Positive solutions of quasi-linear elliptic equations with dependence on the gradient

Abstract

In the present paper we prove a multiplicity theorem for a quasi-linear elliptic problem with dependence on the gradient ensuring the existence of a positive solution and of a negative solution. In addition, we show the existence of the extremal constant-sign solutions: the smallest positive solution and the biggest negative solution. Our approach relies on extremal solutions for an auxiliary parametric problem. Other basic tools used in our paper are sub-supersolution techniques, Schaefer’s fixed point theorem, regularity results and strong maximum principle. In our hypotheses we only require a general growth condition with respect to the solution and its gradient, and an assumption near zero involving the first eigenvalue of the negative $p$-Laplacian operator.

1 Introduction and statement of main results

There is a wide literature on semilinear and quasilinear elliptic equations with gradient dependence on the nonlinear term of the following type

$$\begin{aligned} (P) \quad \left\{ \begin{array}{ll} -\Delta _p u= f(x,u, \nabla u) &{}\quad \text{ in }\quad \Omega \\ u=0 &{}\quad \text{ on } \quad \partial \Omega , \end{array} \right. \end{aligned}$$

where $\Omega$ is a smooth bounded domain in $\mathbb {R}^N$, $1<p<N$, $\Delta _p$ is the $p$-Laplacian operator and $f:\Omega \times \mathbb {R} \times \mathbb {R}^N\rightarrow \mathbb {R}$ is a Carathéodory function.

In this setting, the classical variational methods cannot be applied. This kind of problems are usually studied by means of topological degree, method of sub-supersolutions, fixed point theory and approximation techniques. For instance, in [2] the authors, assuming that $f$ is a $C^1$ function with growth given by

$$\begin{aligned} |f(x,s,\xi )|\le a(s)(1+|\xi |^2), \end{aligned}$$

for some increasing function $a$, obtain a solution of $(P)$ in an ordered interval of sub-supersolutions. In this respect, we also mention [14] where existence results for $(P)$, when $p=2$, are obtained via sub-supersolutions in the Sobolev space $W^{2,q} (\Omega )$ with $q>N$, provided $f(x, s, \xi )$ is Lipschitzian with respect to $\xi$. For the general theory of sub-supersolutions for nonlinear elliptic problems depending on the gradient we refer to [4]. In [15], by combining Krasnoselskii’s fixed point theorem in cones with blow up techniques, the existence of a positive solution of $(P)$ is proved when $f(x,s, \xi )$ is a non negative function and has a suitable growth with respect to $s$ and $\xi$. Recently, in [5] and [16], under different growth in the gradient, the existence of a positive solution is achieved via an approximation on finite dimensional subspaces. An approximation approach in a general functional setting with a $p$-sublinear growth condition in the gradient can be found in [3]. A different approach is proposed in [7] where the authors prove the existence of a positive and a negative solution for $(P)$, when $p=2$, through an iterative method involving Mountain Pass technique assuming that $f(x, s, \xi )$ satisfies Lipschitz conditions on $s$ and $\xi$ in a neighborhood of zero and has a growth like

$$\begin{aligned} |f(x,s,\xi )| \le a_1(1+ |s|^{q}), \ \quad \text{ with } \quad 1 < q < 2^*-1. \end{aligned}$$

It is shown in [11, Theorem 4.3] that if there exist a subsolution $\underline{u}$ and a supersolution $\overline{u}$ belonging to $C^1(\overline{\Omega })$ with $\underline{u}\le \overline{u}$, then the growth condition of Bernstein-Nagumo type

$$\begin{aligned} |f(x,s,\xi )| \le a(x)+ b|\xi |^{p}, \ \quad \text{ with } \quad a\in L^{\frac{p}{p-1}}(\Omega ) \quad \hbox {and} \quad b>0, \end{aligned}$$

for a.a. $x\in \Omega$, all $s\in [\underline{u}(x),\overline{u}(x)]$, all $\xi \in \mathbb {R}^N$, implies the existence of a solution $u\in W^{1,p}_0(\Omega )$ of $(P)$ satisfying $\underline{u}\le u\le \overline{u}$. Different other results based on the sub- supersolution method can be found in [6].

In the present paper, we will prove the existence of a positive and a negative solution for problem $(P)$ by combining sub-supersolution techniques with Schaefer’s fixed point theorem. More precisely, we will consider an auxiliary problem which can be studied through operator theory and sub-supersolutions method and we will obtain solutions of extremal type. This allows us to construct a map whose fixed points are exactly the solutions of our problem $(P)$. Finally, we will show the existence of a fixed point for the constructed map. Moreover, under the same assumptions, we will prove the existence of the smallest positive and of the biggest negative solution of $(P)$. Notice that our assumptions imply that zero is a solution of our problem.

We believe that our result gives a natural approach to the theory of quasilinear elliptic problems with gradient dependence. Furthermore, the hypotheses we assume on the nonlinear reaction term are general and verifiable. An example is provided at the end of our work.

Let us introduce our main results. In the sequel $f:\Omega \times \mathbb {R}\times \mathbb {R}^N\rightarrow \mathbb {R}$ is a Carathéodory function that is, $f(\cdot , s,\xi )$ is measurable for every $s\in \mathbb {R}$ and $\xi \in \mathbb {R}^N$, $f(x,\cdot , \cdot )$ is continuous for almost every $x\in \Omega$. Let $p'$ stand for the conjugate of $p$, i.e., $\frac{1}{p} + \frac{1}{p'} = 1$. As usual, $\lambda _1$ denotes the first eigenvalue of the negative $p$-Laplacian operator on $W^{1,p}_0(\Omega )$. For a later use, we recall that the cone of nonnegative functions $C^1_0(\overline{\Omega })_+=\{u\in C^1_0(\overline{\Omega }): u\ge 0 \ \quad \text{ in } \quad \Omega \}$ has a nonempty interior in the Banach space $C^1_0(\overline{\Omega })=\{u\in C^1(\overline{\Omega }): u=0 \ \quad \text{ on }\quad \partial \Omega \}$ given by

$$\begin{aligned} \mathrm {int}(C^1_0(\overline{\Omega })_+)=\left\{ u\in C^1_0(\overline{\Omega }): u>0 \quad \hbox {in}\quad \Omega , \frac{\partial u}{\partial \nu }<0 \quad \hbox {on} \quad \partial \Omega \right\} , \end{aligned}$$

where $\nu$ stands for the outward normal unit vector to $\partial \Omega$.

