The influence of a metasolution on the behaviour of the logistic equation with nonlocal diffusion coefficient

Abstract

In this paper we use the bifurcation method and fixed point arguments to study a logistic equation with nonlocal diffusion coefficient. We prove the existence of an unbounded continuum of positive solutions that bifurcates from the trivial solution. The global behaviour of this continuum depends strongly on the value of the nonlocal diffusion coefficient at infinity as well as the relative position between the refuge of the species and the weight of the diffusion coefficient. Moreover, we show the complexity of the structure of the set of positive solutions using fixed point arguments.

1 Introduction

In this paper we study the following nonlocal elliptic equation

$$\begin{aligned} \left\{ \begin{array}{ll} -a\left( \displaystyle \int _{\varOmega }q(x)u^p\right) \varDelta u = \lambda u - b(x)u^2&{} \text{ in } \varOmega ,\\ u = 0 &{} \text{ on } \partial \varOmega , \end{array}\right. \end{aligned}$$

(1)

where $\varOmega \subset {\mathbb {R}}^N$, $N \ge 1$, is a bounded and regular domain, $p>0$, $\lambda \in {\mathbb {R}}$, $b\in C^1(\overline{\varOmega })$, is a non-negative and non-trivial function, $a\in C( {\mathbb {R}})$ is a positive function and q(x) is a bounded, non-negative and non-trivial function in $\varOmega$. This equation models the behavior of a species, whose population density is u and its habitat is $\varOmega$. We are assuming that $\varOmega$ is surrounded by inhospitable areas due to homogeneous Dirichlet boundary condition. The reaction term is the classical logistic term: $\lambda$ denotes the growth rate of the species and b represents the limiting effects of crowding in the population u. We will consider two cases:

$(Hb_1$):

$b(x) \ge b_0 >0, \quad \forall x\in \overline{\varOmega },$ for some positive constant $b_0>0$, or

$(Hb_2)$ :

$b(x) \ge 0$ in $\varOmega$ and $b(x) \equiv 0, \forall x\in \varOmega _0\subset \subset \varOmega$, where $\varOmega _0$ is a proper subdomain of $\varOmega$.

For simplicity, we assume that $\varOmega _0$ has only one connected component. In the first case, this limiting effect acts in all the domain. However, when b verifies $(Hb_2)$, there is a region, $\varOmega _0$, where the species grows freely, this set is called a refuge for u.

Finally, in (1) the velocity of the diffusion, the spatial movement of the species, is non-local; that is, it depends on the total population in its habitat. Hence, if a is an increasing function, the species has the tendency to leave crowded zones, while if a is decreasing this means that the species is attracted by the growing population, see for instance [2]. We will assume that a is continuous and positive function defined on ${\mathbb {R}}$, $a(s)>0$ for all $s\in {\mathbb {R}}$, that satisfies

$$\begin{aligned} 0\le a_L \le a(s)\le a_M\le +\infty , \quad \text{ for } s\in {\mathbb {R}},\quad a(0)>0, \end{aligned}$$

where

$$\begin{aligned} a_L = \displaystyle \inf _{s\in [0,\infty )}a(s),\quad a_M = \displaystyle \sup _{s\in [0,\infty )}a(s), \end{aligned}$$

and we define

$$\begin{aligned} a(\infty ) := \lim _{s\rightarrow \infty }a(s). \end{aligned}$$

Also, to simplify some of the proofs of the work, we will assume that $a'(s)=0$ for s, at most, in a discrete set of ${\mathbb {R}}$.

In order to state our main results we need to introduce some notations. Given a domain $D\subset \varOmega$ we denote by $\lambda _1^D$ the principal eigenvalue of the Laplacian operator under homogeneous Dirichlet boundary conditions. We denote $\lambda _1=\lambda _1^\varOmega$.

In the local case, that is, $a\equiv 1$, the main results are (see Sect. 2 and [8]):

• 1. If b verifies $(Hb_1)$, then there exists a positive solution if and only if $\lambda >\lambda _1$. In such case, the positive solution is unique.

• 2. If b verifies $(Hb_2)$, then there exists a positive solution if and only if $\lambda \in (\lambda _1,\lambda _1^{\varOmega _0})$. In this case, the positive solution is unique, and if we denote it by $u_\lambda$, it verifies

  $$\begin{aligned} \lim _{\lambda \rightarrow \lambda _1^{\varOmega _0}}u_\lambda =\mathcal{M}(x), \end{aligned}$$

  where $\mathcal{M}$ is the “metasolution”, that is,

  $$\begin{aligned} \mathcal{M}(x)=\left\{ \begin{array}{ll} +\infty &{}\quad \text{ in } \overline{\varOmega }_0,\\ L(x) &{}\quad \text{ in } \varOmega {\setminus }\overline{\varOmega }_0, \end{array} \right. \end{aligned}$$

  and L is the minimal large solution (see Sect. 2) of

  $$\begin{aligned} \left\{ \begin{array}{ll} -\varDelta L(x) =\lambda _1^{\varOmega _0} L(x) -b(x)L(x)^2&{}\quad \text{ in } \varOmega {\setminus }\overline{\varOmega }_0,\\ L(x)= 0 &{}\quad \text{ on } \partial \varOmega ,\\ L(x) = \infty &{}\quad \text{ on } \partial \varOmega _0. \end{array}\right. \end{aligned}$$

In both cases, if the growth rate is small, then the unique solution is the trivial one. Moreover, when the refuge exists, for large value of the growth rate, the species does not exist because it blows up in $\varOmega _0$.

Now, we recall the known-results of (1). First, it should be noted that Eq. (1) has been studied only for b verifying $(Hb_1)$ in [3, 4] and [10] (see also [12]). In [4] it was proved the existence of positive solution for $\lambda$ large when

$$\begin{aligned} 0<a_L\le a(s)\le a_M<\infty . \end{aligned}$$

This result was improved in [3] showing the existence of positive solution for $\lambda >a_M\lambda _1$. In both papers, the Schauder fixed point theorem was used. In [10], the bifurcation method was employed to study (1) when

$$\begin{aligned} a(s)=c_1+c_2s^\alpha ,\quad c_i>0,\;\alpha >0. \end{aligned}$$

For this particular choice of the function a, the authors in [10] proved the existence of positive solution for $\lambda >c_1\lambda _1$ provided that $2>\max \{\alpha p+1,p-1\}$.

Our results improve the above results. In addition, we assume nor $a_L>0$ neither $a_M<\infty$. Moreover, we do not impose any restriction in p. Furthermore, we study the case $(Hb_2)$, that, to our knowledge, is new in the literature. In fact, the structure of the set of positive solutions could be really complex (see for instance Figs. 5 and 6, that sketch examples of bifurcation diagrams).

Our first main result is concerning to the case b verifying $(Hb_1)$. In this case, the result is analogous to the local case:

• 1. If b verifies $(Hb_1)$, then there exists a positive solution if $\lambda >a(0)\lambda _1$. Moreover, if b is constant and a is increasing, then the positive solution is unique.

However, when b verifies $(Hb_2)$ the results are completely different to the local case. In fact, the results depend strongly on the behaviour of the function a and on the following integral

$$\begin{aligned} I:=\int _\varOmega q(x)\mathcal{M}^p(x)dx. \end{aligned}$$

Hence, we can summarize our main results in this case: Assume that b verifies $(Hb_2)$:

• 1. There exists an unbounded continuum $\mathcal{C}$ in ${\mathbb {R}}\times L^\infty (\varOmega )$ of positive solutions of (1) emanating from the trivial solution at $\lambda =a(0)\lambda _1$.

• 2. If $I=\infty$, then there exists a positive solution if

  $$\begin{aligned} \lambda \in \left( \min \left\{ a(0)\lambda _1,a(\infty )\lambda _1^{\varOmega _0}\right\} , \max \left\{ a(0)\lambda _1,a(\infty )\lambda _1^{\varOmega _0}\right\} \right) . \end{aligned}$$

  In this case, the values $a(\infty )=0$ and $a(\infty )=\infty$ are allowed. In fact, the continuum $\mathcal{C}$ goes to infinity at $\lambda =a(\infty )\lambda _1^{\varOmega _0}$.

• 3. Assume that $I<\infty$ and consider the real equation

  $$\begin{aligned} g(s):=\frac{s}{a^p(s)}=I. \end{aligned}$$

  (2)

  • (a) Assume that (2) does not have positive solution, that is, $g(s)<I$ for all $s\ge 0$. Then, (1) possesses at least one positive solution for $\lambda \in (a(0)\lambda _1,\infty )$.

  • (b) Assume that there exist $s_1<s_2<\dots <s_{m}$, $m\ge 1$, simple roots of (2) and consider

    $$\begin{aligned} \varLambda _i= \lambda _1^{\varOmega _0}\left( \frac{s_i}{I}\right) ^{1/p},\quad i=1,\dots , m. \end{aligned}$$

    Then:

    • i. The unbounded continuum $\mathcal{C}$ of positive solutions of (1) goes to infinity at $\lambda =\varLambda _1$. As consequence, there exists at least one positive solution if

      $$\begin{aligned} \lambda \in ( \min \{a(0)\lambda _1,\varLambda _1\}, \max \{a(0)\lambda _1,\varLambda _1\}). \end{aligned}$$

    • ii. If $m=2k+1$, $k\ge 0$, (1) possesses at least a positive solution for

      $$\begin{aligned} \lambda \in \displaystyle \bigcup _{j=1}^k(\varLambda _{2j},\varLambda _{2j+1}), \end{aligned}$$

      and (1) does not have positive solution for $\lambda$ large.

    • iii. If $m=2k$, $k\ge 1$, (1) possesses at least a positive solution for

      $$\begin{aligned} \lambda \in \displaystyle \bigcup _{j=1}^{k-1}(\varLambda _{2j},\varLambda _{2j+1})\cup (\varLambda _{2k},\infty ). \end{aligned}$$

In Figs. 1, 2, 3, 4, 5 and 6 we show different possibilities of the bifurcation diagrams.

Moreover, we have studied in detail the bifurcation direction from the trivial solution (Sect. 4) and we have detailed the cases when a is increasing and a decreasing, showing the bifurcations diagrams as well as the uniqueness of positive solution.

