1 Introduction

Mathematically, a positione is a particular kind of eigenvalue problem involving a nonlinear function on the reals that is continuous, positive, and monotone. A semipositone is an eigenvalue problem that would be a positone eigenvalue problem except that the nonlinear function is not positive when its argument is zero.

Semipositone problems naturally arise in various studies. For example, consider the Rozenwig–McArthur equations in the analysis of competing species where “harvesting” takes place. The study of positive solutions to these problems, unlike the positone case, turns into a nontrivial question as 0 is not a subsolution, making the method of sub-supersolutions difficult to apply. Semipositone problems, again unlike positone problems, give rise to the interesting phenomenon of symmetry breaking (see [8]).

Consider the nonlinear eigenvalue problems of the form

$$ \textstyle\begin{cases} -\Delta _{p}u=\lambda f(u)&\text{in } \Omega , \\ u=0&\text{on } \partial \Omega . \end{cases} $$
(1.1)

When f is positive and monotone, it is referred to in the literature as a positone problem. The case where f satisfies, \(f(0) < 0\), f is monotone and eventually positive, is referred to in the literature as a semipositone problem. The study of positive solutions to semipostone problems is considerably more challenging, since the range of a solution must include regions where f is negative as well as where f is positive. The study of semipositone problems was first formally introduced by Castro et al. in 1988 (see [7]) in the case of Dirichlet boundary conditions, where several challenging differences were noted in their study when compared to the study of positone problems.

Perera et al. [16] consider the p-superlinear semipositone p-Laplacian problem

$$ \textstyle\begin{cases} -\Delta _{p}u=u^{q-1}-\mu &\text{in } \Omega , \\ u>0&\text{in } \Omega , \\ u=0&\text{on } \partial \Omega \end{cases} $$

and proved the the existence of ground-state positive solutions (see [4–6, 9] for other cases).

Alves et al. [2] prove the existence of a solution for the class of the semipositone problem

$$ \textstyle\begin{cases} -\Delta u =h(x)(f(u)-a)&\text{in } \mathbb{R}^{N}, \\ u>0&\text{in } \mathbb{R}^{N}, \end{cases} $$

via the variational method together with estimates that involve the Riesz potential (see also [1, 10, 11, 21]).

Fu et al. [14] prove the existence of positive solutions for a class of semipositone problems with singular Trudinger–Moser nonlinearities. The proof is based on compactness and regularity arguments.

Castro et al. [6] study the existence of positive weak solutions to the problem (1.1). Here, we refer to [6] and study the existence of positive weak solutions to the problem

$$ \textstyle\begin{cases} -\Delta _{\overrightarrow{p}}u=\lambda f(u)&\text{in } \Omega , \\ u=0&\text{on } \partial \Omega , \end{cases} $$
(1.2)

where \(-\Delta _{\overrightarrow{p}}\) is the anisotropic p-Laplace operator, Ω is an open smooth bounded domain in \(\mathbb{R}^{N}\), \(N\geq 2\) and the function \(f:\mathbb{R}\to \mathbb{R}\) is a differentiable function with \(f(0)<0\) (semipositone), which implies that \(u = 0\) is not a subsolution to (1.2), making the finding of positive solutions rather challenging (see [15]).

We set \(\overrightarrow{p}:=(p_{1},\ldots , p_{N})\), where

$$ \begin{gathered} 1< p_{1}, p_{2},\ldots ,p_{N},\quad \sum^{N}_{i=1} \frac{1}{p_{i}}>1, \end{gathered} $$

\(p_{+}:=\max \{p_{i}: i=1,\ldots , N\}\) and \(p_{-}:=\min \{p_{i}: i=1,\ldots , N\}\).