Our assumptions are:

• $(f_1)$ for every $M>0$, there exist constants $k_M>0$ and $0<\theta _M<\lambda _1$ such that

  $$\begin{aligned} |f(x,s,\xi )|\le k_M+\theta _M|s|^{p-1} \end{aligned}$$

  for a.e. $x\in \Omega$, $s \in \mathbb {R}$ and $\xi \in \mathbb {R}^N$ with $|\xi |\le M$;

• $(f_2)_+$ for every $M>0$ there exists a constant $\eta _M>\lambda _1$ such that

  $$\begin{aligned} \liminf _{s\rightarrow 0^+}\frac{f(x,s,\xi )}{s^{p-1}}\ge \eta _M >\lambda _1 \end{aligned}$$

  uniformly for a.e. $x\in \Omega$ and all $\xi \in \mathbb {R}^N$ with $|\xi |\le M$;

• $(f_3)_+$ for every $M>0$ there exists a constant $\zeta _M>0$ such that

  $$\begin{aligned} \limsup _{s\rightarrow 0^+}\frac{f(x,s,\xi )}{s^{p-1}}\le \zeta _M \end{aligned}$$

  uniformly for a.e. $x\in \Omega$ and all $\xi \in \mathbb {R}^N$ with $|\xi |\le M$.

Assuming $(f_1), (f_2)_+, (f_3)_+$, for every $w\in C^1_0(\overline{\Omega })$, the Dirichlet problem

$$\begin{aligned} (P_w)\qquad \qquad \qquad \left\{ \begin{array}{ll} -\Delta _p u= f(x,u, \nabla w) &{}\quad \text{ in } \quad \Omega \\ u=0 &{}\quad \text{ on } \quad \partial \Omega . \end{array} \right. \end{aligned}$$

has a smallest positive solution $u_w \in C^1_0(\overline{\Omega })$. Then we introduce the map

$$\begin{aligned} T: C^1_0(\overline{\Omega }) \rightarrow C^1_0(\overline{\Omega }), \ w \mapsto u_w, \end{aligned}$$

which is continuous and compact. We notice that the fixed points of $T$ coincide with the solutions of $(P)$. Later on, to apply Schaefer’s fixed point theorem, we need to strengthen the growth condition $(f_1)$. Namely, we will need

$(\widetilde{f_1})$ there exist positive constants $k_0,\theta _0,\theta _1$ with $\theta _0+\theta _1\lambda _1^{1/p'}<\lambda _1$ such that

$$\begin{aligned} |f(x,s,\xi )|\le k_0+\theta _0|s|^{p-1}+\theta _1|\xi |^{p-1} \end{aligned}$$

for a.a. $x\in \Omega$, all $s\in \mathbb {R}$, and $\xi \in \mathbb {R}^N$.

Our first result reads as follows:

Theorem 1.1

Assume $(\widetilde{f_1}), (f_2)_+, (f_3)_+$. Then, problem $(P)$ has a solution $u\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$.

Let us state now the counterpart of the previous theorem on the negative half-line. We formulate in a symmetric way the corresponding hypotheses:

• $(f_2)_-$ for every $M>0$ there exists a constant $\eta _M>\lambda _1$ such that

  $$\begin{aligned} \liminf _{s\rightarrow 0^-}\frac{f(x,s,\xi )}{|s|^{p-2}s}\ge \eta _M >\lambda _1 \end{aligned}$$

  uniformly for a.e. $x\in \Omega$ and all $\xi \in \mathbb {R}^N$ with $|\xi |\le M$;

• $(f_3)_-$ for every $M>0$ there exists a constant $\zeta _M>0$ such that

  $$\begin{aligned} \limsup _{s\rightarrow 0^-}\frac{f(x,s,\xi )}{|s|^{p-2}s}\le \zeta _M \end{aligned}$$

  uniformly for a.e. $x\in \Omega$ and all $\xi \in \mathbb {R}^N$ with $|\xi |\le M$.

On the pattern of Theorem 1.1 we can state

Theorem 1.2

Assume $(\widetilde{f_1}), (f_2)_-,(f_3)_-$. Then, problem $(P)$ has a solution $v\in -\mathrm {int}(C^1_0(\overline{\Omega })_+)$.

By combining Theorems 1.1 and 1.2 we obtain our main multiplicity result.

Theorem 1.3

Assume $(\widetilde{f_1}), (f_2)_{\pm }, (f_3)_{\pm }$. Then, problem $(P)$ has at least two solutions $u$ and $v$, with $u\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$ and $v\in -\mathrm {int}(C^1_0(\overline{\Omega })_+)$.

Corollary 1.1

Under the same assumptions as in Theorem 1.3, problem $(P)$ has the smallest positive solution and the biggest negative solution.

Remark 1.1

Notice that if $u$ is a positive solution of $(P)$, then it is valid the estimate

$$\begin{aligned} f(x, u(x), \nabla u(x)) \le \lambda _1 u^{p-1}(x) \end{aligned}$$

on a set of positive measure in $\Omega$. Indeed, if not, $u$ solves the problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta _p u= m(x) u^{p-1} &{} \text{ in } \quad \Omega \\ u=0 &{} \text{ on } \quad \partial \Omega , \end{array} \right. \end{aligned}$$

for a.e. $x \in \Omega$, where $m(x) = \frac{f(x,u(x), \nabla u(x))}{u^{p-1}(x)} > \lambda _1$. Then the function $u$ has to change sign (see Remark 2.1 below), which contradicts that $u$ is positive. An analogous remark holds for a negative solution $v$ of $(P)$.

2 Auxiliary results

We split the present section in two parts. The first one deals with the sub-supersolution method for an auxiliary problem with fixed gradient in the right-hand side of the elliptic equation. These preliminary results will be used to construct the map $T$ on which relies our fixed point approach for investigating problem $(P)$. The properties of the map $T$ will be studied in the second part of this section.

2.1 Sub-supersolution method

Let us first recall some well known results that are needed in the sequel. In what follows we endow the Sobolev space $W^{1,p}_0(\Omega )$ with the standard norm $\Vert u\Vert =\left( \int _\Omega |\nabla u|^p\,dx\right) ^{\frac{1}{p}}$. As usual, we set $u^+=\max \{u, 0\}$ and $u^-=\max \{0,-u\}$. It is well known if $u\in W^{1,p}_0(\Omega )$, then $u^+, u^-\in W^{1,p}_0(\Omega ).$

Given $m\in L^{\infty }(\Omega )_+$, $m\ne 0$, consider the nonlinear weighted eigenvalue problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta _p u= \hat{\lambda }m(x)|u|^{p-2}u &{} \text{ in } \quad \Omega \\ u=0 &{} \text{ on } \quad \partial \Omega \end{array}. \right. \end{aligned}$$

The least number $\hat{\lambda }>0$, denoted by $\hat{\lambda }_1(m)$, such that the above problem admits a nontrivial solution is called the first eigenvalue of $(-\Delta _p,W^{1,p}_0(\Omega ), m)$. It is well known that $\hat{\lambda }_1(m)$ is positive, isolated, simple and the following variational characterization holds

$$\begin{aligned} \hat{\lambda }_1(m)=\min \left\{ \frac{\Vert u\Vert ^p}{\int _\Omega m|u|^p\,dx}: u\in W^{1,p}_0(\Omega ), u\ne 0\right\} . \end{aligned}$$

We denote by $\phi _{1,m}$ the positive eigenfunction normalized as $\Vert \phi _{1,m}\Vert _p=1$, which is associated to $\hat{\lambda }_1(m)$. One has $\phi _{1,m}\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$.