Fig. 1

Bifurcation diagrams when b verifies $(Hb_2)$, $I=\infty$ and $0<a(\infty )<\infty$. In the left, $a(0)\lambda _1<a(\infty )\lambda _1^{\varOmega _0}$ and in the right $a(\infty )\lambda _1^{\varOmega _0}<a(0)\lambda _1$

Fig. 2

Bifurcation diagrams in the case when b verifies $(Hb_2)$ and $I=\infty$. At the left $a(\infty )=\infty$, at the right $a(\infty )=0$. The first diagram also appears when b verifies $(Hb_1)$

Fig. 3

Possible bifurcation diagrams when b verifies $(Hb_2)$, $I=\infty$ and $a(0)\lambda _1=a(\infty )\lambda _1^{\varOmega _0}$. In the first case, the bifurcation direction is supercritical, while in the second one it is subcritical

Fig. 4

Case b verifies $(Hb_2)$ and $I<\infty$ and $g(s)<I$. To the left: representation of the function g(s) and to the right, its corresponding bifurcation diagram

Fig. 5

Case when b verifies $(Hb_2)$, $I<\infty$ and $g(s)=I$ in an odd number of points. Representation of g and its corresponding bifurcation diagram in the case $a(0)\lambda _1<\varLambda _1$

Fig. 6

Case when b verifies $(Hb_2)$, $I<\infty$ and $g(s)=I$ in an even number of points. Representation of g, and corresponding bifurcation diagrams: in the first case $a(0)\lambda _1>\varLambda _2$ and in the second one, $a(0)\lambda _1<\varLambda _1$

Let us compare our main results with the local case:

• 1. In the case that b verifies $(Hb_1)$ the existence results are very similar (in both case there exists positive solution for $\lambda >a(0)\lambda _1$); however, in the nonlocal case there may be multiplicity of positive solutions, in fact, thanks to the bifurcation direction, we can assure the existence of two positive solutions in some cases.

• 2. Assume that b verifies $(Hb_2)$. In this case, we distinguish two cases: $I=\infty$ and $I<\infty$.

  • (a) $I=\infty$. This case occurs when $Q_+:=\{x\in \varOmega : q(x)>0\}\cap \varOmega _0\ne \emptyset$, that is, the nonlocal diffusion coefficient takes into account the refuge of the species. Unlike the local case, it can exist positive solution for $\lambda$ small (if the diffusion for large values of the population is small, $a(\infty )=0$) or for $\lambda$ large (if the diffusion for large values of the population is large, $a(\infty )=\infty$).

  • (b) $I<\infty$. This case occurs when $Q_+\cap \varOmega _0= \emptyset$, that is, the refuge of the species is not seen by the diffusion. In this case, the structure of the set of positive solutions is complex, but in no case there exists positive solution for $\lambda$ small. Moreover, in order to exist positive solutions for $\lambda$ large is necessary that a goes to infinity faster than $s^{1/p}$. On the contrary, positive solutions do not exist for $\lambda$ large.

The structure of the paper is as follows. First, in Sect. 2, we will introduce some basic notations and terminology necessary along the paper. Section 3 is devoted to recall the local case. In Sect. 4, we apply the bifurcation method to (1). We prove the existence of an unbounded continuum of positive solutions of (1) emanating from the trivial solution at $\lambda =a(0)\lambda _1$. Moreover, we study in detail the direction of this bifurcation. Finally, in Sect. 5 we prove the main results of our paper. For that, we employ an adequate fixed point argument together the bifurcation results.

2 Preliminaries

2.1 An eigenvalue problem

We start the section with some results related to eigenvalue problems that will be needed throughout this paper. Consider the following eigenvalue problem

$$\begin{aligned} \left\{ \begin{array}{ll} - d \varDelta u + c(x)u= \lambda u &{} \text{ in } \varOmega ,\\ u = 0 &{} \text{ on } \partial \varOmega , \end{array}\right. \end{aligned}$$

(3)

where $c\in L^{\infty }(\varOmega )$ and $d>0$ is a positive constant.

We denote the principal eigenvalue of (3) by $\lambda _1[-d\varDelta + c]$, that is, an eigenvalue with a positive eigenfunction associated to it.

Moreover, given a subdomain $D\subseteq \varOmega$, we denote $\lambda _1^D[-d\varDelta + c]$ as the principal eigenvalue in D. In the case $c\equiv 0$ and $d=1$, we denote $\lambda _1^D=\lambda _1^D[-\varDelta ]$. When $D=\varOmega$, we delete the superscript to avoid confusion.

The following result shows some properties of the principal eigenvalue, which are direct consequences of its variational characterization, see for instance [9].

Proposition 1

We have the following properties:

• 1. The map $d\in (0,\infty )\mapsto \lambda _1[-d\varDelta + c]\in {\mathbb {R}}$ is continuous and increasing.

• 2. The map $c\in L^\infty (\varOmega ) \mapsto \lambda _1[-d\varDelta + c]\in {\mathbb {R}}$ is continuous and increasing.

• 3. If $D\subset \varOmega$ is a subdomain, then $\lambda _1[-d\varDelta + c]<\lambda _1^D[-d\varDelta + c]$.

2.2 Sub-supersolution method for nonlocal equation

We introduce the sub-supersolution method for a nonlocal equation of the following general form

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle -a\left( \int _{\varOmega }q(x)u^p\right) \varDelta u = f(x,u)&{} \text{ in } \varOmega ,\\ u = 0 &{} \text{ on } \partial \varOmega , \end{array}\right. \end{aligned}$$

(4)

where $f:\varOmega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is a continuous function. There are different definitions of sub-supersolution of (4) depending on the properties of a and f, see for instance [1]. We present here the sub-supersolution method of [5], Sect. 5, which allows more generality on a and f.

Definition 1

We say that the pair $(\underline{u},\overline{u})$, with $\underline{u}, \overline{u}\in H^1(\varOmega )\cap L^{\infty }(\varOmega )$, is a pair of sub-supersolution of (4) if

• a) $\underline{u}\le \overline{u}$ in $\varOmega$,

• b) $\underline{u}\le 0\le \overline{u}$ on $\partial \varOmega$,

• c)
  $$\begin{aligned}&\displaystyle a\left( \int _{\varOmega }q(x)u^p\right) \int _{\varOmega }\nabla \underline{u}\cdot \nabla \varphi \le \int _{\varOmega }f(x,\underline{u})\varphi \\&\displaystyle a\left( \int _{\varOmega }q(x)u^p\right) \int _{\varOmega }\nabla \overline{u}\cdot \nabla \varphi \ge \int _{\varOmega }f(x,\overline{u})\varphi , \end{aligned}$$

  for all $\varphi \in H_0^1(\varOmega )$, $\varphi \ge 0$ in $\varOmega$ and for all $u\in [\underline{u},\overline{u}]$, where $[\underline{u},\overline{u}]=\{u\in L^\infty (\varOmega ):\underline{u}\le u\le \overline{u}\}$.

The main result reads as follows:

Theorem 1

Assume that there exists a pair of sub-supersolution of (4). Then, there exists a solution $u \in H_0^1(\varOmega )\cap L^{\infty }(\varOmega )$ of (4) such that

$$\begin{aligned} u\in [\underline{u},\overline{u}]. \end{aligned}$$

3 Local logistic equation

In this section we consider the following local logistic problem

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle -\varDelta u = \mu u - b(x)u^2&{} \text{ in } \varOmega ,\\ u = 0 &{} \text{ on } \partial \varOmega . \end{array}\right. \end{aligned}$$

(5)

In the following result we recall the main results concerning to the existence and uniqueness of positive solution of (5) as well as its main properties, see for instance [7, 8] and references therein.

Theorem 2

• 1. If b(x) satisfies $(Hb_1)$, there exists a unique positive solution of (5) if and only if $\mu > \lambda _1$.

• 2. If b(x) satisfies $(Hb_2)$, there exists a unique positive solution of (5) if and only if

  $$\begin{aligned} \lambda _1< \mu < \lambda _1^{\varOmega _0}. \end{aligned}$$

  (6)

  Moreover, if we denote the unique positive solution of (5) by $\theta _{[\mu ,b]}$, we have that the map $\mu \in (\lambda _1,\lambda _1^{\varOmega _0})\rightarrow \theta _{[\mu ,b]}\in C^2({\overline{\varOmega }})$ is continuous, increasing, differentiable and

  $$\begin{aligned} \lim _{\mu \downarrow \lambda _1}\Vert \theta _{[\mu ,b]}\Vert _\infty =0,\quad \lim _{\mu \uparrow \lambda _1^{\varOmega _0}}\Vert \theta _{[\mu ,b]}\Vert _r=\infty , \end{aligned}$$

  (7)

  where $1\le r \le \infty .$ Furthermore,

  $$\begin{aligned} \lim _{\mu \uparrow \lambda _1^{\varOmega _0}}\theta _{[\mu ,b]}(x)\left\{ \begin{array}{ll} +\infty &{} \text{ if } x\in \overline{\varOmega }_0,\\ <\infty &{} \text{ if } x\in \varOmega {\setminus }\overline{\varOmega }_0. \end{array} \right. \end{aligned}$$

  In fact, for any open set $D\subset \varOmega {\setminus } \overline{\varOmega }_0$ it holds that

  $$\begin{aligned}\lim _{\mu \uparrow \lambda _1^{\varOmega _0}}\theta _{[\mu ,b]}=L\quad \text{ in } C^2(\overline{D}), \end{aligned}$$

  where L is the minimal solution of

  $$\begin{aligned} \left\{ \begin{array}{ll} -\varDelta L(x) =\lambda _1^{\varOmega _0} L(x) -b(x)L(x)^2&{}\quad \text{ in } \varOmega {\setminus }\overline{\varOmega }_0,\\ L(x)= 0 &{}\quad \text{ on } \partial \varOmega ,\\ L(x) = \infty &{}\quad \text{ on } \partial \varOmega _0. \end{array}\right. \end{aligned}$$

  (8)

Now, we define a generalized function which will play a fundamental role in our work. Indeed, we define the metasolution (see [8]), that is

$$\begin{aligned} \mathcal{M}(x):= \left\{ \begin{array}{ll} \infty &{} \text{ in } \overline{\varOmega }_0,\\ L(x)&{} \text{ in } \varOmega {\setminus }\overline{\varOmega }_0. \end{array}\right. \end{aligned}$$

(9)

4 Bifurcation results

In this section we apply the bifurcation theory to prove the existence of an unbounded continuum of positive solutions of (1).

