\lambda ^{k+1})/|\Omega |$, (3.35) and Claim 2. $\square$

Fig. 4

Eigenvalues of (EP)

Corollary 3.6

The following statements hold:

• (i) The map $(-\infty , 0)\ni a\mapsto \lambda _{0}(a)$ is increasing and $C^{2}$, and the mapping $a\mapsto u_{\lambda _{0}(a)}$ is $C^{2}((-\infty , 0), H_{0}^{1}(\Omega ))$;

• (ii) For each integer $k\ge 1$, the maps $\mathbb {R}\backslash \{0\}\ni a\mapsto \lambda _{k}(a)$ are increasing and $C^{2}$, and the mappings $a\mapsto u_{\lambda _{k}(a)}$ are $C^{2}(\mathbb {R}\backslash \{0\}, H_{0}^{1}(\Omega ))$.

• (iii) For each integer $k\ge 1$,

  $$\begin{aligned}{} & {} \lambda _{k}(a)\rightarrow (\lambda ^k)^{+} \ \hbox { and} \ \int _{\Omega }u_{\lambda _{k}(a)} dx\rightarrow +\infty \ \text{ as }\ a\rightarrow 0^{+}, \end{aligned}$$

  (3.38)
  $$\begin{aligned}{} & {} \lambda _{k}(a)\rightarrow (\lambda ^{k+1})^{-}, \lambda _{0}(a)\rightarrow \lambda _{1}^{-}, \int _{\Omega }u_{\lambda _{k}(a)} dx\rightarrow -\infty \ \hbox { and}\nonumber \\{} & {} \ \int _{\Omega }u_{\lambda _{0}(a)} dx\rightarrow -\infty \ \text{ as }\ a\rightarrow 0^{-}, \end{aligned}$$

  (3.39)
  $$\begin{aligned}{} & {} \lambda _{k}(a)\rightarrow \lambda _{*}(k)^{-} \ \hbox { and} \ \int _{\Omega }u_{\lambda _{k}(a)} dx\rightarrow 0^{+} \ \text{ as }\ a\rightarrow +\infty \end{aligned}$$

  (3.40)

  and

  $$\begin{aligned} \lambda _{k}(a)\rightarrow \lambda _{*}(k)^{+}, \lambda _{0}(a)\rightarrow -\infty , \int _{\Omega }u_{\lambda _{k}(a)} dx\rightarrow 0^{-} \ \hbox { and} \ \int _{\Omega }u_{\lambda _{0}(a)} dx\rightarrow 0^{-} \ \text{ as }\ a\rightarrow -\infty . \end{aligned}$$

  (3.41)

Proof

(i)–(ii) It follows from item (ii) of Proposition 3.2 and item (i) of Theorem 3.4 (see also item (i) of Corollary 3.5) that, the maps

$$\begin{aligned} (-\infty , 0)\ni a\mapsto (\mu _{|_{(-\infty , \lambda _{1})}})^{-1}(1/a)\in (-\infty , \lambda _{1}) \end{aligned}$$

and

$$\begin{aligned} \mathbb {R}\backslash \{0\}\ni a\mapsto (\mu _{|_{(\lambda ^{k}, \lambda ^{k+1})}})^{-1}(1/a)\in (\lambda ^{k}, \lambda ^{k+1}) \ (k\ge 1) \end{aligned}$$

are well defined, $C^{2}$ and coincides with $\lambda _0(a)$ and $\lambda _k(a)$, respectively. Thus, by using the Chain’s rule, we get

$$\begin{aligned} d\lambda _{k}(a)/da=-\frac{1}{a^{2}\mu '(\lambda _{k}(a))}, \ k\ge 0, \end{aligned}$$

(3.42)

From (3.26) and (3.42), we conclude that $\lambda _0(a)$ and $\lambda _k(a)$ are increasing. The remainder of the proof follows from the identity

$$\begin{aligned} u_{\lambda _{k}(a)}=\sum _{j=1}^{\infty }\frac{\lambda _j}{\lambda _{k}(a)-\lambda _j} \left( \int _{\Omega }\varphi _j dx\right) \varphi _j, \ k\ge 0, \end{aligned}$$

(3.43)

see again item (i) of Theorem 3.4 and item (i) of Proposition 3.2.

(iii) It is a straightforward consequence from equalities

$$\begin{aligned} \lambda _{0}(a)= & {} (\mu _{|_{(-\infty , \lambda _{1})}})^{-1}(1/a),\\ \lambda _k(a)= & {} (\mu _{|_{(\lambda ^{k}, \lambda ^{k+1})}})^{-1}(1/a), \ k\ge 1,\\ \int _{\Omega }u_{\lambda _{k}(a)}dx= & {} \mu (\lambda _{k}(a))=1/a, \ k\ge 0, \end{aligned}$$

and item (ii) of Proposition 3.2. See Figs. 5 and 6. $\square$

Fig. 5

Asymptotic behaviour of $\lambda _k(a)$

Fig. 6

Bifurcation diagram of $\int _{\Omega }u_{\lambda _{k}(a)}dx$ with respect to a

Corollary 3.7

The following statements hold:

• (i) There exists $0<a^{*}$ such that if $a\in (0, a^{*})$, then the eigenfunctions associated to $\lambda _{1}(a)$ have defined sign;

• (ii) If $0<a<(\lambda ^{k+2}-\lambda ^{k+1})/|\Omega |$ and $V_{\lambda _{k+1}}\cap X=\emptyset$, for some integer $k\ge 1$, then the eigenfunctions associated to $\lambda _{k+1}(a)$ change sign;

• (iii) There exists $a_{*}>0$ such that if $a\in (-a_*, 0)$ or $a\in (-(1/\int _{\Omega }e dx)-a_*, (-1/\int _{\Omega }e dx)+a_*)$, then the eigenfunctions associated to $\lambda _{0}(a)$ have defined sign.