As usual, if $m\equiv 1$, set $\lambda _1=\hat{\lambda }_1(m)$ and $\phi _1 = \phi _{1,m}$. The next remark contains useful information on the weighted eigenvalue problems (for the proof and further details we refer to [1]).

Remark 2.1

• (1) If $m_1,m_2\in L^{\infty }(\Omega )_+\setminus \{0\}$ satisfy $m_1\le m_2$ a.e. in $\Omega$, then one has $\hat{\lambda }_1(m_2)\le \hat{\lambda }_1(m_1)$. If in addition $m_1\ne m_2$, then, $\hat{\lambda }_1(m_2)<\hat{\lambda }_1(m_1)$.

• (2) If $u$ is an eigenfunction corresponding to an eigenvalue $\hat{\lambda }\ne \hat{\lambda }_1(m)$, then $u\in C^1_0(\overline{\Omega })$ changes sign.

Remark 2.2

Because of assumptions $(f_2)_{+}$, $(f_3)_{+}$, we get that $f(x,0,\xi )=0$ for a.e. $x\in \Omega$ and every $\xi \in \mathbb {R}^N$. So, in particular, $u=0$ is a solution of $(P)$.

Since in the construction below we will deal with positive solutions, without loss of generality we may assume that $f(x,s,\xi )=0$ for a.e. $x\in \Omega , s\le 0, \xi \in \mathbb {R}^N$.

We are going to consider an auxiliary problem and to prove existence of solutions for it. Namely, for every $w\in C^1_0(\overline{\Omega })$, let us state the Dirichlet problem

$$\begin{aligned} (P_w) \left\{ \begin{array}{ll} -\Delta _p u= f(x,u, \nabla w) &{}\quad \text{ in } \quad \Omega \\ u=0 &{}\quad \text{ on } \quad \partial \Omega . \end{array} \right. \end{aligned}$$

We recall that, for fixed $w\in C^1_0(\overline{\Omega })$, a function $\overline{u}_w \in W^{1,p}(\Omega )$, with $\overline{u}_w\ge 0$ on $\partial \Omega$ (in the sense of trace), is a supersolution for problem $(P_w)$ if

$$\begin{aligned} \int _\Omega |\nabla \overline{u}_w|^{p-2}\nabla \overline{u}_w \nabla v\,dx \ge \int _\Omega f(x, \overline{u}_w, \nabla w)v\,dx \end{aligned}$$

for all $v\in W^{1,p}_0(\Omega )$, $v\ge 0$ a.e. in $\Omega$. A function $\underline{u}_w\in W^{1,p}(\Omega )$, with $\underline{u}_w\le 0$ on $\partial \Omega$ (in the sense of trace), is a subsolution for problem $(P_w)$ if

$$\begin{aligned} \int _\Omega |\nabla \underline{u}_w|^{p-2}\nabla \underline{u}_w \nabla v\,dx \le \int _\Omega f(x, \underline{u}_w, \nabla w)v\,dx \end{aligned}$$

for all $v\in W^{1,p}_0(\Omega )$, $v\ge 0$ a.e. in $\Omega$.

By a solution of problem $(P_w)$ we mean a weak solution, i.e. a function $u\in W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} \int _\Omega |\nabla u|^{p-2}\nabla u \nabla v\,dx = \int _\Omega f(x, u, \nabla w)v\,dx \end{aligned}$$

for all $v\in W^{1,p}_0(\Omega )$.

Before stating the theorem guaranteeing existence of solutions of $(P_w)$, some auxiliary lemmas are required.

Lemma 2.1

Assume $(f_1)$. Then, for every $w\in C^1_0(\overline{\Omega })$ there exists $\overline{u}_w \in \mathrm {int}(C^1_0(\overline{\Omega })_+)$ supersolution of $(P_w)$.

Proof

Let us fix $w\in C^1_0(\overline{\Omega })$ and set $M=\Vert w\Vert _{C^{1}}$. From assumption $(f_1)$, we have that

$$\begin{aligned} |f(x,s, \nabla w(x))|\le k_M+\theta _M|s|^{p-1} \end{aligned}$$

(2.1)

for a.e. $x\in \Omega$ and all $s\in \mathbb {R}$, where $k_M>0$ and $0<\theta _M<\lambda _1$.

Consider the following Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _p u= k_M+\theta _M|u|^{p-1} &{}\quad \text{ in } \quad \Omega \\ u=0 &{} \quad \text{ on } \quad \partial \Omega . \end{array} \right. \end{aligned}$$

Since the embedding of $W^{1,p}_0(\Omega )$ into $L^p(\Omega )$ is compact, the superposition operator $K_M: W^{1,p}_0(\Omega )\rightarrow W^{-1,p'}(\Omega )$ defined by $K_M(u(\cdot ))= k_M+\theta _M|u(\cdot )|^{p-1}$ is completely continuous. Since the $p$-Laplacian operator is strictly monotone, continuous and bounded (see [4]), we have that $\,-\,\Delta _p\,-\,K_M$ is pseudomonotone and bounded. Thanks to the estimate

$$\begin{aligned} \frac{1}{p}\int _\Omega |\nabla u|^p\,dx-\int _\Omega \left[ k_M|u| +\frac{1}{p} \theta _M|u|^{p}\right] dx\ge \frac{1}{p} \left( 1-\frac{ \theta _M}{\lambda _1}\right) \Vert u\Vert ^p- k_M|\Omega |^{\frac{1}{p'}}\lambda _1^{-\frac{1}{p}}\Vert u\Vert , \end{aligned}$$

where $|\Omega |$ denotes the Lebesgue measure of $\Omega$, and using $\theta _M < \lambda _1$ (see $(f_1)$), it follows that $-\Delta _p-K_M$ is coercive, hence surjective. Therefore, there exists a function $\overline{u}_w\in W^{1,p}_0(\Omega )$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _p \overline{u}_w= k_M+\theta _M|\overline{u}_w|^{p-1} &{}\quad \text{ in } \quad \Omega \\ \overline{u}_w=0 &{} \quad \text{ on } \quad \partial \Omega . \end{array} \right. \end{aligned}$$

Let us prove that $\overline{u}_w\ge 0$. Acting as test function with $-\overline{u}_w^-$, we get

$$\begin{aligned} \int _\Omega |\nabla \overline{u}_w^-|^{p}\,dx= -\int _\Omega |\nabla \overline{u}_w|^{p-2}\nabla \overline{u}_w\nabla \overline{u}_w^-\,dx=-\int _\Omega (k_M+\theta _M|\overline{u}_w|^{p-1})\overline{u}_w^-\,dx\le 0, \end{aligned}$$

which implies that $\overline{u}_w^-=0$, so $\overline{u}_w\ge 0$. Notice that $\overline{u}_w\ne 0$. By classical regularity results (see [9, 10, 17]), we have that $\overline{u}_w\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$. Then from (2.1) we infer that $\overline{u}_w$ is a supersolution of $(P_w)$, which completes the proof. $\square$

Lemma 2.2

Assume $(f_2)_+$. Then, for every $w\in C^1_0(\overline{\Omega })$, there exists $\delta =\delta (w)>0$ such that if $0<\varepsilon <\delta$, then $\varepsilon \phi _1$ is a subsolution of $(P_w)$.