4.1 Global bifurcation

In order to write (1) as a fixed point equation, we introduce the operator $\mathcal {L}: L^{\infty }(\varOmega )\rightarrow L^{\infty }(\varOmega )$, defined by

$$\begin{aligned} \mathcal {L} (f):=u, \end{aligned}$$

where u is the unique solution of the linear elliptic equation

$$\begin{aligned} \left\{ \begin{array}{ll} -\varDelta u = f&{} \text{ in } \varOmega ,\\ u = 0 &{} \text{ on } \partial \varOmega .\end{array}\right. \end{aligned}$$

Lemma 1

The operator $\mathcal {L}$ is compact and strictly positive. Moreover, if $f\in L^{\infty }(\varOmega )$, then there exists $C>0$ such that

$$\begin{aligned} \Vert \mathcal {L} f\Vert _{\infty } \le C\Vert f\Vert _{\infty }. \end{aligned}$$

Now, we prove the existence of an unbounded continuum of positive solutions that bifurcates from the trivial solution $u\equiv 0$ at $\lambda =a(0)\lambda _1$.

Theorem 3

There exists an unbounded continuum $\mathcal {C}$ in ${\mathbb {R}}\times C(\overline{\varOmega })$ of positive solutions of (1) emanating from $(\lambda ,u)=(a(0)\lambda _1,0)$.

Proof

Observe that (1) is equivalent to

$$\begin{aligned} u = T_\lambda (u):=\frac{\lambda }{a(0)}\mathcal {L}u + h(\lambda ,u), \end{aligned}$$

where

$$\begin{aligned} h(\lambda ,u) := h_1(\lambda ,u)+h_2(u) \end{aligned}$$

and

$$\begin{aligned} h_1(\lambda ,u)= & {} \lambda \mathcal {L}\left( \left( \displaystyle \frac{1}{a\left( \displaystyle \int _{\varOmega }q(x)(u^+)^p\right) }-\displaystyle \frac{1}{a(0)}\right) u^+\right) , \\ h_2(u)= & {} - \mathcal {L}\left( \displaystyle \frac{b(x)}{a\left( \displaystyle \int _{\varOmega }q(x)(u^+)^p\right) }u^2\right) \end{aligned}$$

and $u^+ = \max \{u,0\}$. Indeed, observe that u is a non-negative and non-trivial solution of (1) if and only if $u=T_\lambda (u)$.

Observe that h is a continuous function and $h(\lambda ,u)=o(\Vert u\Vert _\infty )$ for $\Vert u\Vert _\infty$ near to 0 uniformly on bounded $\lambda$ intervals. Indeed, by Lemma 1, it yields that

$$\begin{aligned} \frac{\Vert h(\lambda ,u)\Vert _\infty }{\Vert u\Vert _{\infty }}\le & {} \left\| h_1(\lambda ,u) \right\| _{\infty }\frac{1}{\Vert u\Vert _{\infty }} +\left\| h_2(u)\right\| _{\infty } \frac{1}{\Vert u\Vert _{\infty }}\\\le & {} C\left( \lambda \left| \displaystyle \frac{1}{a\left( \displaystyle \int _{\varOmega }q(x)(u^+)^p\right) }-\displaystyle \frac{1}{a(0)}\right| + \frac{\Vert b(x)\Vert _{\infty }}{\left| a\left( \displaystyle \int _{\varOmega }q(x)(u^+)^p\right) \right| }\Vert u\Vert _{\infty }\right) \rightarrow 0, \end{aligned}$$

as $\Vert u\Vert _{\infty }\rightarrow 0$. Hence, we can apply Theorem 1.3 in [11] and conclude that there exists a connected component $\mathcal{C}$ of non-negative and non-trivial solutions that emanates from $(a(0)\lambda _1,0)$, which is unbounded. Finally, by the strong maximum principle, any non-negative and non-trivial solution of (1) is positive in $\varOmega$. This concludes the proof. $\square$

Observe that, by elliptic regularity, any solution $u\in L^\infty (\varOmega )$, in fact it belongs to $W^{2,p}(\varOmega )$ for all $p>1$, and then, $u\in C^{1,\alpha }(\overline{\varOmega })$, $\alpha \in (0,1)$.

4.2 Bifurcation direction

In this section we analyze the bifurcation direction from the trivial solution. Recall that the bifurcation direction is called supercritical (resp. subcritical) if for any sequence of positive solutions $(\lambda _n,u_n)$ of (1) such that $\lambda _n\rightarrow a(0)\lambda _1$ and $\Vert u_n\Vert _\infty \rightarrow 0$, then $\lambda _n>a(0)\lambda _1$ (resp. $\lambda _n<a(0)\lambda _1$).

First, we show some important properties of the positive solutions of (1).

Proposition 2

Let $(\lambda ,u)$ be a positive solution of (1).

• 1. If we denote by

  $$\begin{aligned} d=a\left( \int _{\varOmega }q(x)u^p(x)dx\right) , \end{aligned}$$

  then,

  $$\begin{aligned} \frac{u}{d}=\theta _{\left[ \frac{\lambda }{d},b\right] }. \end{aligned}$$

• 2. It holds

  $$\begin{aligned} a\left( \int _{\varOmega }q(x)u^p(x)dx\right) \lambda _1<\lambda <a\left( \int _{\varOmega }q(x)u^p(x)dx\right) \lambda _1^{\varOmega _0}. \end{aligned}$$

Proof

• 1. Dividing the Eq. (1) by $d^2$, we obtain

  $$\begin{aligned} -\varDelta \left( \frac{u}{d}\right) =\frac{\lambda }{d}\left( \frac{u}{d}\right) -b(x)\left( \frac{u}{d}\right) ^2, \end{aligned}$$

  whence the first paragraph follows.

• 2. By the monotonicity of principal eigenvalue, see Proposition 1, we obtain that

  $$\begin{aligned} \begin{array}{rl} \lambda = &{}\lambda _1 \displaystyle \left[ -a\left( \int _{\varOmega }q(x)u^p\right) \varDelta +b(x) u\right] > \lambda _1\left[ -a\left( \int _{\varOmega }q(x)u^p\right) \varDelta \right] = \\ =&{}\displaystyle a\left( \int _{\varOmega }q(x)u^p\right) \lambda _1. \end{array} \end{aligned}$$

  Using again Proposition 1, it becomes apparent that

  $$\begin{aligned} \begin{array}{rl} \lambda = &{}\displaystyle \lambda _1\left[ -a\left( \int _{\varOmega }q(x)u^p\right) \varDelta +b(x) u\right] <\lambda _1^{\varOmega _0} \left[ -a\left( \int _{\varOmega }q(x)u^p\right) \varDelta \right] =\\ =&{}\displaystyle a\left( \int _{\varOmega }q(x)u^p\right) \lambda _1^{\varOmega _0}. \end{array} \end{aligned}$$

$\square$

Based on Proposition 2, the following non-existence result of positive solution of (1) follows.

Proposition 3

Let $(\lambda ,u)$ be a positive solution of (1).

• a) If b(x) verifies $(Hb_1)$, then $\lambda > a_L\lambda _1$.

• b) If b(x) verifies $(Hb_2)$, then $a_L\lambda _1< \lambda < a_M\lambda _1^{\varOmega _0}$.

Remark 1

It is worth emphasizing that in Proposition 3 we assume neither a is bounded nor a strictly positive. Therefore, for example, when a is unbounded the above result reads with $a_M = \infty$.

The following result ascertains the bifurcation direction of the continuum $\mathcal {C}$ that emanates from $(\lambda , u)=(a(0)\lambda _1,0)$.

Theorem 4

Assume that $a\in C^1({\mathbb {R}})$. Denote by $\varphi _1$ the positive eigenfunction associated to $\lambda _1$, normalized by $||\varphi _1||_2=1$. It holds:

• a) If $p>1$, the bifurcation direction is supercritical.

• b) Assume that $p=1$, then:

  $(b_1)$ :

  If

  $$\begin{aligned} a'(0)>-\displaystyle \frac{\displaystyle \int _{\varOmega }b(x)\varphi _1^3}{\lambda _1\displaystyle \int _{\varOmega } q(x)\varphi _1}, \end{aligned}$$

  (10)

  then the bifurcation direction is supercritical.

  $(b_2)$ :

  If

  $$\begin{aligned} a'(0) <-\displaystyle \frac{\displaystyle \int _{\varOmega }b(x)\varphi _1^3}{\lambda _1\displaystyle \int _{\varOmega } q(x)\varphi _1}, \end{aligned}$$

  (11)

  then the bifurcation direction is subcritical.

• c) Assume that $p<1$, then:

  $(c_1)$ :

  If $a'(0) >0$, then the bifurcation direction is supercritical.

  $(c_2)$ :

  If $a'(0) <0$, then the bifurcation direction is subcritical.

Proof

First, we consider the paragraphs a) and b). Thus, assume that $p\ge 1$. We will use the Crandall-Rabinowitz Theorem, see [6]. For that, we define the map:

$$\begin{aligned}&\mathcal{F}:{\mathbb {R}}\times C_0^2(\overline{\varOmega })\rightarrow C^0(\overline{\varOmega }),\\&\quad \mathcal{F}(\lambda , u) = a\left( \int _{\varOmega }q(x)u^p\right) \varDelta u + \lambda u - b(x)u^2. \end{aligned}$$

It is easily seen that $\mathcal{F}\in C^1({\mathbb {R}}\times C^2(\overline{\varOmega });C^0(\overline{\varOmega }))$ and it follows directly that

$$\begin{aligned} \displaystyle \mathcal{F}_{u}(\lambda ,u)(v)= & {} \displaystyle a'\left( \int _{\varOmega }q(x)u^p\right) p\left( \int _{\varOmega }q(x)u^{p-1}v\right) \varDelta u \nonumber \\&+\, \displaystyle a\left( \int _{\varOmega }q(x)u^p\right) \varDelta v+\lambda v - 2b(x)uv, \end{aligned}$$

(12)

$$\begin{aligned} \mathcal{F}_{u\lambda }(\lambda ,u)v= & {} v. \end{aligned}$$

(13)

By definition of $\mathcal{F}$, (12) and (13), we get that

$$\begin{aligned} \mathcal{F}(\lambda , 0) = 0,&\forall \lambda \in {\mathbb {R}},\\ L_0 := \mathcal{F}_u(\lambda , 0)(v)= & {} a(0)\varDelta v + \lambda v,\\ L_1 := \mathcal{F}_{u\lambda }(\lambda , 0)(v)= & {} v. \end{aligned}$$

Then, it follows that

$$\begin{aligned} Ker(L_0) = \{v\in C_0^2(\overline{\varOmega }){\setminus }\{0\}; \ a(0)\varDelta v + \lambda v = 0\} \ne \emptyset . \end{aligned}$$

Since $a(0)\lambda _1$ is a simple eigenvalue of $(-a(0)\varDelta )$, then

$$\begin{aligned} Ker(\mathcal{F}_u(a(0)\lambda _1,0)) = \langle \varphi _1\rangle , \end{aligned}$$

and

$$\begin{aligned} \text{ dim } (Ker (\mathcal{F}_u(a(0)\lambda _1,0))) = \text{ cod } (\text{ Rg } (\mathcal{F}_u(a(0)\lambda _1,0)) = 1. \end{aligned}$$

By the Fredholm Alternative Theorem, it follows that

$$\begin{aligned} \text{ Rg } (\mathcal{F}_u(a(0)\lambda _1,0)) = \left\{ u\in L^2(\varOmega ); \int _{\varOmega }\varphi _1 u = 0\right\} . \end{aligned}$$

On the other hand, it results that

$$\begin{aligned} \mathcal{F}_{u\lambda }(a(0)\lambda _1,0)(\varphi _1) = \varphi _1. \end{aligned}$$

Hence,

$$\begin{aligned} \mathcal{F}_{u\lambda }(a(0)\lambda _1,0)(\varphi _1) \not \in \text{ Rg } (\mathcal{F}_u(a(0)\lambda _1,0)) , \end{aligned}$$

due to $\Vert \varphi _1\Vert _2 =1$.

