Pairs of Positive Solutions for Nonhomogeneous Dirichlet Problems

Abstract

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator. The reaction has a parametric concave term and negative sublinear perturbation. In contrast to the case of a positive perturbation, we show that now for all big values of the parameter $\lambda >0$, we have at least two positive solutions which do not vanish in the domain. In the process we prove a nonlinear maximum principle which is of independent interest.

1 Introduction

Let $\Omega \subseteq \mathbb {R}^N$ be a bounded domain with a $C^2$-boundary $\partial \Omega$. In this paper we study the following nonhomogeneous parametric Dirichlet problem:

In this paper the map $a:\mathbb {R}^N\rightarrow \mathbb {R}^N$ involved in the differential operator, is continuous and strictly monotone, thus maximal monotone too. It exhibits balanced (p-1)-growth and $1<q<p$. In the reaction (right-hand side) we have a parametric “concave” term $x\rightarrow \lambda x^{q-1}$ (since $q<p$) and there is a negative perturbation $-f(z,x)$ which is a Carathéodory function (that is, for all $x\in \mathbb {R}, z\rightarrow f(z,x)$ is measurable and for a.a.$z\in \Omega , x\rightarrow f(z,x)$ is continuous). We assume that $f(z,\cdot )$ is ($q-1$) sublinear as $x\rightarrow 0^+$ and as $x\rightarrow +\infty$. A typical case is when $f(x)=x^{\tau -1}$ for all $x\ge 0$ with $1<\tau <q$. It is well known that if this perturbation enters in the reaction with a positive sign, then the problem has a unique positive solution. This was proved first by Brezis–Oswald [3] for problems driven by the Laplacian and was extended by Diaz–Saa [5] to equations driven by the Dirichlet p-Laplacian and by Fragnelli–Mugnal–Papageorgiou [7] for equations driven by a nonhomogeneous differential operator with Robin boundary condition. The case where the perturbation enters with a negative sign has not been studied. We show that in this case, uniqueness of the solution fails and for big values of the parameter $\lambda >0$, we have at least two positive smooth solutions. However, these solutions do not belong in the interior of the positive cone of $C_{0}^1(\overline{\Omega })=\{u\in C^1(\overline{\Omega }):u|_{\partial \Omega }=0\}$, since the nonlinear Hopf’s lemma cannot be used (see Pucci–Serrin [15], pp. 111, 120). Nevertheless, in Sect. 3, we prove a maximum principle which shows that our solutions are strictly positive in $\Omega$. That result is of independent interest and can be useful in different contexts.

2 Mathematical Background Hypotheses

The analysis of problem ($p_{\lambda }$) will use the Sobolev space $W_0^{1,p}(\Omega )$ and the Banach space $C_0^{1}(\overline{\Omega })$. By $\Vert \cdot \Vert$ we denote the norm of the Sobolev space. On account of the Poincaré inequality, we have $\Vert u\Vert =\Vert Du\Vert _p$ for all $u\in W_0^{1,p}(\Omega )$. The Banach space $C_0^1(\overline{\Omega })=\{u\in C^1(\overline{\Omega }): u|_{\partial \Omega }=0\}$ is ordered with positive (order) cone $C_+=\{u\in C_0^1(\overline{\Omega }): u(z)\ge 0 \text { for all } z\in \overline{\Omega }\}$. This cone has a nonempty interior given by $\text {int}C_+=\{u\in C_+:u(z)>0\quad \text {for all }z\in \Omega ,\quad \dfrac{\partial u}{\partial { n}}|_{\partial \Omega }<0\}$ with $n(\cdot )$ being the outward unit norm on $\partial \Omega$.

If $v,u:\Omega \rightarrow \mathbb {R}$ are measurable functions such that $v(z)\le u(z)$ for a.a $z\in \Omega$, then by [v, u] we denote the order interval in $W_0^{1,p}(\Omega )$ defined by

$$\begin{aligned}{}[v,u]=\{h\in W_0^{1,p}(\Omega ):v(z)\le h(z)\le u(z)\quad \text {for a.a } z\in \Omega \}. \end{aligned}$$

For $x\in \mathbb {R}$, let $x^{\pm }=\max \{\pm x ,0\}$. Then, given $u\in W_0^{1,p}(\Omega )$, we set $u^{\pm }(z)=u(z)^\pm$ for all $z\in \Omega$. We know that $u^{\pm }\in W_0^{1,p}(\Omega ),~u=u^{+}-u^-$ and $|u|=u^{+}+u^{-}$. By $|\cdot |_{N}$ we will denote the Lebesgue measure on $\mathbb {R}^{N}$. Also if X is a Banach space and $\varphi \in C^{1}(X)$, then $K_{\varphi }=\{u\in X: \varphi '(u)=0\}$.