Proof

Let us fix $w\in C^1_0(\overline{\Omega })$ and set $M=\Vert w\Vert _{C^{1}}$. From assumption $(f_2)_+$, for $\sigma >0$ so small that $\eta _M-\sigma >\lambda _1$, there exists $\gamma =\gamma (w)>0$ such that

$$\begin{aligned} f(x,s, \nabla w(x))\ge (\eta _M-\sigma )s^{p-1}>\lambda _1 s^{p-1} \end{aligned}$$

(2.2)

for a.e. $x\in \Omega$ and all $0<s<\gamma$. Set $\delta ={\gamma }{\Vert \phi _1\Vert ^{-1}_{L^\infty }}$. Then, for every $0<\varepsilon <\delta$ and all $\varphi \in W^{1,p}(\Omega ),\ \varphi \ge 0$, we find that

$$\begin{aligned} \int _\Omega |\nabla (\varepsilon \phi _1)|^{p-2}\nabla (\varepsilon \phi _1)\nabla \varphi \,dx = \lambda _1\int _\Omega (\varepsilon \phi _1)^{p-1}\varphi \,dx \le \int _\Omega f(x,\varepsilon \phi _1,\nabla w(x))\varphi \,dx, \end{aligned}$$

that is, $\varepsilon \phi _1$ is a subsolution of $(P_w)$. $\square$

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

Theorem 2.1

Assume $(f_1), (f_2)_+, (f_3)_+$. Then, for every $w\in C^1_0(\overline{\Omega })$ there exists $u_w\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$ which is the smallest positive solution of $(P_w)$.

Proof

Let us fix $w\in C^1_0(\overline{\Omega })$ and set $M=\Vert w\Vert _{C^{1}}$. From Lemmas 2.1 and 2.2, we get that there exist a supersolution $\overline{u}_w$ and for $0<\varepsilon <\delta (w)$ a subsolution $\varepsilon \phi _1$ of $(P_w)$. Since $\overline{u}_w$ and $\varepsilon \phi _1$ belong to $\mathrm {int}(C^1_0(\overline{\Omega })_+)$, it is possible to choose $\varepsilon$ small enough in order that $\overline{u}_w-\varepsilon \phi _1\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$. Consider now the following truncation of $f$:

$$\begin{aligned} f_+(x,s) = \left\{ \begin{array}{lll} f(x,\varepsilon \phi _1(x),\nabla w(x)) &{} \quad \hbox {if} \quad s< \varepsilon \phi _1(x)\\ f(x,s,\nabla w(x)) &{} \quad \hbox {if} \quad \varepsilon \phi _1(x) \le s\le \overline{u}_w(x)\\ f(x,\overline{u}_w(x),\nabla w(x)) &{} \quad \hbox {if} \quad s> \overline{u}_w(x). \end{array} \right. \end{aligned}$$

Denote by $\mathcal E_+:W^{1,p}_0(\Omega )\rightarrow \mathbb {R}$ the associated energy functional, that is

$$\begin{aligned} \mathcal E_+(u)=\frac{1}{p}\Vert u\Vert ^p-\int _\Omega \int _0^{u(x)} f_+(x,t)\,dt\, dx \quad \text{ for } \,\, \text{ all } \quad u\in W^{1,p}_0(\Omega ). \end{aligned}$$

Clearly, $\mathcal E_+$ is sequentially weakly lower semicontinuous, coercive, and continuously differentiable. Hence, it has a global minimum $\tilde{u}_w^\varepsilon$ which is a critical point of $\mathcal E_+$, thus a weak solution of

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _p u= f_+(x,u) &{}\quad \text{ in } \Omega \\ u=0 &{}\quad \text{ on } \partial \Omega . \end{array} \right. \end{aligned}$$

Moreover, by a standard comparison argument one can show that

$$\begin{aligned} \varepsilon \phi _1\le \tilde{u}_w^\varepsilon \le \overline{u}_w, \end{aligned}$$

a.e. in $\Omega$, so that $\tilde{u}_w^\varepsilon$ is a solution of $(P_w)$. Also the strong maximum principle entails $\tilde{u}_w^\varepsilon \in \mathrm {int}(C^1_0(\overline{\Omega })_+)$. Denote by $S_\varepsilon$ the set of $C^1_0$-solutions of $(P_w)$ which lie in the ordered interval $[\varepsilon \phi _1, \overline{u}_w]$. As seen from above, $S_\varepsilon \ne \emptyset$. If we consider in $S_\varepsilon$ the pointwise order, then $S_\varepsilon$ is downward directed and, as can be noticed through Zorn’s Lemma, it has a minimal element $u_w^\varepsilon$ (see [12] for more details). Let us prove that $u_w^\varepsilon$ is the smallest solution of $(P_w)$ in $S_\varepsilon$. To this end, take $v\in S_\varepsilon$. The function $\min \{v, u_w^\varepsilon \}$ is a supersolution of $(P_w)$ and clearly, $\min \{v, u_w^\varepsilon \}\ge \varepsilon \phi _1$. Then, there exists a solution $z$ of $(P_w)$ such that $\varepsilon \phi _1\le z\le \min \{v, u_w^\varepsilon \}$. In particular, it turns out that $z\in S_\varepsilon$, $z\le u_w^\varepsilon$, and from the minimality of $u_w^\varepsilon$ in $S_\varepsilon$, we conclude that $z=u_w^\varepsilon$. So $u_w^\varepsilon \le v$, as we wished.

Fix now a decreasing sequence $\{\varepsilon _n\}_n$ of positive numbers such that $\varepsilon _n\rightarrow 0$. For every $n\in \mathbb {N}$ there exists $u_w^{\varepsilon _n}$ which is the smallest solution of $(P_w)$ in the ordered interval $[\varepsilon _n\phi _1, \overline{u}_w]$. The sequence $\{u_w^{\varepsilon _n}\}_n$ is bounded in $W^{1,p}_0(\Omega )$, so there exists $u_w\in W^{1,p}_0(\Omega )$ such that $u_w^{\varepsilon _n}\rightharpoonup u_w$ in $W^{1,p}_0(\Omega )$. In particular, $u_w^{\varepsilon _n}\rightarrow u_w$ in $L^p(\Omega )$ and $u_w^{\varepsilon _n}(x)\rightarrow u_w(x)$ for a.e. $x \in \Omega$. Notice that $\{ u_w^{\varepsilon _n}\}_n$ is decreasing by construction, so that the convergence $u_w^{\varepsilon _n}\rightarrow u_w$ is uniform.