Then, we can apply the Crandall-Rabinowitz Theorem (see [6]) to conclude that there exist $\epsilon > 0$ and two $C^1$ maps

$$\begin{aligned} \mu : (-\epsilon ,\epsilon ) \rightarrow {\mathbb {R}}, v :(-\epsilon ,\epsilon ) \rightarrow Z, \end{aligned}$$

where Z is the topological complement of $Ker(L_0)$ in $C_0^2(\overline{\varOmega })$, $\mu (0)= 0$, $v(0)=0$ and for each $s\in (-\epsilon ,\epsilon )$

$$\begin{aligned} \left\{ \begin{array}{ll} \lambda (s) = a(0)\lambda _1 + \mu (s), &{}\\ u(s) = s(\varphi _1 + v(s)),&{} \end{array}\right. \end{aligned}$$

(14)

such that $(\lambda (s),u(s))$ are non-trivial solutions of (1), that is

$$\begin{aligned} \mathcal{F}(\lambda (s),u(s))=0 \quad s\in (-\epsilon ,\epsilon ). \end{aligned}$$

Moreover, there exists $\rho >0$ such that if $\mathcal{F}(\lambda ,u)=0$ and $(\lambda ,u)\in B((a(0)\lambda _1,0),\rho )$ then either $u\equiv 0$ or $(\lambda ,u)= (\lambda (s),u(s))$ for some $s\in (-\epsilon ,\epsilon )$; $B((a(0)\lambda _1,0),\rho )$ denotes the ball centered in $(a(0)\lambda _1,0)$ and radius $\rho >0$ in ${\mathbb {R}}\times C_0^2(\overline{\varOmega })$. Observe that u(s) is positive for $s\in (0,\epsilon ).$

On the other hand, the Taylor expansion of the function $a\left( \displaystyle t\right)$ is given by

$$\begin{aligned} a\left( \displaystyle t\right) = a(0) + t a'(0) + \text{ o }(t). \end{aligned}$$

(15)

Replacing (14) and (15) into (1), we obtain that

$$\begin{aligned} -\left( a(0)+s^pa'(0)\int _{\varOmega }q(x)(\varphi _1 +v(s))^p+o(s^p)\right) \varDelta (s\varphi _1 + sv(s)) =\\ (a(0)\lambda _1+\mu (s))(s(\varphi _1+v(s)) - b(x)(s(\varphi _1+v(s)))^2. \end{aligned}$$

Multiplying by $\varphi _1$, integrating in $\varOmega$ and rearranging terms, we have that

$$\begin{aligned}&s^p(a'(0)\lambda _1\int _{\varOmega }q(x)(\varphi _1+ v(s))^p\int _{\varOmega }\varphi _1^2\\&\qquad +\,a'(0)\lambda _1\int _{\varOmega }q(x)(\varphi _1+v(s))^p\int _{\varOmega }\varphi _1v(s)) + o(s^p)\\&\quad = \mu (s)\int _{\varOmega }(\varphi _1+v(s))\varphi _1-s\int _{\varOmega }b(x)(\varphi _1+v(s))^2\varphi _1. \end{aligned}$$

Then, a straightforward manipulation leads to

$$\begin{aligned} \frac{\mu (s)}{s^p} = \frac{a'(0)\lambda _1\displaystyle \int _{\varOmega }q(x)(\varphi _1+v(s))^p+s^{1-p}\int _{\varOmega }b(x)(\varphi _1+v(s))^2\varphi _1}{\displaystyle \int _{\varOmega }(\varphi _1+v(s)) \varphi _1}+o(s). \end{aligned}$$

Now, when $p>1$ we have that $\mu (s)>0$ for for $s > 0$ and small, and then $\lambda (s)>a(0)\lambda _1$ and as consequence the direction is supercritical. This concludes the first paragraph.

When $p=1$, then

$$\begin{aligned} \displaystyle \lim _{s\rightarrow 0}\frac{\mu (s)}{s}=\displaystyle a'(0)\lambda _1\int _{\varOmega }q(x)\varphi _1+\int _{\varOmega }b(x)\varphi _1^3. \end{aligned}$$

Paragraph b) is now an immediate consequence.

Now assume that $p<1$. Suppose that there exists a sequence of positive solutions $\{(\lambda _n,u_n)\}$ of (1) such that

$$\begin{aligned} \lambda _n \rightarrow a(0)\lambda _1 \quad \text{ and } \quad \Vert u_n\Vert _{\infty }\rightarrow 0. \end{aligned}$$

Assume that $a'(0)>0$, it follows that

$$\begin{aligned} a(s)>a(0)\quad 0<s<\epsilon . \end{aligned}$$

(16)

Since $\epsilon>\displaystyle \int _{\varOmega }q(x)u_n^p >0$, for n large, and from (16), we have that

$$\begin{aligned} a\left( \int _{\varOmega }q(x)u_n^p\right) >a(0). \end{aligned}$$

Hence, by Proposition 1 we get

$$\begin{aligned} \begin{array}{rl} \lambda _n=&{} \lambda _1\displaystyle \left[ - a\left( \int _{\varOmega }q(x)u_n^p\right) \varDelta +b(x)u_n\right]>\lambda _1\left[ - a\left( \int _{\varOmega }q(x)u_n^p\right) \varDelta \right] \\ &{}> \displaystyle \lambda _1\left[ - a(0)\varDelta \right] =a(0)\lambda _1. \end{array} \end{aligned}$$

Assume now that $a'(0)<0$. We argue by contradiction: assume that there exists a sequence $(\lambda _n,u_n)$ of positive solutions of (1) such that $\lambda _n\rightarrow a(0)\lambda _1$, $\Vert u_n\Vert _\infty \rightarrow 0$ with $\lambda _n>a(0)\lambda _1$. Let $C>0$, with C large enough, such that

$$\begin{aligned} a'(0)<-\displaystyle \frac{\displaystyle \int _{\varOmega }b(x)\varphi _1^3}{C\lambda _1\displaystyle \int _{\varOmega }q(x) \varphi _1}. \end{aligned}$$

(17)

We fix this value of C in the rest of the proof. Consider the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -a\left( C\displaystyle \int _{\varOmega }q(x)u\right) \varDelta u = u(\lambda -b(x)u) &{} \text{ in } \varOmega ,\\ u=0&{} \text{ on } \partial \varOmega . \end{array}\right. \end{aligned}$$

(18)

Observe that $p=1$ in (18), and then by (17), we can use paragraph $(b_2)$ and conclude that the bifurcation direction for (18) is subcritical. However, we will prove that there exists a sequence of positive solutions $(\mu _n,w_n)$ of (18) with $\mu _n>\lambda _1 a(0)$, $\mu _n\rightarrow \lambda _1 a(0)$ and $\Vert w_n\Vert _\infty \rightarrow 0$.

Take $\mu _n=\lambda _n$. From $a'(0)<0$, it follows that

$$\begin{aligned} a(0)>a(s),\quad 0<s<\epsilon _0. \end{aligned}$$

(19)

Since $p<1$, we obtain that there exists $n_0 \in {\mathbb {R}}$ such that

$$\begin{aligned} q(x)u_n^p \ge Cq(x)u_n \quad \forall n\ge n_0. \end{aligned}$$

Since $a'(0)<0$, then a is decreasing near 0. Then, as $\Vert u_n\Vert _{\infty }\rightarrow 0$, we have that

$$\begin{aligned} a\left( \int _{\varOmega }q(x) u_n^p\right) <a\left( C\int _{\varOmega }q(x)u_n\right) . \end{aligned}$$

(20)

We are going to apply the sub-supersolution method to (18). We take

$$\begin{aligned} \overline{u}=u_n,\quad \underline{u}=\varepsilon \varphi _1, \end{aligned}$$

where $\varepsilon >0$ will be chosen later. First, observe that since $\lambda _n\ge \lambda _0>0$ for some positive $\lambda _0$,

$$\begin{aligned} f(x,s)=s(\lambda _n-b(x)s)>0\quad s\in [\underline{u},\overline{u}]. \end{aligned}$$

Then, since a is decreasing in $[\underline{u},\overline{u}]$ and $f(x,s)>0$, Definition 1 is equivalent to

$$\begin{aligned} -\,a\left( \int _\varOmega C q(x)\underline{u}\right) \varDelta \underline{u}-f(x,\underline{u})\le 0\le -a\left( \int _\varOmega C q(x)\overline{u}\right) \varDelta \overline{u}-f(x,\overline{u}). \end{aligned}$$

We start showing that $\overline{u}=u_n$ is a supersolution of (18). Indeed, by (20)

$$\begin{aligned} -\,a\left( C\int _{\varOmega }q(x)u_n\right) \varDelta (u_n)>-a\left( \int _{\varOmega }q(x)u_n^p\right) \varDelta ( u_n) = u_n(\lambda _n-b(x)u_n). \end{aligned}$$

Now, we are going to see that $\underline{u}=\epsilon \varphi _1$ is a subsolution of (18) provided of $\epsilon$ is small enough. Indeed, $\epsilon \varphi _1$ is subsolution if

$$\begin{aligned} b(x)\epsilon \varphi _1 +a\left( \epsilon \int _{\varOmega }C q(x)\varphi _1\right) \lambda _1\le & {} \lambda _n. \end{aligned}$$

By (19), for $\epsilon$ small, we have that

$$\begin{aligned} b(x)\epsilon \varphi _1 + a\left( \epsilon \int _{\varOmega } Cq(x)\varphi _1\right) \lambda _1\le & {} b_M \epsilon +a(0)\lambda _1 <\lambda _n\\ \Leftrightarrow \epsilon\le & {} \frac{\lambda _n -a(0)\lambda _1}{b_M}, \end{aligned}$$

where $b_M = \displaystyle \max _{x\in \overline{\varOmega }}b(x)$. Therefore, there exists positive solution $w_n$ of (18) for $\lambda _n>a(0)\lambda _1$ such that

$$\begin{aligned} \epsilon \varphi _1\le w_n\le u_n\quad \text{ in } \varOmega . \end{aligned}$$

This is a contradiction. $\square$

5 Logistic equation with nonlocal diffusion

5.1 b verifies $(Hb_1)$

First, we focus on the case where b verifies $(Hb_1)$.