Proof

(i) From identity (3.43) with $k=1$, we conclude that

$$\begin{aligned} \frac{(\lambda _1(a)-\lambda _1)}{\lambda _1}u_{\lambda _{1}(a)}=\left( \int _{\Omega }\varphi _1 dx\right) \varphi _1+\frac{(\lambda _1(a)-\lambda _1)}{\lambda _1}w_a, \end{aligned}$$

(3.44)

where

$$\begin{aligned} w_{a}=\sum _{j=2}^{\infty }\frac{\lambda _j}{\lambda _{1}(a)-\lambda _j} \left( \int _{\Omega }\varphi _j dx\right) \varphi _j. \end{aligned}$$

(3.45)

Since $u_{\lambda _1(a)}, \varphi _1\in C^{2}_{0}(\overline{\Omega })$, we conclude from equality (3.44), that $w_a$ also belongs to $C^{2}_{0}(\overline{\Omega })$. Moreover, it is a consequence of (3.45) and item (iii) of Corollary 3.6 that

$$\begin{aligned} w_{a}\rightarrow u_{\lambda _1} \ \text{ in } \ C^{2}_{0}(\overline{\Omega }), \ \text{ as } \ a\rightarrow 0^{+}, \end{aligned}$$

(3.46)

where $u_{\lambda _1}$ is given in the item (ii) of Proposition 3.1. Thus, by (3.44), (3.46) and item (iii) of Corollary 3.6, we get

$$\begin{aligned} \frac{(\lambda _1(a)-\lambda _1)}{\lambda _1}u_{\lambda _{1}(a)}\rightarrow \left( \int _{\Omega }\varphi _1 dx\right) \varphi _1 \ \text{ in } \ C^{2}_{0}(\overline{\Omega }), \ \text{ as } \ a\rightarrow 0^{+}. \end{aligned}$$

Since $(\int _{\Omega }\varphi _1 dx)\varphi _1$ belongs to the interior of the positive cone in $C^{2}_{0}(\overline{\Omega })$, that is,

$$\begin{aligned} \left( \int _{\Omega }\varphi _1 dx\right) \varphi _1\in int(\mathcal {P})=\{u\in C^{2}_{0}(\overline{\Omega }): u>0 \ \hbox {in } \Omega \hbox {and} \partial u/\partial \eta <0\hbox { on } \partial \Omega \}, \end{aligned}$$

we derive that there exists $a^{*}>0$ such that

$$\begin{aligned} \frac{(\lambda _1(a)-\lambda _1)}{\lambda _1}u_{\lambda _{1}(a)}\in \mathcal {P}, \end{aligned}$$

for all $a\in (0, a^{*})$. Since $\lambda _1(a)$ is increasing and $\lambda _1(a)\rightarrow \lambda _1$ as $a\rightarrow 0^{+}$, by choosing $a_*$ still smaller if necessary, we conclude that $\lambda _1(a)>\lambda _1$ and, consequently,

$$\begin{aligned} u_{\lambda _{1}(a)}\in \mathcal {P}, \end{aligned}$$

(3.47)

for all $a\in (0, a^{*})$. The result follows now from item (i) of Theorem 3.4 and (3.47).

(ii) In this case, item (ii) of Corollary 3.5 implies that $\lambda _{k+1}$ is attained by a nontrivial function u which is orthogonal to $\varphi _1$, that is, u is a changing sign eigenfunction of (EP) associated to $\lambda _{k+1}(a)$. The result follows now from the fact that the eigenspace associated to $\lambda _{k+1}(a)$ has dimension 1.

(iii) Since

$$\begin{aligned} \frac{(\lambda _0(a)-\lambda _1)}{\lambda _1}u_{\lambda _{0}(a)}=\left( \int _{\Omega }\varphi _1 dx\right) \varphi _1+\frac{(\lambda _0(a)-\lambda _1)}{\lambda _1}z_a, \end{aligned}$$

(3.48)

where

$$\begin{aligned} z_{a}=\sum _{j=2}^{\infty }\frac{\lambda _j}{\lambda _{0}(a)-\lambda _j} \left( \int _{\Omega }\varphi _j dx\right) \varphi _j, \end{aligned}$$

(3.49)

we can argue in a similar way as in item (i) to conclude that there exists $a_*>0$ such that, for $a\in (-a_*, 0)$, we have $u_{\lambda _{0}(a)}\in -\mathcal {P}$. The second part follows from identity (3.43) for $k=0$, the fact that $\lambda _{0}(-1/\int _{\Omega }edx)=0$, $e=\sum _{j=1}^{\infty }\left( \int _{\Omega }\varphi _{j} dx\right) \varphi _j$ and item (i) of Corollary 3.6. $\square$

4 An asymptotically linear problem at the origin

In this section, we are interested in using the results of previous section to study the existence of positive solutions to the following class of problems

where $\Omega \subset \mathbb {R}^{N}$ is a smooth bounded domain, $a>0$, $\lambda < \lambda _1(a)$ is a real parameter and f is a continuous function satisfying the following assumptions

(f1):

$\lim _{|t|\rightarrow 0}f(t)/t=0$ and $\lim _{|t|\rightarrow \infty }f(t)/|t|^{p-1}<\infty$, for some $p\in (2, 2^{*})$, where $2^{*}=+\infty$ if $N=1$ or $N=2$, and $2^{*}=2N/(N-2)$ if $N\ge 3$;

(f2):

$\lim _{|t|\rightarrow \infty }F(t)/t^{2}=\infty$, where $F(t):=\int _{0}^{t}f(s) ds$;

(f3):

$\mathbb {R}\backslash \{0\}\ni t\mapsto f(t)/|t|$ is increasing.

A weak solution of (LP) is a function $u\in H_{0}^{1}(\Omega )$ satisfying

$$\begin{aligned} \int _{\Omega }\nabla u\nabla v dx+a\int _{\Omega }u dx\int _{\Omega }v dx=\lambda \int _{\Omega }uv dx+\int _{\Omega }f(u)v dx, \end{aligned}$$

(4.1)

for all $v\in H_{0}^{1}(\Omega )$. Moreover, the energy functional $I_a: H_{0}^{1}(\Omega )\rightarrow \mathbb {R}$ associated to (ALP) is given by

$$\begin{aligned} I_a(u)=\frac{1}{2}\int _{\Omega }|\nabla u|^{2} dx+\frac{a}{2}\left( \int _{\Omega }u dx\right) ^{2}-\frac{\lambda }{2}\int _{\Omega } u^{2} dx-\int _{\Omega }F(u) dx, \end{aligned}$$

where $F(t)=\int _{0}^{t}f(s)ds$ is the primitive of f. Since $f\in C(\mathbb {R})$ and satisfies (f1), one shows that $I_a$ is well defined and $I\in C^{1}(H_{0}^{1}(\Omega ), \mathbb {R})$, with

$$\begin{aligned} I'_a(u)v=\int _{\Omega }\nabla u\nabla v dx+a\int _{\Omega }u dx \int _{\Omega }v dx-\lambda \int _{\Omega } uv dx-\int _{\Omega }f(u)v dx \end{aligned}$$