We want to prove that $u_w\ne 0$. Assume by contradiction that $u_w=0$ and set $z_n=\frac{u_w^{\varepsilon _n}}{\Vert u_w^{\varepsilon _n}\Vert }$. Hence $z_n\in W^{1,p}_0(\Omega )$ and $\Vert z_n\Vert =1$ for every $n\in \mathbb {N}.$ So, $\{z_n\}_n$ converges weakly in $W^{1,p}_0(\Omega )$ to some $z\in W^{1,p}_0(\Omega )$. On account of $z_n\ge 0$ a.e. in $\Omega$, we get $z\ge 0$ a.e. in $\Omega$. Denote

$$\begin{aligned} h_n(x)=\frac{f(x,u_w^{\varepsilon _n}(x), \nabla w(x) )}{(u_w^{\varepsilon _n}(x))^{p-1}}. \end{aligned}$$

The fact that $u_w^{\varepsilon _n}$ is a solution of $(P_w)$ reads as

$$\begin{aligned} -\Delta _p u_w^{\varepsilon _n}= f(x,u_w^{\varepsilon _n}(x), \nabla w(x) ), \end{aligned}$$

so we get

$$\begin{aligned} -\Delta _p z_n= h_n(x) z_n^{p-1}. \end{aligned}$$

(2.3)

Notice also that by invoking assumptions $(f_2)_+$ and $(f_3)_+$ and using the uniform convergence of $\{ u_w^{\varepsilon _n}\}_n$ to zero, we deduce (for a possibly larger $\zeta _{M}$)

$$\begin{aligned} \lambda _1<\eta _M-\sigma \le \frac{f(x,u_w^{\varepsilon _n}(x), \nabla w(x) )}{(u_w^{\varepsilon _n}(x))^{p-1}}\le \zeta _M, \end{aligned}$$

that is

$$\begin{aligned} \lambda _1<\eta _M-\sigma \le h_n(x)\le \zeta _M \end{aligned}$$

a.e. $x\in \Omega$, whenever $n$ is sufficiently large. From the above inequality we get that $h_n$ is bounded in $L^{\infty }(\Omega )$. Consequently, there exists a function $h\in L^{p'}(\Omega )$ such that $h_n\rightharpoonup h$ in $L^{p'}(\Omega )$. By Mazur’s Lemma (see [8, Chapter II]) we derive that

$$\begin{aligned} \lambda _1<\eta _M-\sigma \le h(x)\le \zeta _M \end{aligned}$$

(2.4)

a.e. $x\in \Omega$. On the other hand, (2.3) implies that $\{z_n\}_n$ strongly converges to some $z$ in $W^{1,p}_0(\Omega )$. Here we use that $\{h_n\}_n$ is bounded in $L^\infty (\Omega )$ and the $(S_+)$ property of the $p$-Laplacian operator. The strong convergence ensures that $\Vert z\Vert =1$, so $z\ne 0$. Passing to the limit in (2.3) yields that $z$ verifies

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _p z= h z^{p-1} &{}\quad \text{ in } \quad \Omega \\ z=0 &{}\quad \text{ on }\quad \partial \Omega . \end{array} \right. \end{aligned}$$

Therefore $1$ is an eigenvalue of this weighted eigenvalue problem. From (2.4) and Remark 2.1, we have that $1=\hat{\lambda }_1(\lambda _1)>\hat{\lambda }_1 (h)$ which, again from Remark 2.1, implies that $z$ changes sign, a contradiction. So, we have proved that $u_w\ne 0$.

Classical regularity results enable us to derive that $u_w\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$. In order to conclude the proof it remains to show that $u_w$ is the smallest positive solution of $(P_w)$. It is enough to prove that it is the smallest positive solution in the ordered interval $[0, \overline{u}_w]$. To this end, fix a solution $v$ of $(P_w)$ in $[0, \overline{u}_w]$. In particular, $v$ is a super solution of $(P_w)$ and we can choose $\varepsilon _n>0$ such that $v-\varepsilon _n\phi _1\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$. Then we infer that $\varepsilon _n\phi _1\le u_w^{\varepsilon _n}\le v\le \overline{u}_w$ for $n\in \mathbb {N}$ large enough. Letting $n\rightarrow \infty$, we are led to $u_w\le v$ as we wished. $\square$

2.2 Existence result via Schaefer’s fixed point theorem

Let us recall the well known (see, e.g., [13, Theorem 4.27 ])

Theorem 2.2

(Schaefer’s fixed point theorem) Let $X$ be a Banach space and let $T:X \longrightarrow X$ be a continuous and compact map. Assume that the set

$$\begin{aligned} \left\{ u\in X:\ u=\lambda T(u) \quad \text{ for } \text{ some } \quad \lambda \in [0,1]\right\} \end{aligned}$$

is bounded. Then $T$ has a fixed point.

Throughout the rest of the paper, assumptions $(f_1), (f_2)_\pm , (f_3)_\pm$ hold.

In view of Theorem 2.1, it is well defined the map $T: C^1_0(\overline{\Omega })\rightarrow C^1_0(\overline{\Omega })$ given by

$$\begin{aligned} T(w)=u_w, \end{aligned}$$

where $u_w$ is the smallest positive solution of $(P_w)$ as guaranteed by Theorem 2.1.

It is clear that a fixed point for $T$ will provide a positive solution to the original problem $(P)$. In order to prove the continuity and the compactness of the map $T$ we will need some preliminary results. For any $w\in C^1_0(\overline{\Omega })$, denote by $S_w$ the set of all functions $u\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$ that are solutions of problem $(P_w)$.

Lemma 2.3

If $\{w_n\}_n$ is a bounded sequence in $C^1_0(\overline{\Omega })$ and $\{u_n\}_n$ is a sequence in $C^1_0(\overline{\Omega })$ with $u_n\in S_{w_n}$ for all $n$, then $\{u_n\}_n$ is relatively compact in $C^1_0(\overline{\Omega })$.

Proof

Let $M>0$ satisfy $\Vert w_n\Vert _{C^1}\le M$ for all $n$. We claim that there exists a subsequence of $\{u_n\}_n$ converging in $C^1_0(\overline{\Omega })$. From assumption $(f_1)$ it follows that

$$\begin{aligned} \Vert u_n\Vert ^p&= \int _\Omega f(x,u_n,\nabla w_n)u_n\,dx\\&\le k_M|\Omega |^{\frac{1}{p'}}\Vert u_n\Vert _p+\theta _M\lambda _1^{-1}\Vert u_n\Vert ^p\\&\le k_M|\Omega |^{\frac{1}{p'}} \lambda _1^{-\frac{1}{p}}\Vert u_n\Vert +\theta _M\lambda _1^{-1}\Vert u_n\Vert ^p, \end{aligned}$$

which implies that $\{u_n\}_n$ is bounded in $W^{1,p}_0(\Omega )$ because $\theta _M<\lambda _1$. On the basis of classical regularity results, we get that $\{u_n\}_n$ is bounded in $C^{1,\alpha }(\overline{\Omega })$ for some $\alpha \in (0,1)$ independent of $n$ (due to the assumption that $\{w_n\}_n$ is bounded in $C^1_0(\overline{\Omega })$). Since $C^{1,\alpha }(\overline{\Omega })$ is compactly embedded into $C^{1}(\overline{\Omega })$, we achieve our claim. $\square$

Lemma 2.4

If $\{w_n\}_n$ is a sequence in $C^1_0(\overline{\Omega })$ such that $w_n\rightarrow w$ in $C^1_0(\overline{\Omega })$ and if $\{u_n\}_n$ is a sequence in $C^1_0(\overline{\Omega })$ with $u_n\in S_{w_n}$ for all $n$, then there exist a subsequence $\{u_{n_k}\}_k$ and an element $u\in S_w$ such that $u_{n_k}\rightarrow u$ in $C^1_0(\overline{\Omega })$.