Theorem 5

Assume that b satisfies $(Hb_1)$. Then, there exists a positive solution of (1) if $\lambda > a(0)\lambda _1$. Moreover, if a is increasing and b is constant, then there exists a unique positive solution of (1).

Proof

First, it should be remembered that $b(x)\ge b_0$ for all $x\in \overline{\varOmega }$. We will prove that

$$\begin{aligned} u<\displaystyle \frac{\lambda }{b_0}\quad \text{ in } \varOmega . \end{aligned}$$

(21)

Indeed, we define

$$\begin{aligned} \varOmega _1 =\left\{ x\in \varOmega ; u(x)>\displaystyle \frac{\lambda }{b_0}\right\} . \end{aligned}$$

Thus,

$$\begin{aligned} -\,a\left( \int _{\varOmega }q(x)u^p\right) \varDelta u = \lambda u - b(x)u^2 \le 0\quad \text{ in } \varOmega _1,\quad u=\frac{\lambda }{b_0}\quad \text{ on } \partial \varOmega _1, \end{aligned}$$

implying that $u \le \displaystyle \frac{\lambda }{b_0}$ in $\varOmega _1$. We arrive at a contradiction. Hence $\varOmega _1 = \emptyset$ and we conclude $u\le \lambda /b_0$ in $\varOmega$. The strong maximum principle proves (21).

Hence, by Theorem 3 we know the existence of an unbounded continuum of positive solutions $\mathcal {C}$ that bifurcates from the trivial solution at $\lambda =a(0)\lambda _1$. Moreover, (1) does not possess positive solutions for $\lambda \le a_L\lambda _1$. As consequence of (21), it follows the existence of positive solution for $\lambda >a(0)\lambda _1$.

Assume now that a is increasing and b is constant. Let u and v be positive solutions of (1), with $u \ne v$. We distinguish two cases:

• 1. Assume that $\displaystyle \int _{\varOmega }q(x)u^p = \int _{\varOmega }q(x)v^p$. Then, u and v are positive solutions of

  $$\begin{aligned} -\varDelta v = \displaystyle \frac{\lambda v-bv^2}{ \displaystyle a\left( \int _{\varOmega }q(x)u^p\right) }\quad \text{ in } \varOmega ,\quad v=0\quad \text{ on } \partial \varOmega . \end{aligned}$$

  (22)

  Therefore, since (22) has a unique positive solution, it follows that $u=v$ in $\varOmega$.

• 2. Assume now that $\displaystyle \int _{\varOmega }q(x)u^p < \int _{\varOmega }q(x)v^p$. Observe that since b is constant, it follows by (21) that

  $$\begin{aligned} \lambda v-bv^2>0 \text{ in } \varOmega . \end{aligned}$$

  (23)

  Moreover, since a is increasing and by (23), we obtain that

  $$\begin{aligned} -\,\varDelta v = \displaystyle \frac{\lambda v-bv^2}{ \displaystyle a\left( \int _{\varOmega }q(x)v^p\right) }< \displaystyle \frac{\lambda v-bv^2}{ \displaystyle a\left( \int _{\varOmega }q(x)u^p\right) }. \end{aligned}$$

  Therefore, v is subsolution of (22), which implies that $u>v$, a contradiction.

$\square$

5.2 b verifies $(Hb_2)$

Now, we deal with the case when b verifies $(Hb_2)$. For that, we define

$$\begin{aligned} I: = \displaystyle \int _{\varOmega }q(x)(\mathcal{M}(x))^pdx, \end{aligned}$$

(24)

where $\mathcal{M}(x)$ is defined in (9).

The following result shows (when $I=\infty$) that the structure of the set of positive solutions depends strongly on the behavior of the function a at $\infty$.

Theorem 6

Let $I=\infty$ and assume that b verifies $(Hb_2)$.

• 1. If $0<a(\infty ) < \infty$, then there exists at least one positive solution of (1) for

  $$\begin{aligned} \lambda \in (\min \{a(0)\lambda _1,a(\infty )\lambda _1^{\varOmega _0}\},\max \{a(0)\lambda _1,a(\infty )\lambda _1^{\varOmega _0}\}). \end{aligned}$$

• 2. If $a(\infty ) = \infty$, then there exists at least one positive solution of (1) for $\lambda > a(0)\lambda _1$.

• 3. If $a(\infty ) = 0$, then there exists at least one positive solution of (1) for $\lambda \in (0,a(0)\lambda _1)$.

  Moreover, in all the cases, there exist sequences $(\lambda _n,u_{\lambda _n}),(\lambda _n',u_{\lambda _n'})\in \mathcal {C}$, then

  $$\begin{aligned} \displaystyle \lim _{\lambda _n\rightarrow a(0)\lambda _1}\Vert u_{\lambda _n}\Vert _{\infty } = 0,\quad \displaystyle \lim _{\lambda _n'\rightarrow a(\infty )\lambda _1^{\varOmega _0}}\Vert u_{\lambda _n'}\Vert _{\infty } = \infty . \end{aligned}$$

  (25)

Proof

From Theorem 3 there exists an unbounded continuum $\mathcal {C}$ in ${\mathbb {R}}\times L^{\infty }(\varOmega )$ of positive solutions of (1) that emanates at $\lambda = a(0)\lambda _1$ from the trivial solution. By Proposition 3, (1) does not admit positive solutions for $\lambda \le a_L\lambda _1$.

• 1. Assume that $0<a(\infty )<\infty$, then $0<a_L<a_M<\infty$. Again, by Proposition 3, if there exists positive solution of (1) then $\lambda <a_M\lambda _1^{\varOmega _0}$, and hence, since $\mathcal{C}$ is unbounded, there exists a sequence of positive solutions $(\lambda _n,u_n)\in \mathcal {C}$, with $\lambda _n\rightarrow \lambda ^*\in (0,\infty )$ and $\Vert u_n||_\infty \rightarrow \infty$. We define

  $$\begin{aligned} d_n =\displaystyle a\left( \int _{\varOmega }q(x)u_n^p\right) . \end{aligned}$$

  Then, by Proposition 2

  $$\begin{aligned} \frac{u_n}{d_n}=\theta _{\left[ \frac{\lambda _n}{d_n},b\right] }. \end{aligned}$$

  (26)

  Therefore,

  $$\begin{aligned} d_n = a\left( d_n^p \int _{\varOmega }q(x)\theta _{\left[ \frac{\lambda _n}{d_n},b\right] }^p\right) . \end{aligned}$$

  (27)

  On the other hand, since $u_n$ is a positive solution of (1), then again by Proposition 2 we get

  $$\begin{aligned} \lambda _1<\frac{\lambda _n}{d_n}<\lambda _1^{\varOmega _0}. \end{aligned}$$

  Since $\lambda _n\rightarrow \lambda ^*\in (0,\infty )$, it follows that $d_n$ is bounded above and below. Then, since $\Vert u_n\Vert _\infty \rightarrow \infty$ and using (26), we conclude that $\Vert \theta _{\left[ \frac{\lambda _n}{d_n},b\right] }\Vert _\infty \rightarrow +\infty$, and hence

  $$\begin{aligned} \frac{\lambda _n}{d_n}\rightarrow \lambda _1^{\varOmega _0}. \end{aligned}$$

  We conclude that

  $$\begin{aligned} d_n\rightarrow \frac{\lambda ^*}{\lambda _1^{\varOmega _0}} \quad \text{ and }\quad \theta _{\left[ \frac{\lambda _n}{d_n},b\right] }\rightarrow \mathcal{M}. \end{aligned}$$

  Taking limit in (27), we obtain

  $$\begin{aligned} \frac{\lambda ^*}{\lambda _1^{\varOmega _0}}= a\left( \left( \frac{\lambda ^*}{\lambda _1^{\varOmega _0}}\right) ^pI\right) = a(\infty ) \Rightarrow \lambda ^* = a(\infty )\lambda _1^{\varOmega _0}. \end{aligned}$$

  (28)

• 2. Assume that $a(\infty )=\infty$. In such case, $a_L>0$ and then if $(\lambda ,u)$ is a positive solution of (1) we get by Proposition 3 that $0<a_L\lambda _1<\lambda$. The proof will proceed by contradiction. Suppose that there exists a sequence of positive solutions $(\lambda _n,u_n)\in \mathcal {C}$ of (12) such that $\lambda _n\rightarrow \lambda ^*<\infty$ and $\Vert u_n\Vert _\infty \rightarrow \infty$. With a similar argument to the used in the first paragraph, according to (28), we obtain that $\lambda ^*=a(\infty )\lambda _1^{\varOmega _0}<\infty$. We arrive at a contradiction.

• 3. Assume that $a(\infty )=0$, and hence, $a_M<\infty$ and then by Proposition 3 if $(\lambda ,u)$ is a positive solution of (1) we get that $\lambda <a_M\lambda _1^{\varOmega _0}$. Assume now by contradiction that there exists a sequence of positive solutions $(\lambda _n,u_n)\in \mathcal {C}$ of (12) such that $\lambda _n\rightarrow \lambda ^*$ and $\Vert u_n\Vert _\infty \rightarrow \infty$ with $\lambda ^*>0$. Again, by (28), we can infer that $\lambda ^*=a(\infty )\lambda _1^{\varOmega _0}=0$. A contradiction.