Next, we will introduce the hypotheses on the map $a(\cdot )$. So, let $\theta \in C^{1}(0,\infty )$ be such that

$$\begin{aligned} 0<\widehat{c}\le \frac{\theta '(t)t}{\theta (t)}\le c_{0}\quad \text {and } c_1t^{p-1}\le \theta (t)\le c_2[t^{s-1}+t^{p-1}]\nonumber \\ \text { for all } t>0\text { and some } c_1, c_2>0, 1<s<p. \end{aligned}$$

(1)

Then, the hypotheses on the map $a(\cdot )$ are the following:

$H_{0}$: $a(y)=a_0(|y|)y$ for all $y\in \mathbb {R}^{N}$, with $a_0(t)>0$ for all $t>0$ and

• (i) $a_{0}\in C^{1}(0,\infty )$, $t\rightarrow a_{0}(t)t$ is strictly increasing, $a_{0}(t)t\rightarrow 0^{+}$ as $t\rightarrow 0^+$ and if $l(t)=a_0(t)t$, then $l'(t)t\ge c^{*}l(t)$ for some $c^{*}>0$ all $t>0$;

• (ii) $|\nabla a(y)|\le c_3 \frac{\theta (|y|)}{|y|} \text { for all } y\in \mathbb {R}^{N}{\setminus }\{0\}$, some $c_3>0$;

• (iii) $\frac{\theta (|y|)}{|y|}|\xi |^2\le (\nabla \alpha (y)\xi ,\xi )_{\mathbb {R}^{N}}\,\text { for all } y\in \mathbb {R}^{N}{\setminus }\{0\}$ and all $\xi \in \mathbb {R}^{N}$.

Remark 1

These hypotheses on $a(\cdot )$ are dictated by the nonlinear regularity theory of Lieberman [11]. Also, they lead to the nonlinear maximum principle which we prove in the next section. The hypotheses are not restrictive and include many differential operators of interest (see the Examples below).

From these hypotheses, we see that the primitive function $t\rightarrow G_0(t)$ is strictly increasing and strictly convex. We set $G(y)=G_0(|y|)$ for all $y\in \mathbb {R}^{N}$. Then, the function $G(\cdot )$ is convex, differentiable and $G(0)=0$. Moreover, using the chain rule, we have

$$\begin{aligned} \nabla G(y)=G'_{0}(|y|)\frac{y}{|y|}=a_0(|y|)y=a(y)\quad \text { for all }y\in \mathbb {R}^{N}{\setminus }\{0\}, \nabla G(0)=0. \end{aligned}$$

Therefore, $G(\cdot )$ is the primitive of $a(\cdot )$. Since $G(\cdot )$ is convex and $G(0)=0$, from the properties of convex functions we have

$$\begin{aligned} G(y)\le (a(y),y)_{\mathbb {R}^{N}}\quad \text { for all } y\in \mathbb {R}^{N}. \end{aligned}$$

(2)

From (1) and hypotheses $H_0$, we infer the following properties for the map $a(\cdot )$ (see Papageorgiou–Rădulescu [12]).

Lemma 1

If hypotheses $H_0$ hold, then

• (a) $y\rightarrow a(y)$ is continuous and strictly monotone (thus maximal monotone too);

• (b) $|a(y)|\le c_4[|y|^{s-1}+|y|^{p-1}]$ for some $c_4>0$, all $y\in \mathbb {R}^{N};$

• (c) $\frac{c_1}{p-1}|y|^{p}\le (a(y),y)_{\mathbb {R}^{N}}$   for all $y\in \mathbb {R}^{N}$.

This lemma and (2) lead to the following growth restrictions for the primitive $G(\cdot )$.

Corollary 2

If hypotheses $H_0(i),(ii),(iii)$ hold,

              then $\frac{c_1}{p(p-1)}|y|^{p}\le G(y) \le c_5[|y|^{s-1}+|y|^{p-1}]$ for some $c_5>0$, all $y\in \mathbb {R}^{N}$.

Hypotheses $H_0$ provide a broad framework in which we can fit many differential operators of interest.

Examples:

• (a) $a(y)=|y|^{p-2}y$ with $1<p<\infty$.

This map corresponds to the p-Laplace differential operator defined by

$$\begin{aligned} \Delta _{p}u= \text {div} (|Du|^{p-2}Du)\text { for all } u\in W_0^{1,p}(\Omega ). \end{aligned}$$

• (b) $a(y)=|y|^{p-2}y+|y|^{q-2}y$ with $1<q<p<\infty$.

This map corresponds to the (p, q)-Laplacian defined by

$$\begin{aligned} \Delta _{p}u+\Delta _{q}u\text { for all } u\in W_0^{1,p}(\Omega ). \end{aligned}$$

Such operators arise in many mathematical models of physical processes. We mention the works of Benci–D’Avenia–Fortunato–Pisani [2] (quantum physics), Cherfils–Ilyasov [4] (reaction–diffusion systems) and Bahrauni–Rădulescu–Repovš [1] (transonic flow problems). Some recent results in this direction can be found in the works of Goodrich–Ragusa [8],Goodrich–Ragusa–Scapellato [9], Papageorgiou–Scapellato [13] and Papageorgiou–Zhang [14].

• (c) $a(y)=[1+|y|^{2}]^{\frac{p-2}{2}}y$ with $1<p<\infty$.

This map corresponds to the generalized p-mean curvature differential operator defined by

$$\begin{aligned} \text {div} (1+|Du|^{2})^{\frac{p-2}{2}}Du\text { for all } u\in W_0^{1,p}(\Omega ). \end{aligned}$$

• (d) $a(y)=[1+\frac{|y|^{2}}{(1+|y|^{2p})^{1/2}}]|y|^{p-2}y$ with $1<p<\infty$.