Proof

From Lemma 2.3, there exist a subsequence $\{u_{n_k}\}_k$ and some $u\in C^1_0(\overline{\Omega })$ such that $u_{n_k} \longrightarrow u$ in $C_0^1(\overline{\Omega })$. We wish to prove that $u\in S_w$. We note that

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _p u_{n_k}= f(x,u_{n_k}, \nabla w_{n_k}) &{}\quad \text{ in } \quad \Omega \\ u_{n_k}=0 &{}\quad \text{ on } \quad \partial \Omega \end{array} \right. \end{aligned}$$

for all $k \in \mathbb {N}$. So, for every $\varphi \in W^{1,p}_0(\Omega )$ and $k\in \mathbb {N}$ we obtain

$$\begin{aligned} \int _\Omega |\nabla u_{n_k}|^{p-2}\nabla u_{n_k}\nabla \varphi \,dx= \int _\Omega f(x,u_{n_k}, \nabla w_{n_k})\varphi \,dx. \end{aligned}$$

Passing to the limit as $k\rightarrow \infty$ we deduce

$$\begin{aligned} \int _\Omega |\nabla u|^{p-2}\nabla u\nabla \varphi \,dx= \int _\Omega f(x,u, \nabla w)\varphi \,dx, \end{aligned}$$

i.e. $u$ is a solution of $(P_w)$. In order to prove that $u\ne 0$ it is enough to make use of the same argument as in the corresponding part of the proof of Theorem 2.1. Then classical regularity results and maximum principle imply $u\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$, which completes the proof. $\square$

The following is the key Lemma in our construction.

Lemma 2.5

If $\{w_n\}_n$ is a sequence in $C_0^1(\overline{\Omega })$ such that $w_n \rightarrow w$ in $C^1_0(\overline{\Omega })$. Then, for any $v \in S_w$ there exist $v_n \in S_{w_n}$ such that

$$\begin{aligned} v_n \rightarrow v \quad \hbox {in} \ C^1_0(\overline{\Omega }). \end{aligned}$$

Proof

Fix $n \in \mathbb {N}$, and let $z_n^0$ be the unique solution of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _p u= f(x,v, \nabla w_n) &{}\quad \text{ in } \quad \Omega \\ u=0 &{}\quad \text{ on } \quad \partial \Omega . \end{array} \right. \end{aligned}$$

Exploiting assumption $(f_2)_+$ and bearing in mind that $v\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$ observe that $f(\cdot , v(\cdot ), \nabla w_n(\cdot )) \not \equiv 0$, which leads to $z_n^0 \not = 0$.

Let us prove that $z_n^0$ lies in $\mathrm {int}(C^1_0(\overline{\Omega })_+)$. First, it is straightforward to establish that $\{z_n^0\}_n$ is bounded in $W^{1,p}_0(\Omega )$, thus it is bounded in $L^\infty (\Omega )$ (see [9]), and therefore in $C^{1,\alpha }(\overline{\Omega })$ for some $\alpha \in (0,1)$ (see [10]). Since $C^{1,\alpha }(\overline{\Omega })$ is compactly embedded in $C^1(\overline{\Omega })$, there exists a subsequence $\{z_{n_p}^0\}_p$ strongly convergent in $C_0^1(\overline{\Omega })$ to a solution of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _p u= f(x,v, \nabla w) &{}\quad \text{ in } \quad \Omega \\ u=0 &{}\quad \text{ on } \quad \partial \Omega . \end{array} \right. \end{aligned}$$

Taking into account that $v$ is the unique solution of the above problem, we get

$$\begin{aligned} \lim _{p \rightarrow \infty } z_{n_p}^0 = v \quad \hbox {in} \ C^1_0(\overline{\Omega }). \end{aligned}$$

Actually, as readily seen, the strong convergence is true for the whole sequence

$$\begin{aligned} \lim _{n \rightarrow \infty } z_{n}^0 = v \quad \hbox {in} \ C^1_0(\overline{\Omega }), \end{aligned}$$

which implies the desired assertion.

Now, let us consider the unique positive solution $z_n^1$ of the following problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _p u= f(x,z_n^0, \nabla w_n) &{}\quad \text{ in } \quad \Omega \\ u=0 &{}\quad \text{ on } \quad \partial \Omega . \end{array} \right. \end{aligned}$$

As before we infer that

$$\begin{aligned} \lim _{n \rightarrow \infty } z_n^1 = v \quad \hbox {in} \ C^1_0(\overline{\Omega }). \end{aligned}$$

Inductively, we can define, for each $n \in \mathbb {N}$, $z_n^k$ in $C^1_0(\overline{\Omega })$ as the unique solution of the problem

$$\begin{aligned} (P_n^k) \left\{ \begin{array}{ll} -\Delta _p u= f(x,z_n^{k-1}, \nabla w_n) &{}\quad \text{ in } \quad \Omega \\ u=0 &{}\quad \text{ on } \quad \partial \Omega , \end{array} \right. \end{aligned}$$

and for each $k \in \mathbb {N}$ it follows that

$$\begin{aligned} \lim _{n \rightarrow \infty } z_n^k = v \quad \hbox {in} \ C^1_0(\overline{\Omega }). \end{aligned}$$

Let us now prove that there exists a constant $c>0$ such that $\Vert z_n^k\Vert \le c$ for all $n,k\in \mathbb {N}$. Indeed, setting $M=\max \{\sup _n\Vert w_n\Vert _{C^1},\Vert w\Vert _{C^1} \}$, by hypothesis $(f_1)$ we have that

$$\begin{aligned} \Vert z_n^k\Vert ^p=\int _\Omega f(x, z_n^{k-1}, \nabla w_n)z_n^k\,dx\le \left[ k_M|\Omega |^{\frac{1}{p'}}+ \theta _M \left( \int _\Omega (z_n^{k-1})^{p}\,dx\right) ^{\frac{p-1}{p}}\right] \lambda _1^{-\frac{1}{p}}\Vert z_n^k\Vert , \end{aligned}$$

so

$$\begin{aligned} \Vert z_n^k\Vert ^{p-1}\le \lambda _1^{-\frac{1}{p}}\left[ k_M|\Omega |^{\frac{1}{p'}} + \theta _M\Vert z_n^{k-1}\Vert ^{p-1}_{L^{p}} \right] \le \lambda _1^{-\frac{1}{p}}k_M|\Omega |^{\frac{1}{p'}} + \lambda _1^{-1} \theta _M\Vert z_n^{k-1}\Vert ^{p-1}. \end{aligned}$$

If we set $c_1=\lambda _1^{-\frac{1}{p}}k_M|\Omega |^{\frac{1}{p'}}$ and $c_2= \lambda _1^{-1} \theta _M$, it is easy to see by induction that

$$\begin{aligned} \Vert z_n^k\Vert ^{p-1}\le c_1\left( \sum _{j=0}^{k-1}c_2^j\right) + c_2^k \Vert z_n^0\Vert ^{p-1} \end{aligned}$$

for every $k\in \mathbb {N}$. Since $c_2<1$ (in view of assumption $(f_1)$) and $\{z_n^0\}_n$ is bounded in $W^{1,p}_0(\Omega )$, we deduce that, for every $n, k\in \mathbb {N}$, there holds

$$\begin{aligned} \Vert z_n^k\Vert ^{p-1}\le c_1\left( \sum _{j=0}^{\infty }c_2^j\right) + c_3, \end{aligned}$$

with a constant $c_3>0$. This entails our claim.