Therefore u is a weak solution of (ALP) if, and only if, u is a critical point of $I_a$. Moreover, the Nehari manifold associated to $I_a$ is given by

$$\begin{aligned} \mathcal {N}_a=\left\{ u\in H_{0}^{1}(\Omega )\backslash \{0\}: \int _{\Omega }|\nabla u|^{2}dx+a\left( \int _{\Omega }u dx\right) ^{2}=\lambda \int _{\Omega } u^{2} dx+\int _{\Omega }f(u)u dx\right\} . \end{aligned}$$

In what follows, for each $a\in (0, (\lambda _2-\lambda _1)/|\Omega |)$ and $\lambda <\lambda _1(a)$, we define

$$\begin{aligned} \Vert u\Vert ^{2}_{a}:= \int _{\Omega }|\nabla u|^{2}dx+a\left( \int _{\Omega }u dx\right) ^{2}-\lambda \int _{\Omega } u^{2} dx. \end{aligned}$$

It follows from item (ii) of Corollary 3.5, that $\Vert .\Vert _a$ is a norm in $H_{0}^{1}(\Omega )$, which is equivalent to the usual one. Moreover, for each $u\in H_{0}^{1}(\Omega )$, let us define $\alpha _{u, a}:[0, +\infty )\rightarrow \mathbb {R}$, given by

$$\begin{aligned} \alpha _{u, a}(t)=I_a(tu). \end{aligned}$$

To prove our main result we will make use of a Nehari manifold approach, see [14].

Proposition 4.1

Suppose that f satisfies (f1)–(f3), $a\in (0, (\lambda _2-\lambda _1)/|\Omega |)$ and $\lambda <\lambda _1(a)$. Then,

$(I_{1})$:

For each $u\in H_{0}^{1}(\Omega )\backslash \{0\}$, there exists a unique $t_{u}>0$ such that $\alpha _{u, a}'(t)>0$ in $(0, t_{u})$, $\alpha _{u, a}'(t_{u})=0$ and $\alpha _{u, a}'(t)<0$ in $(t_{u}, +\infty )$;

$(I_{2})$:

There exists $\delta > 0$, such that $t_u \ge \delta$ for each $u\in \mathcal {S}_a:=\{u\in H_{0}^{1}(\Omega ): \Vert u\Vert _a=1\}$ and for each compact set $\mathcal {W}\subset \mathcal {S}_a$ there exists a positive constant $C_{\mathcal {W}}$ such that $t_{u}\le C_\mathcal {W}$ for all $u\in \mathcal {W}$.

Proof

$(I_1)$ For each $u\in H_{0}^{1}(\Omega )\backslash \{0\}$, we have $\alpha _{u, a}(0)=0$. Observe now that,

$$\begin{aligned} \frac{I_a(tu)}{t^{2}}=\frac{1}{2}\Vert u\Vert ^{2}_a-\int _{[u\ne 0]}\left[ \frac{F(tu)}{(tu)^{2}}\right] u^{2} dx. \end{aligned}$$

(4.2)

By (f1), passing to the limit as $t\rightarrow 0^{+}$, we get

$$\begin{aligned} \lim _{t\rightarrow 0^{+}}\frac{I_a(tu)}{t^{2}}=\frac{1}{2}\Vert u\Vert ^{2}_a>0. \end{aligned}$$

Therefore, $\alpha _{u, a}(t)>0$ for $t>0$ small enough. On the other hand, by (f1), we know that $f(0)=0$. Moreover, combining (f1) and (f3), we conclude that $f(t)>0$ if $t>0$, and $f(t)<0$ if $t<0$. Thus, $F(t)\ge 0$ for all $t\in \mathbb {R}$. Therefore, by (f2) and Fatou’s Lemma, we obtain

$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{I_a(tu)}{t^{2}}=\frac{1}{2}\Vert u\Vert ^{2}_a-\liminf _{t\rightarrow \infty }\int _{[u\ne 0]}\left[ \frac{F(tu)}{(tu)^{2}}\right] u^{2} dx=-\infty , \end{aligned}$$

showing that, for t large enough,

$$\begin{aligned} \alpha _{u, a}(t)=t^{2}\left[ \frac{I_a(tu)}{t^{2}}\right] <0. \end{aligned}$$

Since $\alpha _{u, a}$ is $C^1$, there exists a critical point $t_{*}$ for $\alpha _{u, a}$, which is global maximum point. The uniqueness of $t_u$ is a consequence of (f3). The proof of item ($I_1$) is complete.

$(I_2)$ Suppose by contradiction that there exists $\{u_{n}\}\subset \mathcal {S}_a$ such that $t_{u_{n}}\rightarrow 0$. Thus,

$$\begin{aligned} 1= \Vert u_{n}\Vert ^{2}_a=\int _{[u_n\ne 0]}\left[ \frac{f(t_{u_{n}}u_{n})}{t_{u_{n}}u_{n}}\right] u_{n}^{2} dx. \end{aligned}$$

(4.3)

Up to a subsequence, there exists $u\in H_{0}^{1}(\Omega )$ such that

$$\begin{aligned}{} & {} u_{n}\rightharpoonup u \ \hbox { in}\ H_{0}^{1}(\Omega ),\\{} & {} u_{n}\rightarrow u \ \hbox { in}\ L^{p}(\Omega ),\\{} & {} u_{n}(x)\rightarrow u(x) \ \hbox {a.e. in} \Omega \hbox {and} |u_{n}(x)|\le h(x), \end{aligned}$$

for some $h\in L^{p}(\Omega )$. From (4.3), (f1) and Lebesgue Dominated Convergence Theorem, passing to the limit in (4.3) as $n\rightarrow \infty$, we get a contradiction. Therefore, there exists $\delta >0$ such that $t_{u}\ge \delta$ for all $u\in \mathcal {S}_a$. To prove the second part of $(I_2)$, suppose by contradiction that there exist a compact set $\mathcal {W}\subset \mathcal {S}_a$ and a sequence $\{u_{n}\}\subset \mathcal {W}$ satisfying $t_{u_n}\rightarrow \infty$. Thus, there exists $u\in \mathcal {W}$ with

$$\begin{aligned} u_{n}\rightarrow u \ \hbox { in}\ H_{0}^{1}(\Omega ). \end{aligned}$$

(4.4)

From (4.4) and $u\in \mathcal {W}$, we have

$$\begin{aligned} |[u\ne 0]|>0 \ \hbox { and} \ \chi _{[u_{n}\ne 0]}(x)\rightarrow 1 \ \text{ a.e. } \text{ in }\ [u\ne 0]. \end{aligned}$$