This map corresponds to the following differential operator which arises in problems of plasticity theory

$$\begin{aligned} \text {div}\left( \left( 1+\frac{|Du|^{2}}{(1+|Du|^{2p})^{1/2}}\right) |Du|^{p-2}Du\right) \text {for all }u\in W_0^{1,p}(\Omega ). \end{aligned}$$

Let $A:W_0^{1,p}(\Omega )\rightarrow W_0^{1,p}(\Omega )^*=W^{-1,p'}(\Omega )~(\frac{1}{p}+\frac{1}{p'}=1)$ be the nonlinear operator defined by

$$\begin{aligned} \langle A(u),h \rangle = \int _{\Omega } (a(Du),Dh)_{\mathbb {R}^{N}}\mathrm{d}z \text { for all } u,h\in W_0^{1,p}(\Omega ). \end{aligned}$$

This operator is continuous and strictly monotone, thus maximal monotone too. Moreover, if we consider the integral functional $j:W_0^{1,p}(\Omega )\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} j(u)= \int _{\Omega }G(Du)\mathrm{d}z\text { for all } u\in W_0^{1,p}(\Omega ), \end{aligned}$$

then $j\in C^{1}( W_0^{1,p}(\Omega ))$ and $j'(u)=A(u)$ for all $u\in W_0^{1,p}(\Omega )$.

Now we introduce our hypotheses on the perturbation f(z, x):

$H_{1}$: $f:\Omega \times \mathbb {R}\rightarrow \mathbb {R}$ is a Carathéodory function such that $f(z,0)=0$ for a.a.$z\in \Omega$ and

• (i) $0\le f(z,x) \le a(z)[1+x^{p-1}]$ for a.a.$z\in \Omega$, all $x\ge 0$, with $a\in L^{\infty }(\Omega )$;

• (ii) $\lim \limits _{x\rightarrow +\infty }\dfrac{f(z,x)}{x^{q-1}}=0\,\text { uniformly for a.a. } z\in \Omega$ ;

• (iii) $\lim \limits _{x\rightarrow 0^+}\dfrac{f(z,x)}{x^{q-1}}=+\infty \text { uniformly for a.a. } z\in \Omega$ ;

• (iv) there exists $\mu \in (1,q)$ such that for all $\rho >0$, we can find $\widehat{\xi }_{\rho }^{\lambda }>0$ for which we have $\lambda x^{q-1}-f(z,x)+\widehat{\xi }_{\rho }^{\lambda }x^{\mu -1}\ge 0$ for a.a.$z\in \Omega$, all $x\in [0,\rho ]$.

Remark 2

In hypothesis $H_1(iv)$ we need $\mu \in (1,q)$. This is a consequence of hypothesis $H_1(iii)$ and of the fact that the perturbation f(z, x) enters in the reaction with a negative sign. However, this prohibits us from having a nonlinear Hopf’s lemma (see Pucci–Serrin [15], p. 120), since hypothesis (1.1.5) in [15] is no longer true. Therefore, we see that the negative sign in the perturbation changes the geometry and is a source of difficulties. Nevertheless, in the next section we prove a maximum principle which shows that the positive solutions of problem $(p_{\lambda })$ do not vanish in $\Omega$. This maximum principle extends Theorem 1.1 of Zhang [16].

3 A Maximum Principle

In this section we prove a nonlinear maximum principle. Our result was inspired by the work of Zhang [16] (Theorem 1.1) and we extend the result of [16]. The hypotheses of Zhang [16] on $a(\cdot )$ are more restrictive and do not cover the important case of the (p, q)-Laplacian (see (12) in [16]). The result is of independent interest.

Proposition 3

If $u\in C_{+}{\setminus } \{0\},~\widehat{\xi }>0$ and $\mu \in (1,q)$ satisfy

$$\begin{aligned} -\text {diva}(Du)+\widehat{\xi }u^{\mu -1}\ge 0\quad \text {in }\Omega , \end{aligned}$$

then $u(z)>0$    for all $z\in \Omega$.

Proof

We argue by contradiction. So, suppose we can find $z_1,z_2\in \Omega$ and $\rho >0$ such that $\overline{B}_{2\rho }(z_2)\subseteq \Omega$ ($B_{2\rho }(z_2)=\{z\in \Omega : |z-z_{2}|<2\rho \}$), $z_{1}\in \partial B_{2\rho }(z_{1}),~u(z_1)=0,~u|_{B_{2\rho }(z_{2})}>0$. By varying $z_{2}$ with $z_1$ fixed, we see that we can choose $\rho >0$ small.

Since $u(z_{1})=0=\min \limits _{\overline{\Omega }}u$ and $z_1\in \Omega$, we have

$$\begin{aligned} Du(z_1)=0. \end{aligned}$$

(3)

Let $m_{\rho }=\min [u(z): z\in \partial B_{\rho }(z_2)]>0$. As $\rho \rightarrow 0^+,~z_2$ converges to $z_1$ (which we fixed) and so $m_{\rho }\rightarrow 0^+$ and $\frac{m_{\rho }}{\rho }\rightarrow 0^+$ (by L’Hopital’s rule).