By standard arguments as already used before, we have that the net $\{z_n^k\}_{n,k}$ is relatively compact in $C_0^1(\overline{\Omega })$. Then up to a subnet, we can assume that there exists $u \in C_0^1(\overline{\Omega })$ such that

$$\begin{aligned} \lim _{n,k\rightarrow \infty } z_n^k = u, \end{aligned}$$

that is, there exists $N \in \mathbb {N}$ such that

$$\begin{aligned} \Vert z_n^k - u\Vert _{C^1}< \varepsilon \end{aligned}$$

(2.5)

for all $n,k >N$. At this point, for every $k > N$, we obtain that $\limsup _{n \rightarrow \infty }\Vert z_n^k - u\Vert _{C^1} \le \varepsilon$. We are thus able to select a subsequence $\{z_{n_p}\}_p$ with the property

$$\begin{aligned} \lim _{p \rightarrow \infty }\Vert z_{n_p}^k - u\Vert _{C^1} = \Vert \lim _{p \rightarrow \infty }z_{n_p}^k - u\Vert _{C^1}=\Vert v - u\Vert _{C^1} \le \varepsilon , \end{aligned}$$

thereby $u=v$.

Consequently, according to (2.5), for all $n,k>N$ we have

$$\begin{aligned} \Vert z_n^k - v\Vert _{C^1}< \varepsilon . \end{aligned}$$

(2.6)

Now, for each $n \in \mathbb {N}$ there exist $\{k_s=k_s(n)\}_s \subseteq \mathbb {N}$ and $v_n \in C_0^1(\overline{\Omega })$ satisfying

$$\begin{aligned} \lim _{s \rightarrow \infty } z_n^{k_s} = v_n \ \hbox {in } C_0^1(\overline{\Omega }). \end{aligned}$$

Clearly, $v_n$ is a solution of $(P_{w_n})$. Let us prove that

$$\begin{aligned} \lim _{n \rightarrow \infty } v_n = v \ \hbox {in } C_0^1(\overline{\Omega }). \end{aligned}$$

If not, we can construct a subsequence $\{n_p\}_p$ with $n_p >N$ such that $\Vert v_{n_p}-v\Vert _{C^1}>\varepsilon >0$ for all $p \in {\mathbb {N}}$ and some $\varepsilon >0$. This amounts to saying that

$$\begin{aligned} \varepsilon < \Vert v_{n_p}-v\Vert _{C^1} = \Vert \lim _{s \rightarrow \infty } z_{n_p}^{k_s} - v\Vert _{C^1} = \lim _{s \rightarrow \infty }\Vert z_{n_p}^{k_s} - v\Vert _{C^1}. \end{aligned}$$

In particular, for any $p \in \mathbb {N}$ there exists $s_p=s_p(n_p)\in \mathbb {N}$ such that $k_{s_p}>N$ and

$$\begin{aligned} \Vert z_{n_p}^{k_{s_p}} - v\Vert _{C^1} > \varepsilon , \end{aligned}$$

against (2.6).

Recall that $v\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$. Since the convergence of $\{v_n\}_n$ to $v$ is uniform on compact subsets of $\Omega$, it follows that $v_n>0$ in $\Omega$ whenever $n$ is sufficiently large. Also, since $\{\frac{\partial v_n}{\partial \nu }\}_n$ converges uniformly to $\frac{\partial v}{\partial \nu }$ on $\partial \Omega$ we obtain that $v_n \in \mathrm {int}(C^1_0(\overline{\Omega })_+)$ for $n$ large enough. This completes the proof. $\square$

3 Proofs of main results

In the present section, in order to apply Schaefer’s theorem, we replace $(f_1)$ with the stronger assumption $(\widetilde{f_1})$.

Proof of Theorem 1.1

We wish to prove that the map $T$ introduced in the previous section is continuous and compact. First, let us note that the map $T$ is compact, i.e. for any sequence $\{w_n\}_n$ bounded in $C^1_0(\overline{\Omega })$, $\{T(w_n)\}_n$ is relatively compact in $C^1_0(\overline{\Omega })$. This follows readily from Lemma 2.3 applied to $u_n=T(w_n)\in S_{w_n}$.

Let us prove now that $T$ is continuous. Let $\{w_n\}_n$ be a sequence in $C_0^1(\overline{\Omega })$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } w_n = w \quad \hbox {in } C^1_0(\overline{\Omega }) \end{aligned}$$

and, for every $n \in \mathbb {N}$, set $u_n = T(w_n)$. By Lemma 2.4, there exist a subsequence $\{u_{n_k}\}_k$ and $u\in S_w$ that fulfill

$$\begin{aligned} \lim _{k \rightarrow \infty } u_{n_k} =u \quad \hbox {in } C^1_0(\overline{\Omega }). \end{aligned}$$

We need to check that $u$ is the smallest positive solution of $(P_{w})$. Fix a positive solution $v$ of $(P_w)$. By Lemma 2.5, there exists $v_n\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$ (positive) solution of $(P_{w_n})$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } v_n = v \quad \hbox {in} \ C^1_0(\overline{\Omega }). \end{aligned}$$

Since $u_{n_k}$ is the smallest positive solution of $(P_{w_{n_k}})$, we have that

$$\begin{aligned} u_{n_k} \le v_{n_k} \quad \hbox {for} \,\,\hbox {all} \quad \ k \in \mathbb {N}. \end{aligned}$$

Passing to the limit yields $u \le v$. This means that

$$\begin{aligned} \lim _{k \rightarrow \infty } T(w_{n_k}) =T(w) \quad \hbox {in} \ C^1_0(\overline{\Omega }). \end{aligned}$$

In fact, the whole sequence $T(w_n)$ converges to $T(w)$. If not, there exists a subsequence $\{w_{n_p}\}_p$ such that $\Vert T(w_{n_p})-T(w)\Vert _{C^1}\ge \varepsilon$ for every $p\in \mathbb {N}$ and for some $\varepsilon >0$. Arguing as above, we find a subsequence $u_{n_{p_r}}=T(w_{n_{p_r}})$ converging to $T(w)$, which gives rise to a contradiction. It is thus proven that $T$ is continuous.