Let p̅ denote the harmonic means \(\overline{p}= N/ (\sum^{N}_{i=1}\frac{1}{p_{i}} )\), and define

$$ \begin{aligned} p^{\star }:= \frac{N}{ (\sum^{N}_{i=1}\frac{1}{p_{i}} )-1}= \frac{N\overline{p}}{N-\overline{p}} \quad \text{and}\quad p_{\infty}:=\max \bigl\{ p_{+},p^{\star }\bigr\} . \end{aligned} $$

Here and after, we assume \(p_{+}< p^{\star}\). Thus, \(p_{\infty}=p^{\star}\):

\((H_{1})\):

Suppose there exist \(q\in (p_{+}-1,p^{\star}-1)\), \(A>0\), \(B>0\) such that

$$ \textstyle\begin{cases} A(u^{q}-1)\leq f(u)\leq B(u^{q}-1)&\text{for } u>0, \\ f(u)=0&\text{for } u\leq -1. \end{cases} $$
(1.3)
\((H_{2})\):

Assume an Ambrosetti–Rabinowitz-type condition, i.e., that there exist \(\theta >p_{+}\) and \(M\in \mathbb{R}\) such that

$$ uf(u)\geq \theta F(u)+M, $$
(1.4)

where

$$ F(u)= \int _{0}^{u}f(s)\,ds . $$

Remark 1.1

Equation (1.3) implies that there exist positive real numbers \(A_{1}\), \(B_{1}\) such that

$$ F(u)\leq B_{1}\bigl( \vert u \vert ^{q+1}+1\bigr)\quad \text{for all } u\in \mathbb{R} $$
(1.5)

and

$$ F(u)\geq A_{1}\bigl( \vert u \vert ^{q+1}+1\bigr)\quad \text{for all } u\in \mathbb{R}. $$
(1.6)

With respect to the above, the main result of this paper is Theorem 1.2. Our result extends the result of [5, Theorem 1.1] and [6, Theorem 1.1].

Theorem 1.2

There exists \(\lambda ^{*}>0\) such that if \(\lambda \in (0,\lambda ^{*})\), then the problem (1.2) has a positive weak solution \(u_{\lambda}\in L^{\infty}(\overline{\Omega})\).

The rest of the paper is organized as follows. In Sect. 2, the suitable function space that is the anisotropic Sobolev space is recalled and necessary facts are also recalled. In Sect. 3, we study the Mountain-Pass Theorem and Palais–Smale condition for the problem. In Sect. 4, we present the proof of the main result, Theorem 1.2, which shows the existence of a positive solution of the problem (1.2).

2 Function spaces

Here, we define the anisotropic Sobolev spaces (see [18–20] and references therein), to which the solutions for our problems naturally belong, by

$$ \textstyle\begin{cases} W^{1,\overrightarrow{p}} (\Omega ):= \{u\in W^{1,1}(\Omega ): \int _{\Omega } \vert \frac{\partial u}{\partial x_{i}} \vert ^{p_{i}}< \infty , i=1,\ldots , N \}, \\ W^{1,\overrightarrow{p}}_{0}(\Omega )=W^{1,\overrightarrow{p}}( \Omega )\cap W^{1,1}_{0}(\Omega ) \end{cases} $$
(2.1)

with the norm

$$ \Vert u \Vert _{W^{1,\overrightarrow{p}}(\Omega )}:= \int _{ \Omega } \bigl\vert u(x) \bigr\vert \,dx +\sum _{i=1}^{N} \biggl( \int _{\Omega } \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}}. $$

We consider \(W^{1,\overrightarrow{p}}_{0}(\Omega )\) endowed with the norm

$$ \begin{aligned} \Vert u \Vert _{W^{1,\overrightarrow{p}}_{0}(\Omega )}&:=\sum _{i=1}^{N} \biggl( \int _{\Omega } \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}} \\ &=\sum_{i=1}^{N} \Vert u \Vert _{W^{1,p_{i}}_{0}(\Omega )}. \end{aligned} $$

We recall the following theorem [13, Theorem 1].

Theorem 2.1

Let \(\Omega \subset \mathbb{R}^{N}\) be an open bounded domain with a Lipschitz boundary. If

$$ p_{i}>1, \quad \textit{for all } i=1,\ldots ,N,\qquad \sum _{i=1}^{N} \frac{1}{p_{i}}>1, $$

then for all \(r\in [1,p^{*}]\), there is a continuous embedding \(W_{0}^{1,\overrightarrow{p}}(\Omega )\subset L^{r}(\Omega )\). For \(r< p^{*}\), the embedding is compact.