We introduce the annulus (ring) $R\subseteq \Omega$ defined by

$$\begin{aligned} R=\{z\in \Omega : \rho<|z-z_{2}|< 2\rho \}. \end{aligned}$$

We set

$$\begin{aligned} \eta = -\ln \frac{m_{\rho }}{\rho }+\frac{N-1}{\rho }>0 \quad (\text {for }\rho >0 \text { small}). \end{aligned}$$

(4)

We consider the function

$$\begin{aligned} v_{\rho }(t)= \frac{m_{\rho }[\rho ^{\frac{\eta t}{c^*}}-1]}{\rho ^{\frac{\eta \rho }{c^*}}-1} \quad \text {for all }0\le t\le \rho . \end{aligned}$$

Since $m_{\rho }\eta \rightarrow 0^+$ as $\rho \rightarrow 0^+$, for $\rho \in (0,1)$ small we have

$$\begin{aligned}&0\le v_{\rho }(t)<1 \quad \text { and }0< v'_{\rho }(t)<1 \quad \text { for all } t\in [0,\rho ],\end{aligned}$$

(5)

$$\begin{aligned}&v''_{\rho }(t)=\frac{\eta }{c^*}v'_{\rho }(t)\quad \text { for all } t\in [0,\rho ]. \end{aligned}$$

(6)

To simplify things, we may assume that $z_{2}=0$. Let $r=|z|$, $s=2\rho -r$. For $s\in [0,\rho ]$ and $r\in [\rho ,2\rho ]$, we define

$$\begin{aligned}&y(r)=v_{\rho }(2\rho -r)=v_{\rho }(s),\\&\quad \Rightarrow \,y'(r)=-v'_{\rho }(s)\quad \text { and } y''(r)=v''_{\rho }(s). \end{aligned}$$

We set $y(z)=y(r)$ for all $z\in \Omega$ with $|z|=r$. Then, $y\in C^{2}(R)$ and using the function $l(\cdot )$ from hypothesis $H_0(i)$, we have

$$\begin{aligned} \text {div}a(Dy)&=l'(v'_{\rho }(s))v''_\rho (s)-\frac{N-1}{r}l(v'_{\rho }(s))\\&=\frac{\eta }{c^*}l'(v'_{\rho }(s))v'_\rho (s)-\frac{N-1}{r}l(v'_{\rho }(s))\quad (\text {see } (6))\\&\ge [\eta -\frac{N-1}{r}]l(v'_\rho (s))\quad (\text {see hypothesis }H_0(i))\\&\ge (-\ln \frac{m_{\rho }}{\rho })l(v'_{\rho }(s))\quad (\text {see }(4)\text { and recall } r\ge \rho )\\&\ge (-\ln \frac{m_{\rho }}{\rho })\frac{c_1}{p-1}v'_{\rho }(s)^{p-1}\quad (\text {see Lemma }1)\\&\ge \widehat{\xi }v'(s)^{\mu -1}\quad \text {for } \rho \in (0,1)\text { small} \end{aligned}$$

(note that $v'_{\rho }(0)>0$ and $v'_{\rho }(\cdot )$ is increasing, see (6), (5)),

$$\begin{aligned} \Rightarrow \text {div}a(Dy)+\widehat{\xi }y^{\mu -1}\le 0\quad \text {in } R. \end{aligned}$$

(7)

Note that $y\le u$ on $\partial R$ and by hypothesis

$$\begin{aligned} -\text {diva}(Du)+\widehat{\xi }u^{\mu -1}\ge 0\quad \text {in } R. \end{aligned}$$

(8)

Then, from (7), (8) and Theorem 3.4.1, p. 61, of Pucci–Serrin [15] (the weak comparison principle), we have

$$\begin{aligned} y(z)\le u(z)\quad \text {for all } z\in \overline{R}. \end{aligned}$$

Then, we have

$$\begin{aligned}&\lim \limits _{\tau \rightarrow 0}\dfrac{y(z_1+\tau (z_2-z_1))-y(z_1)}{\tau }\le \lim \limits _{\tau \rightarrow 0}\dfrac{u(z_1+\tau (z_2-z_1))-u(z_1)}{\tau }\\&\quad (\text {recall } u(z_1)=0 \text { and }y\ge 0)\\&\quad \Rightarrow \, 0< v'(0)\le Du(z_1)=0,\text { a contradiction}. \end{aligned}$$

So, we conclude that $u(z)>0$ for all $z\in \Omega$. $\square$

4 Positive Solutions

In this section we show that for $\lambda >0$ big, problem ($p_\lambda$) admits a pair of positive solutions. We start by producing one positive solution.

Proposition 4

If hypotheses $H_0,H_1$ hold, then for all $\lambda >0$ big problem ($p_\lambda$) has a positive solution $u_{\lambda }\in C_+{\setminus } \{0\}$, $0< u_{\lambda }(z)$ for all $z\in \Omega$.