Now we check that the set

$$\begin{aligned} \{w\in C^1_0(\overline{\Omega }):\ w=\lambda T(w)\ \quad \text{ for } \,\, \text{ some }\quad \lambda \in [0,1]\} \end{aligned}$$

(3.1)

is bounded in $C^1_0(\overline{\Omega })$. So let $w\in C^1_0(\overline{\Omega })$ and $\lambda \in [0,1]$ be such that $w=\lambda T(w)$. We may assume that $\lambda >0$ (otherwise, $w=0$). Then, $w$ is a solution of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _p w= \tilde{f}_\lambda (x,w, \nabla w) &{}\quad \text{ in } \quad \Omega \\ w=0 &{}\quad \text{ on } \quad \partial \Omega , \end{array} \right. \end{aligned}$$

where $\tilde{f}_\lambda (x,s,\xi )=\lambda ^{p-1}f(x,\lambda ^{-1}s,\xi )$. By $(\widetilde{f_1})$, the Carathéodory function $\tilde{f}_\lambda$ satisfies the growth condition

$$\begin{aligned} |\tilde{f}_\lambda (x,s,\xi )|\le k_0+\theta _0|s|^{p-1}+\theta _1|\xi |^{p-1} \end{aligned}$$

(3.2)

for a.a. $x\in \Omega$, all $s\in \mathbb {R}$, and all $\xi \in \mathbb {R}^N$. Notice that the coefficients in (3.2) are independent of $\lambda \in (0,1]$. We claim that there is $M>0$ independent of $w$ and $\lambda$ such that

$$\begin{aligned} \Vert w\Vert \le M. \end{aligned}$$

(3.3)

To see this, acting with the test function $w$ and using (3.2), we obtain the following estimate

$$\begin{aligned} \Vert w\Vert ^p&= \int _\Omega \tilde{f}_\lambda (x,w,\nabla w)w\,dx \le \int _\Omega \big (k_0w+\theta _0|w|^p+\theta _1|\nabla w|^{p-1}|w|\big )\,dx \\&\le k_0|\Omega |^{\frac{1}{p'}}\lambda _1^{-\frac{1}{p}}\Vert w\Vert +\theta _0\lambda _1^{-1}\Vert w\Vert ^p+\theta _1\lambda _1^{-\frac{1}{p}}\Vert w\Vert ^{p}. \end{aligned}$$

Since $1>\theta _0\lambda _1^{-1}+\theta _1\lambda _1^{-\frac{1}{p}}$ (see $(\widetilde{f_1})$) and $p>1$, we get (3.3). In view of (3.2), (3.3), classical regularity theory (cf. [9, 10]) yields $M'>0$ independent of $w$ and $\lambda$ such that $\Vert w\Vert _{C^1}\le M'$. This establishes the boundedness of the set in (3.1).

Observe that any fixed point of the map $T$ is a solution of problem $(P)$ in the interior of the cone of nonnegative functions by construction, the thesis follows by applying Schaefer’s fixed point theorem (Theorem 2.2). $\square$

Mutatis mutandis we can show that the map assigning to every $w\in C^1_0(\overline{\Omega })$ the biggest negative solution of $(P_w)$ (its existence can be proved similarly to that of the smallest positive solution) is also continuous and compact. Finally, we can conclude as in the case of positive solutions to achieve the results in Theorems 1.2 and 1.3.

Proof of Corollary 1.1

From Theorem 1.1 we know that there exists a solution $u\in \mathrm {int}(C^1_0(\overline{\Omega })_+$) which can be regarded as a supersolution of $(P)$.

By virtue of $(f_2)_+$ we can find $\varepsilon _0>0$ such that

$$\begin{aligned} -\Delta _p(\varepsilon \phi _1)=\lambda _1(\varepsilon \phi _1)^{p-1} \le f(x,\varepsilon \phi _1,\varepsilon \nabla \phi _1) \end{aligned}$$

provided $0<\varepsilon \le \varepsilon _0$. This expresses that $\varepsilon \phi _1$ is a subsolution of $(P)$ for all $0<\varepsilon \le \varepsilon _0$. In addition, since $u, \phi _1\in \mathrm {int}(C^1_0(\overline{\Omega })_+$), we may choose $\varepsilon _0$ so small that $u-\varepsilon \phi _1\in \mathrm {int}(C^1_0(\overline{\Omega })_+)$ for every $0<\varepsilon \le \varepsilon _0$.

We are thus in the position to apply [4, Theorem 3.22], which yields the existence of the smallest positive solution $u_\varepsilon$ of problem $(P)$ in the ordered interval $[\varepsilon \phi _1,u]$ for all $0<\varepsilon \le \varepsilon _0$. Set $u_n:=u_{\frac{1}{n}}$. Thanks to the choice of $u_n$, we note that the sequence $\{u_n\}_{n}$ is decreasing. So, there exists $u_0\in C^1_0(\overline{\Omega })_+$ such that $u_n\rightarrow u_0$ in $C^1_0(\overline{\Omega })$ and $u_0$ is a solution of $(P)$. On the basis of hypotheses $(f_2)_+$ and $(f_3)_+$ we also infer that $u_0\not =0$ (see the proof of Theorem 2.1). This enables us to apply the strong maximum principle to obtain that $u_0\in \mathrm {int}(C^1_0(\overline{\Omega })_+$).

Now we show that $u_0$ is the smallest positive solution. To this end, let $v$ be a positive solution of problem $(P)$. The nonlinear regularity theory and strong maximum principle ensure that $v\in \mathrm {int} (C^1_0(\overline{\Omega })_+$). It follows that

$$\begin{aligned} \frac{1}{n}\phi _1\le \min \{u,v\}\le u \end{aligned}$$

whenever $n$ is sufficiently large. By [4, Theorem 3.22] we see that there exists a solution $v_n$ of $(P)$ with $v_n\in [\frac{1}{n}\phi _1, \min \{u,v\}]$ because $\min \{u,v\}$ is a supersolution. Then the minimality property of $u_n$ entails that $u_n\le v_n\le v$. Letting $n\rightarrow \infty$ leads to $u_0\le v$.

In an analogous way we can proceed to justify the existence of the biggest negative solution of problem $(P)$. $\square$

Finally, we provide a simple example of nonlinearity $f(x,s,\xi )$ which fulfills our hypotheses.

Example 3.1

Let $g:\overline{\Omega }\times \mathbb {R}^N\rightarrow \mathbb {R}$ be a continuous, positive function. Then, for any continuous function $f: \overline{\Omega }\times \mathbb {R} \times \mathbb {R}^N\rightarrow \mathbb {R}$ satisfying the growth condition $(\widetilde{f_1})$ and

$$\begin{aligned} f(x,s,\xi )= |s|^{p-2}s(\lambda _1+g(x,\xi )) \quad \hbox {for } \ |s| \hbox { small}, \end{aligned}$$

Theorem 1.3 applies. Indeed, it is readily seen that hypotheses $(f_2)_{\pm }, (f_3)_{\pm }$ hold true.