Since, from (f3), $f(t)/t\ge 0$ for all $t\ne 0$, we get

$$\begin{aligned} 1=\Vert u_{n}\Vert ^{2}_a\ge \int _{[u\ne 0]}\left[ \frac{f(t_{u_{n}}u_{n})}{t_{u_n}u_{n}}\right] \chi _{[u_{n}\ne 0]}(x)u_{n}^{2} dx. \end{aligned}$$

Again, by (f3), we obtain $f(t)t\ge 2F(t)\ge 0$, and so

$$\begin{aligned} 1\ge 2\int _{[u\ne 0]}\left[ \frac{F(t_{u_{n}}u_{n})}{(t_{u_n}u_{n})^{2}}\right] \chi _{[u_{n}\ne 0]}(x)u_{n}^{2}dx. \end{aligned}$$

Passing to the limit as $n\rightarrow \infty$ in the previous inequality, and invoking (f2) and Fatou’s Lemma, we get a contradiction. $\square$

Proposition 4.2

Suppose that f satisfies (f3). Then, functional $I_a$ is bounded from below on $\mathcal {N}_a$.

Proof

By (f3), we conclude that $(1/2)f(t)t-F(t)> 0$, for all $t\ne 0$. Thus,

$$\begin{aligned} I_a(u)=I_a(u)-\frac{1}{2}I'_a(u)u=\int _{\Omega }\left( \frac{1}{2}f(u)u-F(u)\right) dx> 0 \end{aligned}$$

for all $u\in \mathcal {N}$. $\square$

From now on, we denote $c_{a}:=\inf _{u\in \mathcal {N}_a}I_a(u)$. From the proof of Proposition 4.2, it is clear that $c_{a}\ge 0$ and, if $c_{a}$ is attained, the inequality is strict.

Proposition 4.3

Suppose that f satisfies (f1)–(f3), $a\in (0, (\lambda _2-\lambda _1)/|\Omega |)$ and $\lambda <\lambda _1(a)$. Then, there exists $u_{a}\in \mathcal {N}$ such that

$$\begin{aligned} I_a(u_{a})=c_{a}. \end{aligned}$$

Proof

Let $\{u_{n}\}\subset \mathcal {N}$ be a sequence satisfying $I_a(u_{n})\rightarrow c_{a}$. We are going to show that $\{u_{n}\}$ is bounded in $H_{0}^{1}(\Omega )$. In fact, suppose by contradiction that, for some subsequence, $\Vert u_{n}\Vert _a\rightarrow \infty$. Defining $v_{n}=u_{n}/\Vert u_{n}\Vert _a$, we have

$$\begin{aligned}{} & {} v_{n}\rightharpoonup v \ \hbox { in}\ H_{0}^{1}(\Omega ),\\{} & {} v_{n}\rightarrow v \ \hbox { in}\ L^{p}(\Omega ),\\{} & {} v_{n}(x)\rightarrow v(x) \ \hbox {a.e. in} \Omega \hbox {and} |v_{n}(x)|\le h(x), \end{aligned}$$

for some $h\in L^{p}(\Omega )$. Hence, there exists $d\in \mathbb {R}$ such that, if $v=0$,

$$\begin{aligned} d\ge I_a(t_{v_n}v_{n})\ge I_a(tv_{n})=\frac{t^{2}}{2}-\int _{\Omega }F(tv_{n}) dx\rightarrow \frac{t^{2}}{2}, \end{aligned}$$

for all $t>0$, which is a clear contradiction. On the other side, if $v\ne 0$, by (f2)–(f3) and Fatou’s Lemma, one has

$$\begin{aligned} \frac{I_a(u_n)}{\Vert u_{n}\Vert ^{2}_a}=\frac{1}{2}-\int _{[v\ne 0]}\left[ \frac{F(\Vert u_{n}\Vert _av_{n})}{(\Vert u_{n}\Vert _av_{n})^{2}}\right] \chi _{[v_n\ne 0]}(x)v_{n}^{2} dx\rightarrow -\infty , \ \hbox { as}\ n\rightarrow \infty , \end{aligned}$$

leading us to another contradiction, since $I_a(u_{n})/\Vert u_n\Vert ^{2}_a\rightarrow 0$ as $n\rightarrow \infty$. Therefore, $\{u_{n}\}$ is bounded and, up to a subsequence, there exists $u_{a}\in H_{0}^{1}(\Omega )$ such that

$$\begin{aligned} u_{n}\rightharpoonup u_{a} \ \hbox { in}\ H_{0}^{1}(\Omega ) \end{aligned}$$

and

$$\begin{aligned} u_{n}\rightarrow u_{a} \ \hbox { in}\ L^{s}(\Omega ), \end{aligned}$$

for $s\in [1, 2^{*})$. From Lebesgue Dominated Convergence Theorem, we obtain

$$\begin{aligned} \int _{\Omega }f(u_{n})u_{n}dx\rightarrow \int _{\Omega }f(u_{a})u_{a}dx. \end{aligned}$$

Claim 1: $u_{a}\in \mathcal {N}_a$

Observe that,

$$\begin{aligned} \Vert u_{a}\Vert ^{2}_a\le \liminf _{n\rightarrow \infty }\Vert u_{n}\Vert ^{2}_a=\liminf _{n\rightarrow \infty }\int _{\Omega }f(u_{n})u_{n}dx=\int _{\Omega }f(u_{a})u_{a}dx. \end{aligned}$$

Suppose by contradiction that $u_{a}\not \in \mathcal {N}_a$, that is

$$\begin{aligned} \Vert u_{a}\Vert ^{2}_a<\int _{\Omega }f(u_{a})u_{a}dx. \end{aligned}$$

(4.5)

We first show that $u_{a}\ne 0$. In fact, otherwise $u_{a}=0$, and then

$$\begin{aligned} \Vert u_{n}\Vert ^{2}_a=\int _{\Omega }f(u_{n})u_{n}dx\rightarrow 0. \end{aligned}$$

(4.6)

On the other hand, by (f1) and continuous embedding,

$$\begin{aligned} \Vert u_{n}\Vert ^{2}_a\le \int _{\Omega }f(u_{n})u_{n}dx\le \varepsilon \Vert u_{n}\Vert ^{2}_a+C\Vert u_{n}\Vert ^{p}_a, \end{aligned}$$

which means

$$\begin{aligned} (1-\varepsilon )\Vert u_{n}\Vert ^{2}_a\le C\Vert u_{n}\Vert ^{p}_a, \end{aligned}$$

for some $\varepsilon$ small enough. This shows that

$$\begin{aligned} \Vert u_{n}\Vert _a\ge \left( \frac{1-\varepsilon }{C}\right) ^{1/(p-2)}. \end{aligned}$$