Proof

Let $\varphi _{\lambda }: W_{0}^{1,p}(\Omega )\rightarrow \mathbb {R}$ be the $C^{1}$-functional defined by

$$\begin{aligned} \varphi _{\lambda }(u)=\int _{\Omega } G(Du)\mathrm{d}z+\int _{\Omega } F(z,u^+)\mathrm{d}z-\frac{\lambda }{q}\Vert u^+\Vert _{q}^{q}\quad \text {for all } u\in W_{0}^{1,p}(\Omega ). \end{aligned}$$

Since $q<p$, using Corollary 2, we see that $\varphi _{\lambda }(\cdot )$ is coercive. Also, from the Sobolev embedding theorem, we see that $\varphi _{\lambda }(\cdot )$ is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find $u_{\lambda }\in W_{0}^{1,p}(\Omega )$ such that

$$\begin{aligned} \varphi _{\lambda }(u_{\lambda })=\inf [\varphi _{\lambda }(u):u\in W_{0}^{1,p}(\Omega )]. \end{aligned}$$

(9)

Let $\Omega _{0}\subseteq \Omega$ be open subset such that $\overline{\Omega }_{0}\subseteq \Omega$. Consider a function $y\in C_{c}^{1}(\Omega )$ such that

$$\begin{aligned} 0\le y \le 1\text { and } y|_{\overline{\Omega }_0}\equiv 1 \end{aligned}$$

(such a function is called “cut-off function” and is obtained by mollification, see, for example, Evans [6], p. 310). Hypotheses $H_1(i)(ii)$ imply that given $\varepsilon >0$, we can find $c_6=c_6(\varepsilon )>0$ such that

$$\begin{aligned} 0\le F(z,x)\le \frac{\varepsilon }{q}x^{q}+c_6\quad \text {for a.a. }z\in \Omega ,\text { all }x\ge 0. \end{aligned}$$

(10)

Then, we have

$$\begin{aligned} \varphi _{\lambda }(y)&\le \int _{\Omega } G(Dy)\mathrm{d}z-\frac{\lambda -\varepsilon }{q}\Vert y\Vert _{q}^{q}+c_6|\Omega |_{N}\quad (\text {see }(10))\\&\le c_7 -\frac{\lambda -\varepsilon }{q}\int _{\Omega _0}y^q \mathrm{d}z\quad \text {for some } c_7>0\quad (\text {see Corollary }2). \end{aligned}$$

Therefore, we can find $\lambda _*>\varepsilon$ such that

$$\begin{aligned}&c_7< \frac{\lambda -\varepsilon }{q}\int _{\Omega _0}y^q \mathrm{d}z \quad \text {for all }\lambda > \lambda _*,\\&\quad \Rightarrow \,\varphi _{\lambda }(y)<0,\\&\quad \Rightarrow \,\varphi _{\lambda }(u_\lambda )<0=\varphi _{\lambda }(0)\quad (\text {see }(9)),\\&\quad \Rightarrow \,u_\lambda \ne 0. \end{aligned}$$

From (9) we have

$$\begin{aligned}&\varphi '_{\lambda }(u_\lambda )=0\nonumber \\&\Rightarrow \,\langle A(u_\lambda ),h\rangle =\int _{\Omega }[\lambda u_{\lambda }^{+}-f(z,u_\lambda ^+)]h \mathrm{d}z\quad \text {for all }h\in W_0^{1,P}(\Omega ). \end{aligned}$$

(11)

In (11) we choose $h=-u_\lambda ^-\in W_0^{1,P}(\Omega )$, and using Lemma 1 we obtain

$$\begin{aligned} \frac{c_1}{p-1}\Vert Du_\lambda ^{-}\Vert _{p}^{p}\le 0,\quad \Rightarrow \quad u_\lambda \ge 0,~u_\lambda \ne 0. \end{aligned}$$

So, $u_\lambda$ is a positive solution of ($p_\lambda$). Invoking Theorem 7.1, p. 286 of Ladyzhenskaya–Uraltseva [10], we have that $u_\lambda \in L^{\infty }(\Omega )$. Then, the nonlinear regularity theory of Lieberman [11] implies that $u_{\lambda }\in C_{+}{\setminus } \{0\}$. Let $\rho =\Vert u_\lambda \Vert _\infty$ and let $\widehat{\xi }_{\rho }^{\lambda }>0$ be as postulated by hypothesis $H_{1}(iv)$. We have

$$\begin{aligned} -\,\text {diva}(Du_{\lambda })+\widehat{\xi }_{\rho }^{\lambda }u_{\lambda }^{\mu -1}\ge 0\text { in }\Omega , \quad \Rightarrow \quad 0< u_\lambda (z)\text { for all } z\in \Omega \,(\text {see Proposition } 3). \end{aligned}$$

$\square$

Using this first solution, we can produce a second one.

Proposition 5

If hypotheses $H_0,H_1$ hold and $\lambda >\lambda _*$, then problem ($p_\lambda$) has a second positive solution $\widehat{u}_{\lambda }\in$ $C_+{\setminus }\{0\},~ \widehat{u}_{\lambda }\ne u_{\lambda }$ and $0<\widehat{u}_{\lambda }(z)$ for all $z\in \Omega$.

Proof

Let $k_{\lambda }:\Omega \times \mathbb {R}\rightarrow \mathbb {R}$ be the Carathéodory function defined by

$$\begin{aligned} k_{\lambda }(z,x)=\left\{ \begin{array}{lll} \lambda (x^+)^{q-1}-f(z,x^+)\,\,\,&{}&{}\text{ if } x\le u_{\lambda }(z) \\ \lambda u_{\lambda }(z)^{q-1}-f(z,u_\lambda (z))\,\,\,&{}&{}\text{ if } u_\lambda (z)< x. \end{array}\right. \end{aligned}$$

(12)

We set $K_\lambda (z,x)=\int _{0}^{x}k_\lambda (z,s)ds$ and consider the $C^1$-functional $\widehat{\varphi }_{\lambda }: W_{0}^{1,p}(\Omega )\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} \widehat{\varphi }_{\lambda }(u)=\int _{\Omega }G(Du)\mathrm{d}z-\int _{\Omega }K_{\lambda }(z,u)\mathrm{d}z\text { for all }u\in W_{0}^{1,p}(\Omega ). \end{aligned}$$

Claim 1: $K_{\widehat{\varphi }_{\lambda }}\subseteq [0,u]\cap C_+$.