$\square$

Now, we handle the case $I<\infty$. In this case, the structure of the set of positive solutions of (1) depends on the real equation

$$\begin{aligned} g(s)=I, \end{aligned}$$

(29)

where the function $g:[0,\infty )\rightarrow [0,\infty )$ is defined by

$$\begin{aligned} g(s) := \frac{s}{(a(s))^p}. \end{aligned}$$

(30)

The first result deals with the case that (29) does not have solutions. First, we need to prove the following result

Lemma 2

Assume that $I<\infty$ and that there exists a sequence $(\lambda _n,u_n)$ of positive solutions of (1) such that $\Vert u_n\Vert _\infty \rightarrow \infty$ and $\lambda _n\rightarrow \lambda _*<\infty$. Then, $\lambda _*>0$ and there exists $s_*>0$ such that $g(s_*)=I$, in fact,

$$\begin{aligned} s_*=\left( \frac{\lambda _*}{\lambda _1^{\varOmega _0}}\right) ^p I. \end{aligned}$$

Proof

With the same notation as in Proposition 2, we have that

$$\begin{aligned} \lambda _1<\frac{\lambda _n}{d_n}<\lambda _1^{\varOmega _0}, \end{aligned}$$

where

$$\begin{aligned} d_n= a\left( \int _\varOmega q(x)u_n(x)^pdx\right) . \end{aligned}$$

Then,

$$\begin{aligned} u_n=d_n\theta _{\left[ \frac{\lambda _n}{d_n},b\right] }, \end{aligned}$$

(31)

and hence,

$$\begin{aligned} d_n=a\left( d_n^p\int _{\varOmega }q(x)\theta _{\left[ \frac{\lambda _n}{d_n},b\right] }^p\right) . \end{aligned}$$

(32)

Since $\lambda _*<\infty$, $d_n$ is bounded above, and then, by (31), if $\Vert u_n\Vert _\infty \rightarrow \infty$ we have that

$$\begin{aligned} \frac{\lambda _n}{d_n}\rightarrow \lambda _1^{\varOmega _0}. \end{aligned}$$

Passing to the limit in (32), we obtain:

$$\begin{aligned} \frac{\lambda _*}{\lambda _1^{\varOmega _0}}=a\left( \left( \frac{\lambda _*}{\lambda _1^{\varOmega _0}}\right) ^pI\right) . \end{aligned}$$

Then,

$$\begin{aligned} a(s_*)=\left( \frac{s_*}{I}\right) ^{1/p}, \end{aligned}$$

and consequently, $g(s_*)=I$. This concludes the result. $\square$

Theorem 7

Let $I<\infty$. If there does not exist solution of (29), then there exists at least one positive solution of (1) for $\lambda > a(0)\lambda _1$.

Proof

We know that there exists an unbounded continuum $\mathcal{C}$ of positive solutions emanating from $(\lambda ,u)=(a(0)\lambda _1,0)$. Assume by contradiction that there exists a sequence $(\lambda _n,u_n)\in \mathcal {C}$ of positive solutions such that $\lambda _n\rightarrow \lambda _*<\infty$ and $\Vert u_n\Vert _\infty \rightarrow \infty$. Then, by Lemma 2 there exists a positive solution of (29), a contradiction. $\square$

The next result will be the cornerstone in the rest of the work.

Proposition 4

Assume that $(\lambda ,u)$ is a positive solution of (1). Then,

$$\begin{aligned} g\left( \int _\varOmega q(x)u^p(x)dx\right) <I. \end{aligned}$$

Proof

Observe that

$$\begin{aligned} g\left( \int _{\varOmega }q(x)u^p(x)dx\right)= & {} \displaystyle \frac{\displaystyle \int _{\varOmega }q(x)u^p(x)dx}{\left( a\left( \displaystyle \int _{\varOmega }q(x)u^p(x)dx\right) \right) ^p} \\= & {} \int _\varOmega q(x)\left( \frac{u(x)}{a\left( \displaystyle \int _{\varOmega }q(x)u^p(x)dx\right) }\right) ^pdx\\= & {} \int _\varOmega q(x)\theta ^p_{\left[ \frac{\lambda }{d},b\right] }<\int _\varOmega q(x)\mathcal{M}^p(x)dx=I, \end{aligned}$$

where

$$\begin{aligned} d=a\left( \displaystyle \int _{\varOmega }q(x)u^p(x)dx\right) . \end{aligned}$$

$\square$

In the following result, we show a non-existence result.

Proposition 5

If there exists $\overline{s}>0$ such that

$$\begin{aligned} g(s)>I\quad \text{ for } s>\overline{s}, \end{aligned}$$

(33)

then, there exists $\overline{\lambda }>0$ such that (1) does not possess positive solution for $\lambda >\overline{\lambda }$.

Proof

Assume by contradiction that there exists a sequence of positive solutions $(\lambda _n,u_n)$ of (1) for $\lambda _n\rightarrow \infty$. We know by Proposition 2 that

$$\begin{aligned} \lambda _1<\frac{\lambda _n}{a\left( \displaystyle \int _{\varOmega }q(x)u_n^p(x)dx\right) }<\lambda _1^{\varOmega _0}. \end{aligned}$$

Hence, $a\left( \displaystyle \int _{\varOmega }q(x)u_n^p(x)dx\right) \rightarrow \infty$, which implies that $\displaystyle \int _{\varOmega }q(x)u_n^p (x)dx\rightarrow \infty$. Then by (33),

$$\begin{aligned} g\left( \displaystyle \int _{\varOmega }q(x)u_n^p(x)dx\right) >I, \end{aligned}$$

this is a contradiction with Proposition 4. $\square$

The following result shows that the $Proj_{{\mathbb {R}}}(\mathcal{C})$ is bounded when there exists solutions of (29). Here, given $(\lambda ,u)\in \mathcal{C}$ we denote $Proj_{{\mathbb {R}}}(\lambda ,u)=\lambda$.

Proposition 6

If there exists $s^*>0$ such that $g(s^*) = I$, then $Proj_{{\mathbb {R}}}(\mathcal{C})\subset (0,\varLambda ^*)$ for some $\varLambda ^*<\infty .$ In fact, if we denote $s_1>0$ the least solution of (29) and

$$\begin{aligned} \varLambda _1=\lambda _1^{\varOmega _0}\left( \frac{s_1}{I}\right) ^{1/p}, \end{aligned}$$

then, there exists a sequence $(\lambda _n,u_n)\in \mathcal{C}$ such that $\Vert u_n\Vert _\infty \rightarrow \infty$ and $\lambda _n\rightarrow \varLambda _1$.

Proof

We define the continuous map

$$\begin{aligned} \mathcal {H}: {\mathbb {R}}\times L^{\infty }(\varOmega )\rightarrow {\mathbb {R}},\quad \mathcal {H}(\lambda ,u)=\int _{\varOmega }q(x)u^p(x)dx. \end{aligned}$$

Hence, since $\mathcal{C}$ is connected, we obtain that $\mathcal{H}(\mathcal{C})$ is a connected set in ${\mathbb {R}}$.

Assume by contradiction that there exists a sequence $(\lambda _n,u_{\lambda _n})\in \mathcal{C}$ such that $\lambda _n\rightarrow +\infty$. We know by Proposition 2 that

$$\begin{aligned} \lambda _n<\lambda _1^{\varOmega _0}a\left( \int _\varOmega q(x)u_{\lambda _n}(x)^p dx\right) , \end{aligned}$$

then $a\left( \int _\varOmega q(x)u_{\lambda _n}(x)^pdx\right) \rightarrow \infty$. We conclude that $\int _\varOmega q(x)u_{\lambda _n}^p(x)dx\rightarrow +\infty$; that is,

$$\begin{aligned} \mathcal{H}(\lambda _n,u_{\lambda _n})\rightarrow +\infty . \end{aligned}$$

On the other hand, $\mathcal{H}(a(0)\lambda _1,0)=0$, and hence, we can conclude that

$$\begin{aligned}{}[0,+\infty )\subset \mathcal{H}(\mathcal{C}). \end{aligned}$$

Thus, there exists $\lambda ^*$ such that

$$\begin{aligned} s^*=\int _{\varOmega }q(x)u_{\lambda ^*}^p(x)dx. \end{aligned}$$

Then, by Proposition 4, we get that

$$\begin{aligned} g(s^*)=g\left( \int _{\varOmega }q(x)u_{\lambda ^*}^p(x)dx\right) <I, \end{aligned}$$

a contradiction.

Since $\mathcal{H}(\mathcal{C})$ is a connected set in ${\mathbb {R}}$ and observe that $\mathcal{H}(a(0)\lambda _1,0)=0$, it follows by Proposition 4 that

$$\begin{aligned} \mathcal{H}(\mathcal{C})\subset [0,s_1). \end{aligned}$$

(34)

On the other hand, we know that there exists a sequence of positive solutions $(\lambda _n,u_{\lambda _n})\in \mathcal{C}$ of (12) such that $\lambda _n\rightarrow \lambda _*<\infty$ and $\Vert u_{\lambda _n}\Vert _\infty \rightarrow \infty$. Moreover, observe that by the proof of Lemma 4 we get that

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathcal{H}(\lambda _n,u_{\lambda _n})=\lim _{n\rightarrow \infty }\int _{\varOmega }q(x)u_{\lambda _n}^p(x)dx=\lim _{n\rightarrow \infty }d_n^p\int _\varOmega q(x)\theta ^p_{\left[ \frac{\lambda _n}{d_n},b\right] }= \left( \frac{\lambda _*}{\lambda _1^{\varOmega _0}}\right) ^pI, \end{aligned}$$

and then by (34)

$$\begin{aligned} \left( \frac{\lambda _*}{\lambda _1^{\varOmega _0}}\right) ^pI\le s_1=\left( \frac{\varLambda _1}{\lambda _1^{\varOmega _0}}\right) ^pI. \end{aligned}$$

This implies that $\lambda _*=\varLambda _1$. $\square$

Now, we study the case where $I<\infty$ and there exist positive solutions of (29). In the first result, we show the existence of positive solutions between $a(0)\lambda _1$ and $\varLambda _1$.