(4.7)

By comparing (4.6) and (4.7), we get a contradiction. Therefore, $u_a\ne 0$ and, as a consequence, (4.5) is equivalent to

$$\begin{aligned} \alpha _{u_{a}, a}'(1)<0. \end{aligned}$$

It follows from $(I_1)$ that there exists a unique $0<t_{a}<1$ satisfying $t_{a}u_{a}\in \mathcal {N}_a$. By (f3),

$$\begin{aligned} c_{a}\le I_a(t_{a}u_{a})= & {} \int _{\Omega }\left[ \frac{1}{2}f(t_{a}u_{a})(t_{a}u_{a})-F(t_{a}u_{a})\right] dx\\< & {} \int _{\Omega }\left[ \frac{1}{2}f(u_{a})u_{a}-F(u_{a})\right] dx\\= & {} \lim _{n\rightarrow \infty }\int _{\Omega }\left[ \frac{1}{2}f(u_{n})u_{n}-F(u_{n})\right] dx\\= & {} \lim _{n\rightarrow \infty } I_a(u_{n})=c_{a}, \end{aligned}$$

a clear contradiction. So, $u_{a}\in \mathcal {N}_a$.

Claim 2: $u_{n}\rightarrow u_{a}$ in $H_{0}^{1}(\Omega )$

By Claim 1, we obtain

$$\begin{aligned} \Vert u_{n}-u_{a}\Vert ^{2}_a= & {} \Vert u_{n}\Vert ^{2}_a-2\langle u_n, u_a\rangle _a+\Vert u_{a}\Vert ^{2}_a\\= & {} \Vert u_{n}\Vert ^{2}_a-\Vert u_{a}\Vert ^{2}_a+o_{n}(1)\\= & {} \int _{\Omega }f(u_{n})u_{n}dx-\int _{\Omega }f(u_{a})u_{a}dx+o_{n}(1)=o_{n}(1), \end{aligned}$$

where $\langle .,.\rangle _a$ denotes the inner product associated with $\Vert .\Vert _a$. Therefore, Claim 2 is proved. Finally, from Claim 2,

$$\begin{aligned} c_{a}=\lim _{n\rightarrow \infty }I_a(u_{n})=I_a(u_{a}). \end{aligned}$$

$\square$

Before stating the main result of this section, we point out that a ground state solution $u_a$ for the problem (ALP) is a weak solution that, among all the weak solutions of (ALP), has the least energy, that is,

$$\begin{aligned} I_a(u_a)=\min \{I_a(u): u \ \text{ is } \text{ a } \text{ weak } \text{ solution } \text{ of } ({ ALP})\}. \end{aligned}$$

Theorem 4.4

Suppose that f satisfies (f1)–(f3), $a\in (0, (\lambda _2-\lambda _1)/|\Omega |)$ and $\lambda <\lambda _1(a)$. Then, problem (ALP) has a nontrivial ground-state solution $u_a$. Furthermore, if $\lambda <\lambda _1$, then

$$\begin{aligned} u_{a}\rightarrow u_*\ \hbox {in} H_{0}^{1}(\Omega ) \hbox {, as} a\rightarrow 0^{+}, \end{aligned}$$

where $u_*$ is a nontrivial ground-state solution of

Proof

Since, by $(I_1)$, $(I_2)$ and [14], the mapping $m_a:\mathcal {S}_a\rightarrow \mathcal {N}_a$ is an homeomorphism, where $m_a(w)=t_w w$ and $m_a^{-1}(w)=w/\Vert w\Vert _a$, there exists $v_{a}\in \mathcal {S}_a$ such that $v_a=m_{a}^{-1}(u_a)$. It follows from Corollary 10 in [14], that

$$\begin{aligned} \Psi _a(v_a)=\inf _{v\in \mathcal {S}}\Psi _a(v)=c_{a}. \end{aligned}$$

Thus, $v_a$ is a minimum point of $\Psi _a\in C^{1}(\mathcal {S}_a, \mathbb {R})$, with $\Psi _a:=I_a\circ m_a$. So, $v_a$ is a critical point of $\Psi _a$ and, again by Corollary 10 in [14], we conclude that $u_{a}=m_a(v_{a})$ is a nontrivial critical point of $I_a$ and, as a consequence, $u_a$ is a ground-state solution of (ALP).

Let $a_n\rightarrow 0^+$ be an arbitrary sequence. We denote $u_n:=u_{a_{n}}$, $I_{n}:=I_{a_{n}}$ and $c_n:=c_{a_{n}}$. Moreover, in the sequel,

$$\begin{aligned} I(u)=\frac{1}{2}\int _{\Omega }|\nabla u|^{2} dx-\frac{\lambda }{2}\int _{\Omega } u^{2} dx-\int _{\Omega }F(u) dx \end{aligned}$$

is the energy functional of the limit problem (LP), $\mathcal {N}$ is the Nehari set associated with I and $c_*:=\inf _{u\in \mathcal {N}}I(u)$. Since $\lambda <\lambda _1$, under (f1)–(f3), and following the same arguments, we can prove versions of Propositions 4.1 and 4.2 for the problem (LP), with $H_{0}^{1}(\Omega )$ equipped with the norm $\Vert u\Vert ^{2}:=\int _{\Omega }|\nabla u|^{2} dx$, see [14].

By arguing as in the proof of Proposition 4.3, we can prove that $\{u_n\}$ is bounded in $(H_{0}^{1}(\Omega ), \Vert .\Vert )$. Therefore, passing to a subsequence, there exists $u_{*}\in H_{0}^{1}(\Omega )$ such that

$$\begin{aligned} u_{n}\rightharpoonup u_{*} \ \hbox { in}\ H_{0}^{1}(\Omega ) \end{aligned}$$

(4.8)

and

$$\begin{aligned} u_{n}\rightarrow u_{*} \ \hbox { in}\ L^{s}(\Omega ), \end{aligned}$$

(4.9)

com $s\in [1, 2^{*})$. From Lebesgue Dominated Convergence Theorem, we obtain

$$\begin{aligned} \int _{\Omega }f(u_{n})u_{n}dx\rightarrow \int _{\Omega }f(u_{*})u_{*}dx. \end{aligned}$$

Claim 1: $u_{*}$ is a nontrivial solution of (LP).