Let $u\in K_{\widehat{\varphi }_{\lambda }}$. We have

$$\begin{aligned}&\widehat{\varphi }'_{\lambda }(u)=0,\nonumber \\&\quad \Rightarrow \langle A(u),h\rangle =\int _{\Omega }K_\lambda (z,u)h \mathrm{d}z\quad \text {for all } h\in W_{0}^{1,p}(\Omega ). \end{aligned}$$

(13)

In (13) we use the test function $h=-u^{-}\in W_{0}^{1,p}(\Omega )$. Then, from (12) and Lemma 1, we have

$$\begin{aligned} \frac{c_1}{p-1}\Vert Du^{-}\Vert _{p}^{p}\le 0,\quad \Rightarrow \quad u\ge 0. \end{aligned}$$

Next, we test (13) with $h=[u-u_\lambda ]^+\in W_{0}^{1,p}(\Omega )$. We obtain

$$\begin{aligned} \langle A(u),(u-u_\lambda )^{+}\rangle&=\int _{\Omega }[\lambda u_\lambda ^{q-1}-f(z,u_\lambda )](u-u_\lambda )^{+}\mathrm{d}z \quad (\text {see }(12))\\&=\langle A(u_\lambda ),(u-u_\lambda )^{+}\rangle \quad (\text {since }u_\lambda \text { is a solution of } (p_\lambda )),\\&\Rightarrow u\le \widehat{u}_{\lambda }\quad (\text {from the monotonicity of }A). \end{aligned}$$

We have proved that $u\in [0,u_\lambda ]$. Moreover, the nonlinear regularity theory of Lieberman [11] implies that $u\in C_+$. Therefore, we conclude that $K_{\widehat{\varphi }_{\lambda }}\subseteq [0,u]\cap C_+$. This proves Claim 1.

Claim 2: We can find $\rho _0>0$ such that

$$\begin{aligned} 0< m_0\le \widehat{\varphi }^{+}_{\lambda }(u)\quad \text {for all } u\in W_{0}^{1,p}(\Omega ),~\Vert u\Vert =\rho _0. \end{aligned}$$

Hypotheses $H_1(i),(iii)$ imply that given $\eta >\lambda$, we can find $c_8=c_8(\eta )>0$ such that

$$\begin{aligned} f(z,x)\ge \eta x^{q-1}-c_8x^{p-1}\quad \text {for a.a.}z\in \Omega ,\text { all }x\ge 0. \end{aligned}$$

(14)

It follows that

$$\begin{aligned} \lambda x^{q-1}-f(z,x)\le c_8x^{p-1}-(\eta -\lambda ) x^{q-1}\quad \text {for a.a.}z\in \Omega ,\text { all }x\ge 0. \end{aligned}$$

Since $q<p$ and $\eta >\lambda$, we see that we can find $\delta \in (0,1)$ small such that

$$\begin{aligned}&\lambda x^{q-1}-f(z,x)\le 0\quad \text {for a.a.}z\in \Omega ,\text { all }0\le x\le \delta ,\end{aligned}$$

(15)

$$\begin{aligned}&\quad \Rightarrow \frac{\lambda }{q}x^{q}-F(z,x)\le 0\quad \text {for a.a.}z\in \Omega ,\text { all }0\le x\le \delta . \end{aligned}$$

(16)

Let $u\in W_{0}^{1,p}(\Omega )$ and introduce the set $\Omega _{\delta }^{u}=\{z\in \Omega :u(z)>\delta \}.$ Using Corollary 2, we have

$$\begin{aligned} \widehat{\varphi }_{\lambda }(u)\ge \frac{c_1}{p(p-1)}\Vert Du\Vert _{p}^{p}-\int _{\Omega } K_\lambda (z,u)\mathrm{d}z. \end{aligned}$$

(17)

We estimate the integral in the right-hand side of (17). We have

$$\begin{aligned} \int _{\Omega }K_{\lambda }(z,u)\mathrm{d}z=\int _{\Omega {\setminus }\Omega _\delta ^{u}}K_{\lambda }(z,u)\mathrm{d}z +\int _{\Omega _\delta ^{u}}K_{\lambda }(z,u)\mathrm{d}z. \end{aligned}$$

(18)

We examine the first integral in the right-hand side of (18). Then,

$$\begin{aligned} \int _{\Omega {\setminus }\Omega _\delta ^{u}}K_{\lambda }(z,u)\mathrm{d}z=\int _{(\Omega {\setminus }\Omega _\delta ^{u})\cap \{u\le u_{\lambda }\}}K_{\lambda }(z,u)\mathrm{d}z+\int _{(\Omega {\setminus }\Omega _\delta ^{u})\cap \{u_\lambda < u\}}K_{\lambda }(z,u)\mathrm{d}z \end{aligned}$$