Theorem 8

Let $I<\infty$ and assume that there exists $s_0>0$ solution of (29). Then, there exists at least a positive solution of (1) for

$$\begin{aligned} \lambda \in (\min \{a(0)\lambda _1,\varLambda _1\},\max \{a(0)\lambda _1,\varLambda _1\}). \end{aligned}$$

Moreover, there exist sequences $(\lambda _n,u_{\lambda _n}),(\lambda _n',u_{\lambda _n'})\in \mathcal {C}$, then

$$\begin{aligned} \displaystyle \lim _{\lambda _n\rightarrow a(0)\lambda _1}\Vert u_{\lambda _n}\Vert _{\infty } = 0,\quad \displaystyle \lim _{\lambda _n'\rightarrow \varLambda _1}\Vert u_{\lambda _n'}\Vert _{\infty } = \infty . \end{aligned}$$

Proof

We know the existence of an unbounded continuum $\mathcal{C}$ of positive solutions emanating at $\lambda =a(0)\lambda _1$ from the trivial solution. Proposition 6 concludes the result. $\square$

Now, we need to introduce and study the following map.

Proposition 7

Fix $\lambda >0$ and define the function $h_\lambda :[0,\infty )\rightarrow [0,\infty )$ by

$$\begin{aligned} h_\lambda (s) := \int _{\varOmega }q(x)\left( \theta _{\left[ \frac{\lambda }{a(s)},b\right] }\right) ^p. \end{aligned}$$

(35)

The function $h_\lambda$ is well defined in the set

$$\begin{aligned} \varLambda _\lambda =\left\{ s\in [0,\infty ): \frac{\lambda }{\lambda _1^{\varOmega _0}}<a(s)<\frac{\lambda }{\lambda _1}\right\} . \end{aligned}$$

Moreover, $h_\lambda$ is continuous in $\varLambda _\lambda$ and

• 1. Let $s^n,s_n\in \varLambda _\lambda$ such that $s^n\rightarrow s^*$ and $s_n\rightarrow s_*>0$ such that $a(s^*)=\displaystyle \frac{\lambda }{\lambda _1^{\varOmega _0}}$ and $a(s_*)=\displaystyle \frac{\lambda }{\lambda _1}$. Then,

  • (a)
    $$\begin{aligned} \lim _{s^n\rightarrow s^*}h_\lambda (s^n) = I. \end{aligned}$$

    (36)
  • (b)
    $$\begin{aligned} \lim _{s_n\rightarrow s_*}h_\lambda (s_n) = 0. \end{aligned}$$

    (37)

• 2. It holds that

  $$\begin{aligned} h_\lambda (s)\le I,\quad \forall s\in [0,\infty ). \end{aligned}$$

Proof

$h_\lambda$ is well defined and is continuous in $\varLambda _\lambda$ by Theorem 2.

• 1.

  • (a) Again, by Theorem 2, it follows that

    $$\begin{aligned} \lim _{s^n\rightarrow s^*}h_\lambda (s^n)=\lim _{s^n\rightarrow s^*}\int _{\varOmega }q(x)\left( \theta _{\left[ \frac{\lambda }{a(s^n)},b\right] }\right) ^p = \int _{\varOmega }q(x)(\mathcal{M}(x))^p = I. \end{aligned}$$

  • (b) Analogously,

    $$\begin{aligned} \lim _{s_n\rightarrow s_*}h_\lambda (s_n)=\lim _{s_n\rightarrow s_*}\int _{\varOmega }q(x)\left( \theta _{\left[ \frac{\lambda }{a(s_n)},b\right] }\right) ^p = 0. \end{aligned}$$

• 2. By definition and Theorem 2, we get

  $$\begin{aligned} h_\lambda (s) = \int _{\varOmega }q(x)\left( \theta _{\left[ \frac{\lambda }{a(s)},b\right] }\right) ^p \le \int _{\varOmega }q(x)(\mathcal{M}(x))^p = I. \end{aligned}$$

$\square$

In the following result we prove that Eq. (1) is in fact equivalent to find a fixed point to the real equation $h_\lambda (s)=g(s)$. Specifically,

Proposition 8

If there exists $s^*>0$ such that $h_\lambda (s^*)=g(s^*)$, then there exists at least one positive solution of (1) for $\lambda \in (a(s^*)\lambda _1,a(s^*)\lambda _1^{\varOmega _0})$.

Conversely, if u is a positive solution of (1), then there exists $s^*>0$ such that $h_\lambda (s^*)=g(s^*)$, in fact,

$$\begin{aligned} s^*=\int _\varOmega q(x)u^p(x)dx. \end{aligned}$$

Proof

By (30) and (35), we obtain

$$\begin{aligned} \frac{s^*}{(a(s^*))^p}=g(s^*)=h_\lambda (s^*)=\int _{\varOmega }q(x)\left( \theta _{\left[ \frac{\lambda }{a(s^*)},b\right] }\right) ^p. \end{aligned}$$

Hence,

$$\begin{aligned} s^* = (a(s^*))^p\int _{\varOmega }q(x)\left( \theta _{\left[ \frac{\lambda }{a(s^*)},b\right] }\right) ^p = \int _{\varOmega }q(x)\left( a(s^*)\theta _{\left[ \frac{\lambda }{a(s^*)},b\right] }\right) ^p. \end{aligned}$$

(38)

Observe that

$$\begin{aligned} u = a(s^*)\theta _{\left[ \frac{\lambda }{a(s^*)},b\right] } \end{aligned}$$

is solution of (1). Indeed, using (38)

$$\begin{aligned} -\,a\left( \int _{\varOmega }q(x)u^p\right) \varDelta u= & {} -\,a\left( \int _{\varOmega }q(x)\left( a(s^*)\theta _{\left[ \frac{\lambda }{a(s^*)},b\right] }\right) ^p\right) \varDelta (a(s^*)\theta _{\left[ \frac{\lambda }{a(s^*)},b\right] })\\= & {} (a(s^*))^2\left( \frac{\lambda }{a(s^*)}\theta _{\left[ \frac{\lambda }{a(s^*)},b\right] } - b(x) (\theta _{\left[ \frac{\lambda }{a(s^*)},b\right] })^2\right) \\= & {} \lambda (a(s^*)\theta _{\left[ \frac{\lambda }{a(s^*)},b\right] })-b(x)\left( a(s^*)\theta _{\left[ \frac{\lambda }{a(s^*)},b\right] }\right) ^2\\= & {} \lambda u-b(x)u^2. \end{aligned}$$

Conversely, assume that u is a positive solution of (1). Then, denoting by $d= a(\int _\varOmega q(x)u^p(x)dx)$, we get

$$\begin{aligned} \frac{u}{d}=\theta _{[\lambda /d,b]}, \end{aligned}$$

and then,

$$\begin{aligned} \int _\varOmega q(x)u^p(x)dx=d^p\int _\varOmega q(x)\theta ^p_{[\lambda /d,b]}. \end{aligned}$$

Hence,

$$\begin{aligned} \displaystyle \frac{\displaystyle \int _\varOmega q(x)u^p(x)dx}{\displaystyle \left( a\left( \displaystyle \int _\varOmega q(x)u^p(x)dx\right) \right) ^p}=\int _\varOmega q(x)\theta ^p_{\left[ \mu ,b\right] }, \end{aligned}$$

where

$$\begin{aligned} \mu =\frac{\lambda }{a\left( \displaystyle \int _\varOmega q(x)u^p(x)dx\right) }. \end{aligned}$$

This concludes the result. $\square$

The following result shows that if $g(s)<I$ between two roots of (29) $\alpha _1<\alpha _2$, that is $g(\alpha _1)=g(\alpha _2)$, then we can find positive solution in the interval $\lambda \in (\varLambda _1,\varLambda _2)$.

Proposition 9

If $g(s)<I$ for all $s\in (\alpha _1,\alpha _2)$ with

$$\begin{aligned} g(\alpha _1)=g(\alpha _2)= I, \end{aligned}$$

then, there exists at least a positive solution of (1) for $\lambda \in (\varLambda _1,\varLambda _2)$, where $\varLambda _i = \displaystyle \left( \frac{\alpha _i}{I}\right) ^{\frac{1}{p}}\lambda _1^{\varOmega _0}$, $i\in \{1,2\}$.

Proof

Since $g(\alpha _1)=g(\alpha _2)=I$, we know that

$$\begin{aligned} a(\alpha _1)=\left( \frac{\alpha _1}{I}\right) ^{\frac{1}{p}}<\left( \frac{\alpha _2}{I}\right) ^{\frac{1}{p}} = a(\alpha _2). \end{aligned}$$

(39)

Take $\lambda \in (\varLambda _1,\varLambda _2)$, that is

$$\begin{aligned} \frac{\lambda }{\lambda _1^{\varOmega _0}}\in \left( \left( \frac{\alpha _1}{I}\right) ^{\frac{1}{p}},\left( \frac{\alpha _2}{I}\right) ^{\frac{1}{p}}\right) =(a(\alpha _1),a(\alpha _2)). \end{aligned}$$

By continuity of a, there exists $s_*\in (\alpha _1,\alpha _2)$ such that $a(s_*)=\displaystyle \frac{\lambda }{\lambda _1^{\varOmega _0}}$.

Observe that $g(s)<I$ for all $s\in (\alpha _1,\alpha _2)$ is equivalent to

$$\begin{aligned} \left( \frac{s}{I}\right) ^{1/p}<a(s)\quad \text{ for } \text{ all } s\in (\alpha _1,\alpha _2). \end{aligned}$$

We define the sets

$$\begin{aligned} \overline{S}:=\left\{ \overline{s}_j\in [\alpha _1,\alpha _2];\;a(\overline{s}_j)=\displaystyle \frac{\lambda }{\lambda _1^{\varOmega _0}}\right\} \quad \text{ and }\quad \underline{S}:=\left\{ \underline{s}_j;\;a(\underline{s}_j)=\displaystyle \frac{\lambda }{\lambda _1}\right\} . \end{aligned}$$

Since $a'(s)=0$ in, at most, a discrete set, we can order $\underline{S}$ and $\overline{S}$ such that

$$\begin{aligned} \overline{s_1}<\overline{s_2}<\dots \quad \text{ and }\quad \underline{s_1}<\underline{s_2}<\dots \end{aligned}$$

We know that $s_*\in \overline{S}$, and so, $\overline{S}\ne \emptyset$.