Since $u_{n}\in \mathcal {N}_{a_n}$ and $a_n\rightarrow 0^+$,

$$\begin{aligned} \Vert u_{*}\Vert ^{2}\le & {} \liminf _{n\rightarrow \infty }\left[ \Vert u_{n}\Vert ^{2}+a_n\left( \int _{\Omega }u_{n} dx\right) ^{2}\right] \\= & {} \liminf _{n\rightarrow \infty }\left[ \lambda \int _{\Omega }u_n^{2} dx+\int _{\Omega }f(u_{n})u_{n}dx\right] \\= & {} \lambda \int _{\Omega }u_*^{2} dx+\int _{\Omega }f(u_*)u_*dx. \end{aligned}$$

Thus, supposing by contradiction that $u_{*}\not \in \mathcal {N}$, we conclude that

$$\begin{aligned} \Vert u_{*}\Vert ^{2}<\lambda \int _{\Omega }u_*^{2} dx+\int _{\Omega }f(u_{*})u_{*}dx. \end{aligned}$$

(4.10)

Suppose that $u_{*}=0$, since $u_{n}\in \mathcal {N}_{a_n}$, we get

$$\begin{aligned} \Vert u_{n}\Vert ^{2}=-a_n\left( \int _{\Omega }u_{n} dx\right) ^{2}+\lambda \int _{\Omega }u_n^{2} dx+\int _{\Omega }f(u_{n})u_{n}dx\rightarrow 0. \end{aligned}$$

(4.11)

Now, by (f1) and continuous embedding,

$$\begin{aligned} \left( 1-\frac{\lambda }{\lambda _1}\right) \Vert u_{n}\Vert ^{2}\le \int _{\Omega }f(u_{n})u_{n}dx\le \varepsilon \Vert u_{n}\Vert ^{2}+C\Vert u_{n}\Vert ^{p}. \end{aligned}$$

Since $\lambda <\lambda _1$, for some $\varepsilon$ small enough, we obtain

$$\begin{aligned} \Vert u_{n}\Vert \ge \left( \frac{1-\varepsilon -\lambda /\lambda _1}{C}\right) ^{1/(p-2)}>0. \end{aligned}$$

(4.12)

By comparing (4.11) and (4.12), we get a contradiction. Therefore, $u_*\ne 0$ and, passing to the limit as $n\rightarrow \infty$ in

$$\begin{aligned} \int _{\Omega }\nabla u_n\nabla v dx+a_n\int _{\Omega }u_n dx\int _{\Omega }v dx=\lambda \int _{\Omega }u_nv dx+\int _{\Omega }f(u_n)v dx, \end{aligned}$$

we conclude from (4.8) and (4.9), that

$$\begin{aligned} \int _{\Omega }\nabla u_*\nabla v dx=\lambda \int _{\Omega }u_*v dx+\int _{\Omega }f(u_*)v dx, \end{aligned}$$

for all $v\in H_{0}^{1}(\Omega )$. Thus $u_*$ is a nontrivial weak solution of (LP).

Claim 2: $u_{n}\rightarrow u_*$ in $H_{0}^{1}(\Omega )$ and $I(u_*)=c_*:=\inf _{u\in \mathcal {N}}I(u)$.

By Claim 1,

$$\begin{aligned} u_*\in \mathcal {N}, \end{aligned}$$

(4.13)

and

$$\begin{aligned} \Vert u_{n}-u_{*}\Vert ^{2}= & {} \Vert u_{n}\Vert ^{2}-2\int _{\Omega } \nabla u_n \nabla u_*dx+\Vert u_{*}\Vert ^{2}\\= & {} \Vert u_{n}\Vert ^{2}-\Vert u_{*}\Vert ^{2}+o_{n}(1)\\= & {} \int _{\Omega }f(u_{n})u_{n}dx-\int _{\Omega }f(u_{*})u_{*}dx+o_{n}(1)=o_{n}(1), \end{aligned}$$

Therefore, $u_{n}\rightarrow u_{*}$ in $H_{0}^{1}(\Omega )$. To show that $c_n\rightarrow c_{*}$, let $u_0$ be a ground-state solution of (LP), whose existence, under assumptions (f1)–(f3) and $\lambda <\lambda _1$, is proved in Theorem 16 of [14]. Consequently

$$\begin{aligned} u_0\in \mathcal {N}. \end{aligned}$$

(4.14)

It follows from Proposition 4.1 that, for each $n\in \mathbb {N}$, there exists a unique $t_{n}>0$ such that

$$\begin{aligned} t_{n}u_0\in \mathcal {N}_{a_n}. \end{aligned}$$

(4.15)

From identity

$$\begin{aligned} \Vert u_{0}\Vert ^{2}_{a_n}= \int _{[u_0\ne 0]}\left[ \frac{f(t_n u_{0})}{t_n u_{0}}\right] u_{0}^{2} dx, \end{aligned}$$

(4.16)

combined with (f1)–(f3), we conclude that, up to a subsequence, $t_{n}\rightarrow t_0>0$. We are going to show that $t_0=1$. In fact, since $a_n\rightarrow 0^+$, passing to the limit as $n\rightarrow \infty$ in (4.16), we get

$$\begin{aligned} t_0 u_0\in \mathcal {N}. \end{aligned}$$

(4.17)

By comparing (4.14) and (4.17), we conclude that $t_0=1$.

Finally, by (4.13) and (4.15),

$$\begin{aligned} c_*\le I(u_{*})= & {} \int _{\Omega }\left[ \frac{1}{2}f(u_{*})u_{*}-F(u_{*})\right] dx\\= & {} \lim _{n\rightarrow \infty }\int _{\Omega }\left[ \frac{1}{2}f(u_{n})u_{n}-F(u_{n})\right] dx\\= & {} \lim _{n\rightarrow \infty } I_n(u_{n})\\= & {} c_{n}\le I_{n}(t_n u_{0}), \end{aligned}$$

Now, since $t_n\rightarrow 1$ and $u_0$ is a ground-state solution of (LP), passing to the limit as $n\rightarrow \infty$, we get

$$\begin{aligned} I(u_*)=c_*. \end{aligned}$$

This proves Claim 2. The result now follows from Claim 1 and Claim 2. $\square$

5 Open questions, comments and biological meaning

The main goal of this section is briefly to point out some questions which remain without answer, as well as conjecture some rather likely situations involving the problems considered in previous sections.