(19)

Using (12), we see that

$$\begin{aligned} \int _{(\Omega {\setminus }\Omega _\delta ^{u})\cap \{u\le u_\lambda \}}K_{\lambda }(z,u)\mathrm{d}z=\int _{(\Omega {\setminus }\Omega _\delta ^{u})\cap \{u\le u_{\lambda }\}}\left[ \frac{\lambda }{q}(u^+)^{q}-F(z,u^{+})\right] \mathrm{d}z\le 0 \quad (\text {see }(16)). \end{aligned}$$

(20)

Similarly, using once again (12), we obtain

$$\begin{aligned} \int _{(\Omega {\setminus }\Omega _\delta ^{u})\cap \{u_\lambda< u\}}K_{\lambda }(z,u)\mathrm{d}z&=\int _{(\Omega {\setminus }\Omega _\delta ^{u})\cap \{u_\lambda< u\}}\left[ \lambda u_{\lambda }^{q-1}-f(z,u_\lambda )\right] (u-u_{\lambda })\mathrm{d}z\nonumber \\&\le \int _{(\Omega {\setminus }\Omega _\delta ^{u})\cap \{u_\lambda< u\}}\left[ (\lambda -\eta )u_{\lambda }^{q-1}+c_8u_\lambda ^{p-1}\right] (u-u_{\lambda })\mathrm{d}z\quad (\text {see }(14))\nonumber \\&\le c_8\int _{(\Omega {\setminus }\Omega _\delta ^{u})\cap \{u_\lambda < u\}}u^{p}\mathrm{d}z\quad (\text {since } \eta >\lambda ). \end{aligned}$$

(21)

Returning to (19) and using (20) and (21), we obtain

$$\begin{aligned} \int _{\Omega {\setminus }\Omega _\delta ^{u}}K_{\lambda }(z,u)\mathrm{d}z\le c_8\int _{(\Omega {\setminus }\Omega _\delta ^{u})\cap \{u_\lambda < u\}}u^{p}\mathrm{d}z. \end{aligned}$$

Since $u_{\lambda }\in C_+{\setminus }\{0\}$, from the absolute continuity of the Lebesgue integral, we see that given $\varepsilon >0$, we can choose $\delta \in (0,1)$ even smaller if necessary so that

$$\begin{aligned} \int _{\Omega {\setminus }\Omega _\delta ^{u}}K_{\lambda }(z,u)\mathrm{d}z\le \varepsilon . \end{aligned}$$

(22)

Next, we estimate the second integral in the right-hand side of (18). Using (12), we have

$$\begin{aligned} \int _{\Omega _\delta ^{u}}K_{\lambda }(z,u)\mathrm{d}z= & {} \int _{\Omega _\delta ^{u}\cap \{u\le u_{\lambda }\}}\left[ \frac{\lambda }{q}(u^+)^{q}-F(z,u^+)\right] \mathrm{d}z\nonumber \\&+\int _{\Omega _\delta ^{u}\cap \{u_\lambda < u\}}\left[ \lambda u_{\lambda }^{q-1}-f(z,u_\lambda )\right] (u-u_{\lambda })\mathrm{d}z. \end{aligned}$$

(23)

Since $F\ge 0$ (see hypothesis $H_1(i)$), we have

$$\begin{aligned} \int _{\Omega _\delta ^{u}\cap \{u\le u_{\lambda }\}}\left[ \frac{\lambda }{q}(u^+)^{q}-F(z,u^+)\right] \mathrm{d}z\le \frac{\lambda }{q}\int _{\Omega _\delta ^{u}\cap \{u\le u_\lambda \}}(u^+)^{q}\mathrm{d}z. \end{aligned}$$

(24)

Similarly, since $f\ge 0$, we have

$$\begin{aligned} \int _{\Omega _\delta ^{u}\cap \{u_{\lambda }< u\}}\left[ \lambda u_\lambda ^{q-1}-f(z,u)\right] (u-u_\lambda )\mathrm{d}z\le \lambda \int _{\Omega _\delta ^{u}\cap \{u_\lambda < u\}}u^{q}\mathrm{d}z. \end{aligned}$$

(25)

We return to (23) and use (24) and (25). We obtain

$$\begin{aligned} \int _{\Omega _\delta ^{u}}K_{\lambda }(z,u)\mathrm{d}z&\le \lambda \int _{\Omega _\delta ^{u}}|u|^{q} \mathrm{d}z\nonumber \\&\le \lambda c_9 \int _{\Omega _\delta ^{u}}|u|^{p} \mathrm{d}z\nonumber \\&~~~~~\text {for some } c_9>0\quad (\text {since } \delta>0 \text { and }q<p )\nonumber \\&\le \lambda c_9 |\Omega _{\delta }^{u}|_{N}^{1-\frac{p}{r}}\left[ \int _{\Omega _\delta ^{u}}|u|^{r} \mathrm{d}z\right] ^{p/r}\nonumber \\&~~~~~\text { with } p<r<p^*\quad (\text {by H}\ddot{\mathrm{o}}\text {lder's inequality })\nonumber \\&=\lambda c_9 |\Omega _{\delta }^{u}|_{N}^{1-\frac{p}{r}}\Vert u\Vert _{r}^{p}\nonumber \\&\le \lambda c_{10} |\Omega _{\delta }^{u}|_{N}^{1-\frac{p}{r}}\Vert u\Vert ^{p}\\&~~~~~\text {for some } c_{10}>0\quad (\text {since } W_{0}^{1,p}(\Omega )\hookrightarrow L^{r}(\Omega )).\nonumber \end{aligned}$$