Now, we separate the proof in two different cases:

• 1. We suppose $\underline{S}\cap [\alpha _1,\alpha _2]=\emptyset$. Then, we take $\overline{s}_{j_0}=\displaystyle \max _{j\in {\mathbb {N}}} \overline{s}_j\in [\alpha _1,\alpha _2]$. By definition,

  $$\begin{aligned} a(\overline{s}_{j_0})=\displaystyle \frac{\lambda }{\lambda _1^{\varOmega _0}}. \end{aligned}$$

  Since $\underline{S}\cap [\alpha _1,\alpha _2]=\emptyset$, $a(s)<\frac{\lambda }{\lambda _1}$ for all $s\in [\alpha _1,\alpha _2]$ and hence there exists $\delta >0$ such that

  $$\begin{aligned} \frac{\lambda }{\lambda _1^{\varOmega _0}}<a(s)<\frac{\lambda }{\lambda _1}, \quad s\in (\overline{s}_{j_0},\alpha _2+\delta ). \end{aligned}$$

  (40)

  By Proposition 7, we obtain

  $$\begin{aligned} \lim _{s\downarrow \overline{s}_{j_0} }h_\lambda (s) = I. \end{aligned}$$

  By Proposition 7, $h_\lambda (\alpha _2+\delta )\le I$ and $h_\lambda (\alpha _2)<I$. Then,

  $$\begin{aligned} h_\lambda (\overline{s}_{j_0})-g(\overline{s}_{j_0})>0\quad \text{ and }\quad h_\lambda (\alpha _2)-g(\alpha _2)<0, \end{aligned}$$

  and hence there exists $s^*\in (\overline{s}_{j_0},\alpha _2)\subset (\alpha _1,\alpha _2)$ such that

  $$\begin{aligned} g(s^*)=h_\lambda (s^*). \end{aligned}$$

  Moreover, by (40) observe that $s^*$ is such that $\lambda \in (\lambda _1 a(s^*),\lambda _1^{\varOmega _0}a(s^*))$. Hence, by Proposition 8 there exists at least a positive solution of (1).

• 2. Assume that $\underline{S}\cap [\alpha _1,\alpha _2]\ne \emptyset$. We take $\overline{s}_1<\overline{s}_2$.

  • (a) Suposse $\underline{S}\cap (\overline{s}_1,\overline{s}_2)\ne \emptyset$. Take

    $$\begin{aligned} \underline{s}_{j_0}=\min \{ \underline{S}\cap (\overline{s}_1,\overline{s}_2)\}. \end{aligned}$$

    Consider now $[\overline{s}_1,\underline{s}_{j_0}]$. It is clear that

    $$\begin{aligned} \frac{\lambda }{\lambda _1^{\varOmega _0}}<a(s)<\frac{\lambda }{\lambda _1}\quad \text{ for } \text{ all } s\in [\overline{s}_1,\underline{s}_{j_0}]. \end{aligned}$$

    Then, by Proposition 7, we obtain

    $$\begin{aligned} \lim _{s\downarrow \overline{s}_{1} }h_\lambda (s) = I,\quad \text{ and } \quad \lim _{s\rightarrow \underline{s}_{j_0} }h_\lambda (s) = 0. \end{aligned}$$

    Moreover, $g(\underline{s}_{j_0})>0$ and $g(\overline{s}_{1} )<I$. Hence,

    $$\begin{aligned} h_\lambda (\overline{s}_{1})-g(\overline{s}_{1})>0\quad \text{ and }\quad h_\lambda (\underline{s}_{j_0} )-g(\underline{s}_{j_0} )<0, \end{aligned}$$

    and then, there exists $s^*\in (\overline{s}_1, \underline{s}_{j_0} )\subset (\overline{s}_{1},\overline{s}_{2})$ such that $h_\lambda (s^*)=g(s^*)$. Hence, by Proposition 8 there exists at least a positive solution of (1).

  • (b) Suposse $\underline{S}\cap (\overline{s}_1,\overline{s}_2)=\emptyset$. Then, we take $\overline{s}_2<\overline{s}_3$. If $\underline{S}\cap (\overline{s}_2,\overline{s}_3)\ne \emptyset$, then we can repeat the previous reasoning. If $\underline{S}\cap (\overline{s}_2,\overline{s}_3) =\emptyset$ we consider $\overline{s}_3<\overline{s_4}$. Hence, we can continue this argument until $\underline{S}\cap [\overline{s}_{m_0},\overline{s}_{m_0+1}]\ne \emptyset$, for some $m_0$.

  This completes the proof. $\square$

Proposition 10

If $g(s)<I$ for all $s\in (\alpha _1,+\infty )$ with

$$\begin{aligned} g(\alpha _1)= I. \end{aligned}$$

Then there exists positive solution of (1) for $\lambda \in (\varLambda _1,+\infty )$, where

$$\begin{aligned} \varLambda _1 = \displaystyle \left( \frac{\alpha _1}{I}\right) ^{\frac{1}{p}}\lambda _1^{\varOmega _0}. \end{aligned}$$

Proof

Take $\lambda >\varLambda _1$, then $\lambda /\lambda _1^{\varOmega _0}>\left( \frac{\alpha _1}{I}\right) ^{\frac{1}{p}}=a(\alpha _1)$. Since $g(s)<I$ for $s>\alpha _1$ it follows that $\lim _{s\rightarrow \infty }a(s)=\infty$. Then, there exist $s^*,s^{**}>\alpha _1$ such that $\lambda /\lambda _1^{\varOmega _0}=a(s^*)$ and $\lambda /\lambda _1=a(s^{**})$. Now, the argument carried out in the above Proposition can be adapted to cover this case. $\square$

Now, we are ready to prove the main result of this section.

Theorem 9

Assume that b verifies $(Hb_2)$, $I<\infty$ and that there exist $s_1<s_2<\dots <s_{m}$, $m\ge 1$, simple roots of (29) and consider

$$\begin{aligned} \varLambda _i= \lambda _1^{\varOmega _0}\left( \frac{s_i}{I}\right) ^{1/p},\quad i=1,\dots , m. \end{aligned}$$

Then:

• 1. From $\lambda =a(0)\lambda _1$ an unbounded continuum of positive solutions of (1) bifurcates from the trivial solution and it goes to infinity at $\lambda =\varLambda _1$. As consequence, there exists at least a positive solution of (1) if

  $$\begin{aligned} \lambda \in (\min \{a(0)\lambda _1,\varLambda _1\}, \max \{a(0)\lambda _1,\varLambda _1\}). \end{aligned}$$

• 2. If $m=2k+1$, $k\ge 0$, (1) possesses at least a positive solution for

  $$\begin{aligned} \lambda \in \displaystyle \bigcup _{j=1}^k(\varLambda _{2j},\varLambda _{2j+1}), \end{aligned}$$

  and (1) does not have positive solution for $\lambda$ large.

• 3. If $m=2k$, $k\ge 1$, (1) possesses at least a positive solution for

  $$\begin{aligned} \lambda \in \displaystyle \bigcup _{j=1}^{k-1}(\varLambda _{2j},\varLambda _{2j+1})\cup (\varLambda _{2k},\infty ). \end{aligned}$$

Moreover, if a is increasing then for any $\varLambda _{2j+1}$, $j=0,\ldots , k-1$, there exists a sequence of positive solutions $(\lambda _n,u_{\lambda _n})$ of (1) such that $\lambda _n\rightarrow \varLambda _{2j+1}$ and

$$\begin{aligned} \Vert u_{\lambda _n}\Vert _\infty \rightarrow \infty . \end{aligned}$$

Proof

• 1. The first paragraph follows by Theorem 8.

• 2. Assume now that $m=2k+1$. Then, $g(s_{2j})=g(s_{2j+1})$ and $g(s)<I$ for $s\in (s_{2j},s_{2j+1})$ for $j=1,\dots , k$. Then, we can apply Proposition 9 for $\alpha _1=s_{2j}$ and $\alpha _2=s_{2j+1}$. Furthermore, according to Proposition 5, (1) does not possess positive solutions for $\lambda$ large.

• 3. When $m=2k$, the result follows by Propositions 9 and 10 with $\alpha _1=s_{2j}$ and $\alpha _2=s_{2j+1}$ and $\alpha _1=s_{2k}$, respectively.

Finally, assume that a is increasing. We are going to show that a bifurcation to infinity occurs at $\varLambda _{2j+1}$. Indeed, take $\lambda _n\uparrow \varLambda _{2j+1}$. Then,

$$\begin{aligned} \frac{\lambda _n}{\lambda _1^{\varOmega _0}}\uparrow a(s_{2j+1}). \end{aligned}$$

Then, for each n, take the unique $s_n<s_{2j+1}$ (recall that a is increasing) such that

$$\begin{aligned} a(s_n)=\frac{\lambda _n}{\lambda _1^{\varOmega _0}}. \end{aligned}$$

It is apparent that $\frac{\lambda _n}{\lambda _1}>a(s_{2j+1})$ for n large, and then there exists a unique $s^n>s_{2j+1}$ such that

$$\begin{aligned} a(s^n)=\frac{\lambda _n}{\lambda _1}. \end{aligned}$$

Hence, $g(s_n)-h_{\lambda _n}(s_n)<0$ and $g(s^n)-h_{\lambda _n}(s^n)>0$. We can conclude the existence of $s^*_n\in (s_n,s^n)$ such that

$$\begin{aligned} g(s^*_n)=h_{\lambda _n}(s_n), \end{aligned}$$

and by Proposition 8 there exists a positive solution. In fact, the positive solution is

$$\begin{aligned} u_n = a(s^*_n)\theta _{\left[ \frac{\lambda _n}{a(s^*_n)},b\right] }. \end{aligned}$$

But, observe that when $\lambda _n\rightarrow \varLambda _{2j+1}$ then $s_n\rightarrow s_{2j+1}$ and hence $s^*_n\rightarrow s_{2j+1}$. Thus,

$$\begin{aligned} \frac{\lambda _n}{a(s^*_n)}\rightarrow \frac{\varLambda _{2j+1}}{a(s_{2j+1})}=\lambda _1^{\varOmega _0}. \end{aligned}$$

This ends the proof. $\square$

Remark 2

The condition that the roots $s_j$ are simple is only necessary to write Theorem 9 more clearly and to have that if $g(s)<I$ for $s\in (s_j,s_{j+1})$, then $g(s)>I$ for $s\in (s_{j+1},s_{j+2})$. However, if, for instance, $g(s)<I$ for $s\in (s_j,s_{j+1}) \cup (s_{j+1},s_{j+2})$, then it can be proved the existence of positive solution for $\lambda \in (\varLambda _j,\varLambda _{j+1}) \cup (\varLambda _{j+1},\varLambda _{j+2}).$