5.1 Dead-core solutions

Among other things, Theorem 2.5 guarantees the existence of a positive number $a_*$ for which the solution $u_{a_*}$ belongs to the boundary of the positive cone $\mathcal {P}$ in $C^{1, \beta }_{0}(\overline{\Omega })$. As we have mentioned in the proof of Theorem 2.5, at least one of the following two class of phenomena holds true for functions in $\partial \mathcal {P}$:

(i):

“Dead-zones”, that is, is observed the existence of sets $\Omega _*\subset \Omega$, such that $u_{a_*}\equiv 0$ in $\Omega _*$;

(ii):

Normal derivative vanishing, that is, there exists $\Gamma _*\subset \partial \Omega$ with positive $(N-1)$-dimensional Lebesgue measure such that $\partial u_{a_{*}}/\partial \eta =0$ on $\Gamma _*$.

Since examples of the second situation have been more common in the literature (see [1]), a very reasonable question to ask now is: “What kind of functions f will produce dead-core solutions of (LP) for some positive a?”. Next considerations shed some light on this open question.

Lemma 5.1

Let $f\in L^{\infty }(\Omega )$, $f\ge 0$, f is not constant and $\Omega _{*}\subsetneq \Omega$ be a smooth domain. If a nontrivial and nonnegative weak solution $u_{a}$ of (LP) vanishes in $\Omega _*$ for some $a>0$, then f coincides with nontrivial constant c in $\Omega _*$. Furthermore, the following equalities hold

$$\begin{aligned} c= a\int _{\Omega \backslash \Omega _*}u_{a} dx \ \hbox { in}\ \Omega _*, \end{aligned}$$

(5.1)

and

$$\begin{aligned} a=\frac{c}{\int _{\Omega \backslash \Omega _{*}}(f-c)e dx}. \end{aligned}$$

(5.2)

Proof

Suppose that, for some $a> 0$, the weak solution $u_{a}$ of (LP) is such that $u_a=0$ in $\Omega _*$. By choosing $\varphi \in C_{0}^{\infty }(\Omega _*)$ as a test function, we get

$$\begin{aligned} a\int _{\Omega \backslash \Omega _*}u_{a} dx\int _{\Omega _*}\varphi dx=\int _{\Omega _*}f(x)\varphi dx, \end{aligned}$$

that is,

$$\begin{aligned} \int _{\Omega _*}\left( f(x)-a\int _{\Omega \backslash \Omega _*}u_{a} dx\right) \varphi dx=0, \end{aligned}$$

for all $\varphi \in C_{0}^{\infty }(\Omega _*)$. Therefore,

$$\begin{aligned} f\equiv a\int _{\Omega \backslash \Omega _*}u_{a} dx=:c \ \hbox { in}\ \Omega _*. \end{aligned}$$

(4.3)

By (4.3), f is a nontrivial constant in $\Omega _*$ because $a> 0$ and $u_a$ is nonnegative and nontrivial. Moreover, combining (4.3) and Theorem 2.2, we get

$$\begin{aligned} c = \left( \frac{a\alpha _f}{1+a\int _{\Omega }e dx }\right) \int _{\Omega }e dx = \left( \frac{a}{1+a\int _{\Omega }e dx} \right) \int _{\Omega }fe dx. \end{aligned}$$

(4.4)

Finally, since by (4.3) we have $c>0$, we can isolate a, to obtain

$$\begin{aligned} a=\frac{c}{\int _{\Omega }fe dx-c\int _{\Omega }e dx}=\frac{c}{\int _{\Omega \backslash \Omega _*}(f-c)e dx}. \end{aligned}$$

(4.5)

$\square$

It seems that the converse of previous lemma is not true for any non-constant and nonnegative $f\in L^{\infty }(\Omega )$, which is constant in some smooth domain $\Omega _*\subset \Omega$. This fact can be observed in the following particular case:

$$\begin{aligned} \left\{ \begin{array}{ll} -u''+a\int _{0}^{1}u(s) ds=f(x) &{} \text{ in } (0, 1),\\ u(0)=u(1)=1, \end{array}\right. \end{aligned}$$

(4.6)

where $f(x)=1$ in [0, 1/2] and $f(x)=4x-1$ in [1/2, 1]. The solution of (4.6) is

$$\begin{aligned} u_a(x)=\left[ \frac{7}{12}-\frac{11a}{16(a+12)}\right] x+\frac{11a}{16(a+12)}x^{2}-g(x), \end{aligned}$$

with $g(x)=x^{2}/2$ in [0, 1/2] and

$$\begin{aligned} g(x)=\frac{2}{3}x^3-\frac{x^2}{2}+\frac{x}{2}-\frac{1}{12} \ \hbox { in}\ [1/2, 1]. \end{aligned}$$

In this particular case, we have

$$\begin{aligned} \Omega _*=(0, 1/2), \ c=1 \ \text{ and } \ e(x)=\frac{1}{2}x(1-x), \ x\in [0, 1]. \end{aligned}$$

Thus, choosing

$$\begin{aligned} a_*=\frac{c}{\int _{\Omega \backslash \Omega _{*}}(f-c)e dx}=\frac{1}{\int _{1/2}^{1}(4x-2)\frac{x(1-x)}{2} dx}=32, \end{aligned}$$

it follows that

$$\begin{aligned} u_{a_*}(x)=-\frac{5}{12}x \ \hbox { in}\ [0, 1/2]. \end{aligned}$$

That is, $\Omega _*=(0, 1/2)$ is not a dead-zone of $u_{a_*}$. More generally, it does not exist $a>0$ for which (0, 1/2) is a dead-zone of $u_a$, since

$$\begin{aligned} u_a(x)=\left[ \frac{7}{12}-\frac{11a}{16(a+12)}\right] x+\left[ \frac{11a}{16(a+12)}-\frac{1}{2}\right] x^{2} \ \hbox { in}\ [0, 1/2]. \end{aligned}$$

In this sense, it remains open the question about the existence and characterization of functions $f\in L^{\infty }(\Omega )$ for which (LP) admits dead-core solutions for some $a>0$.