(26)

We return to (18) and use (22) and (26). Then,

$$\begin{aligned} \int _{\Omega }K_{\lambda }(z,u)\mathrm{d}z\le \varepsilon +\lambda c_{10} |\Omega _{\delta }^{u}|_{N}^{1-\frac{p}{r}}\Vert u\Vert ^{p}. \end{aligned}$$

(27)

From (17) and (27), we have

$$\begin{aligned} \widehat{\varphi }_{\lambda }(u)\ge \left[ \frac{c_1}{p(p-1)}-\lambda c_{10} |\Omega _{\delta }^{u}|_{N}^{1-\frac{p}{r}}\right] \Vert u\Vert ^{p}-\varepsilon . \end{aligned}$$

If $\Vert u\Vert \rightarrow 0$, then $|u(z)|\rightarrow 0$ for a.a.$z\in \Omega$ and $|\Omega _{\delta }^{u}|_{N}\rightarrow 0$ uniformly for $\delta \in (0,1)$ small. So, we can find $\rho _0\in (0,\Vert u_\lambda \Vert )$ small such that if $\Vert u\Vert =\rho _0$, then

$$\begin{aligned} \lambda c_{10} |\Omega _{\delta }^{u}|_{N}^{1-\frac{p}{r}}\le \frac{c_1}{2p(p-1)}. \end{aligned}$$

Hence, for $\Vert u\Vert =\rho _0$, we have

$$\begin{aligned} \widehat{\varphi }_{\lambda }(u)\ge \frac{c_1}{2p(p-1)}\rho _0^{p}-\varepsilon . \end{aligned}$$

Recall that $\varepsilon >0$ is arbitrary. So, we choose $\varepsilon \in (0,1)$ small so that

$$\begin{aligned} \widehat{\varphi }_{\lambda }(u)\ge m_{0}>0\quad \text { for all }\Vert u\Vert =\rho _0. \end{aligned}$$

(28)

This proves Claim 2.

Consider the set $\overline{B}_{\rho _0}=\{u\in W_{0}^{1,p}(\Omega ):\Vert u\Vert \le \rho _0\}$. From the reflexivity of $W_{0}^{1,p}(\Omega )$ and the Eberlein–Smulian theorem, we have that $\overline{B}_{\rho _0}$ is sequentially weakly compact. Also $\widehat{\varphi }_{\lambda }(\cdot )$ is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find $\widehat{u}_{\lambda }\in \overline{B}_{\rho _0}$ such that

$$\begin{aligned} \widehat{\varphi }_{\lambda }(\widehat{u}_{\lambda })=\inf [\widehat{\varphi }_{\lambda }(u):u\in \overline{B}_{\rho _0}]. \end{aligned}$$

From (12), for $\lambda >0$ big, we have

$$\begin{aligned}&\widehat{\varphi }_{\lambda }(\widehat{u}_{\lambda })<0=\widehat{\varphi }_{\lambda }(0),\\&\quad \Rightarrow 0<\Vert \widehat{u}_{\lambda }\Vert< \rho _0\quad (\text {see } (28))\\&\quad \Rightarrow \widehat{u}_{\lambda }\ne u_{\lambda }\quad (\text {recall }\rho _{0}<\Vert u_{\lambda }\Vert )\text { and }\widehat{u}_{\lambda }\ne 0. \end{aligned}$$

Moreover, from Claim 1 and (12), we have that

$$\begin{aligned} \widehat{u}_{\lambda }\in C_+{\setminus } \{0\} \quad \text {is a positive solution of }(p_{\lambda }) \end{aligned}$$

Finally, as for $u_{\lambda }$, using Proposition 3, we have $0<\widehat{u}_{\lambda }(z)$ for all $z\in \Omega$. $\square$

So, summarizing the situation for problem ($p_{\lambda }$), we can state the following multiplicity theorem for problem ($p_\lambda$).

Theorem 6

If hypotheses $H_0,H_1$ hold, then for all $\lambda >0$ big problem ($p_\lambda$) has at least two positive solutions $u_{\lambda }, \widehat{u}_{\lambda }\in C_+{\setminus } \{0\},u_{\lambda }\ne \widehat{u}_{\lambda }$ and $0<u_{\lambda }(z), \widehat{u}_{\lambda }(z)$ for all $z\in \Omega$.

Remark 3

If for a.a.$z\in \Omega$, the quotient $x\rightarrow \frac{f(z,x)}{x^{p-1}}$ is strictly decreasing on ${\mathop {\mathbb {R}}\limits ^{\circ }}_{+}=(0,\infty )$, then if the reaction is $\lambda x^{q-1}+f(z,x)$, the problem has a unique positive solution. However, if the reaction is $\lambda x^{q-1}-f(z,x)$ as in ($p_\lambda$), then we no longer have uniqueness of the positive solution and in fact for $\lambda >0$ big enough we can guarantee the existence of at least two positive smooth solutions which do not vanish in $\Omega$.