5.2 Sign of eigenfunctions

We have proved in item (iii) of Corollary 3.7 that if a is close enough from zero or $-1/\int _{\Omega }e dx$, and negative, then the eigenfunctions associated to $\lambda _0(a)$ have defined sign, more specifically, we have proved that $u_{\lambda _{0}(a)}$ is negative. We actually believe that $u_{\lambda _0(a)}$ is negative for all $a<0$, at least, it is which seems to indicate some particular unidimensional cases of (EP). For instance, consider the problem

where $a<0$ is given. First, suppose that $a<-1/\int _{0}^{1}e(s) ds$, where e is such that $-e''(x)=1$ in (0, 1) and $e(0)=e(1)=0$, that is,

$$\begin{aligned} e(x)=\frac{1}{2}x(1-x) \ \text{ and } \ \int _{0}^{1}e(s) ds=1/12. \end{aligned}$$

We are interested in eigenfunctions of (UEP) associated to $\lambda _0(a)$. Since, by Proposition 3.5, $\lambda _0(a)<0$ whenever $a<-1/\int _{0}^{1}e(s) ds=-12$, we are going to consider $\lambda <0$ in (UEP). For each fixed $r\in \mathbb {R}$, by considering the auxiliary problem

$$\begin{aligned} \left\{ \begin{array}{ll} -u''+ar=\lambda u &{} \text{ in } (0, 1),\\ u(0)=u(1)=0, \end{array}\right. \end{aligned}$$

(5.7)

we can easily solve the equation in order the get the solution

$$\begin{aligned} u_r(x)= & {} \frac{ar}{-\lambda }\left[ \frac{(\exp {(\sqrt{-\lambda }})-1)}{(\exp {(\sqrt{-\lambda })}-\exp {(-\sqrt{-\lambda })})}\exp {(-\sqrt{-\lambda }}x)\right. \\{} & {} \left. +\frac{(1-\exp {(-\sqrt{-\lambda }}))}{(\exp {(\sqrt{-\lambda })}-\exp {(-\sqrt{-\lambda })})}\exp {(\sqrt{-\lambda }}x)-1\right] . \end{aligned}$$

Observe that $u_r$ is a solution (UEP) if, and only if,

$$\begin{aligned} r=\int _{0}^{1}u_r(s) ds, \end{aligned}$$

or equivalently,

$$\begin{aligned} a\left[ \frac{4-(2+\sqrt{-\lambda })\exp {(-\sqrt{-\lambda })}-(2-\sqrt{-\lambda })\exp {(\sqrt{-\lambda })}}{\lambda \sqrt{-\lambda }(\exp {(\sqrt{-\lambda })}-\exp {(-\sqrt{-\lambda })})}\right] =1, \end{aligned}$$

(5.8)

that is, $\lambda _0(a)$ is the unique negative root of (5.8). Thus,

$$\begin{aligned}&u_{\lambda _0(a)}(x)=\frac{1}{-\lambda _0(a)}\left[ \frac{(\exp {(\sqrt{-\lambda _0(a)}})-1)}{(\exp {(\sqrt{-\lambda _0(a)})}-\exp {(-\sqrt{-\lambda _0(a)})})}\exp {(-\sqrt{-\lambda _0(a)}}x)\right] \\&\quad +\left[ \frac{(1-\exp {(-\sqrt{-\lambda _0(a)}}))}{(\exp {(\sqrt{-\lambda _0(a)})}-\exp {(-\sqrt{-\lambda _0(a)})})}\exp {(\sqrt{-\lambda _0(a)}}x)-1\right] , \ x\in [0, 1]. \end{aligned}$$

Therefore, $u_{\lambda _0(a)}$ is negative for all $a<-12$. If $a=-12$, by Proposition 3.5, we know that $\lambda _0(-12)=0$ and, to find $u_{\lambda _0(-12)}$ we have to study the problem (UEP) with $\lambda =0$. But, in this case, a straightforward calculation shows us that

$$\begin{aligned} u_{\lambda _0(-12)}(x)=-e(x)=\frac{1}{2}x(x-1), \ \theta \in [0, 1], \end{aligned}$$

which is clearly negative. Finally, if $-12<a<0$, then, by Proposition 3.5, $0<\lambda _0(a)<\lambda _1=\pi ^{2}$. By arguing as in previous cases, we conclude that for each given $a\in (-12, 0)$, $\lambda _0(a)$ is the unique solution in $(0, \pi ^{2})$ of

$$\begin{aligned}&a\left[ \frac{(7\lambda +1)\cot {(\sqrt{\lambda })}\exp (1)}{4\lambda (1+\lambda )\sqrt{\lambda }\exp (1)}+\right. \\&\quad + \left. \frac{(\exp {2}-4\lambda \cos {(2\sqrt{\lambda })}+(\lambda -1)\cos {(3\sqrt{\lambda })}\exp (1)+8\sqrt{\lambda }\sin ^{3}(\sqrt{\lambda })\exp (1))\csc {\sqrt{\lambda }}}{4\lambda (1+\lambda )\sqrt{\lambda }\exp (1)}\right] = 1, \end{aligned}$$

and

$$\begin{aligned}&u_{\lambda _0(a)}(x)=\frac{1}{\lambda _0(a)}\left[ \left( \frac{\cos {(\sqrt{\lambda _0(a)})}\exp (1)-\cos {(2\sqrt{\lambda _0(a)})}}{\sin {(\sqrt{\lambda _0(a)})}\exp (1)}\right) \exp (x)\sin {(\sqrt{\lambda _0(a)}x)}+\right. \\&\quad +\left. \cos {(2\sqrt{\lambda _0(a)}x)}-\exp (x)\cos (\sqrt{\lambda _0(a)}x)\right] \ x\in [0, 1], \end{aligned}$$

is negative.

5.3 Sign of ground state solutions

When $a=0$ an argument used in [14] shows that ground-state solutions of (ALP) have defined sign. Unfortunately, such a sort of reasoning cannot be used when $a\ne 0$, since the presence of the integral term $\int _{\Omega }u dx$ in the functional $I_a(u)$, prevents us to get the identity $I_a(u)=I_a(u^+)+I_{a}(u^-)$ as u changes sign. Taking into account the asymptotic behaviour proved in Theorem 4.4, we think that under some additional assumption of regularity on function f, the ground-state solutions of (ALP) should preserve sign for a small enough. Another relevant question to be investigated is about the existence of ground state solutions of (ALP) when $a>0$ is not small.

5.4 Biological interpretation

In this subsection, we make some quick considerations about possible biological interpretations of our previous results. Initially, according to the model studied in Sect. 2, more specifically, in item (iii) of Corollary 2.3, it is observed that when one considers the model with a non-negative function f depending only on $x\in \Omega$, the parameter a represents how the spatial distribution of the species is affected by the total population. In fact, the greater the parameter a, the greater the influence of the total population on the spatial distribution of the individuals within its domain $\Omega$. In particular, when $f\ge 0$ and $a\rightarrow \infty$, we have proved that $u_a\rightarrow P_{X}u_f$ and, consequently, $\int _{\Omega } u_a dx\rightarrow 0$. This fact reveals that when the influence of the total population (on the spatial distribution of the individuals) grows, then the species tends to be extinct. The same phenomenon can be seen in the eigenvalue problem presented in Sect. 3, item (iii) of Corollary 3.6.