Definition 2.2

An element \(u\in W_{0}^{1,\overrightarrow{p}}(\Omega )\) is called a weak solution to (1.2), if

$$ \sum_{i=1}^{N} \int _{\Omega } \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}-2} \frac{\partial u}{\partial x_{i}}\frac{\partial \phi}{\partial x_{i}}\,dx = \lambda \int _{\Omega }f(u)\phi \,dx $$
(2.2)

for all \(\phi \in W_{0}^{1,\overrightarrow{p}}(\Omega )\).

Associated to (1.2) we have the functional \(J_{\lambda}:W_{0}^{1,\overrightarrow{p}}(\Omega )\to \mathbb{R}\) defined by

$$ J_{\lambda}(u):=\sum_{i=1}^{N} \frac{1}{p_{i}} \int _{\Omega } \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx -\lambda \int _{\Omega }F\bigl(u(x)\bigr) \,dx . $$
(2.3)

Remark 2.3

\(J_{\lambda}\) is a functional of class \(C^{1}\) and the critical points of the functional \(J_{\lambda}\) are the weak solutions of (1.2) (see [17] for a similar argument).

By the Mountain-Pass Theorem we can prove the existence of one solution of (1.2) and then we show for the proper value of λ that the solution is positive.

3 Mountain-Pass Theorem and Palais–Smale condition

The next two lemmas prove that \(J_{\lambda}\) satisfies the geometric hypotheses of the Mountain-Pass Theorem.

Lemma 3.1

Assume \(\phi \in W_{0}^{1,\overrightarrow{p}}(\Omega )\) denotes a positive differentiable function with \(\|\phi \|_{W_{0}^{1,\overrightarrow{p}}(\Omega )}=1\). There exists \(\lambda _{1}>0\) such that if \(\lambda \in (0,\lambda _{1})\), then \(J_{\lambda}(c\lambda ^{-r}\phi )\leq 0\), where \(r=\frac{1}{q+1-p_{+}}>0\), \(c=((N+1)p_{-}^{-1}A_{1}^{-1}\|\phi \|_{q+1}^{-q-1})^{r}\) and \(A_{1}\) is given by (1.6).

Proof

Since

$$ \Vert \phi \Vert _{W^{1,\overrightarrow{p}}_{0}(\Omega )}=\sum_{i=1}^{N} \biggl( \int _{\Omega } \biggl\vert \frac{\partial \phi}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}}=1, $$

then \(\int _{\Omega } \vert \frac{\partial \phi}{\partial x_{i}} \vert ^{p_{i}}\,dx \leq 1\) for all \(i=1,\ldots ,N\). Also, \(p_{i}>1\), therefore

$$ \int _{\Omega } \biggl\vert \frac{\partial \phi}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \leq \biggl( \int _{\Omega } \biggl\vert \frac{\partial \phi}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}}, $$

hence,

$$ \sum_{i=1}^{N} \biggl( \int _{\Omega } \biggl\vert \frac{\partial \phi}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)\leq \sum_{i=1}^{N} \biggl( \int _{\Omega } \biggl\vert \frac{\partial \phi}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}}. $$

Let \(s=c\lambda ^{-r}\), then by (1.6), we have

$$ \begin{aligned} J_{\lambda}(s\phi )& = \sum _{i=1}^{N} \int _{\Omega } \frac{ \vert \frac{\partial (s\phi )}{\partial x_{i}} \vert ^{p_{i}}}{p_{i}}\,dx - \lambda \int _{\Omega }F(s\phi ) \,dx \\ & = \sum_{i=1}^{N}\frac{s^{p_{i}}}{p_{i}} \int _{\Omega } \biggl\vert \frac{\partial \phi}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx -\lambda \int _{\Omega }F(s \phi ) \,dx \\ & \leq \frac{\sum_{i=1}^{N}s^{p_{i}}}{p_{-}}-\lambda \int _{\Omega }F(s\phi ) \,dx \\ & \leq \frac{\sum_{i=1}^{N}s^{p_{i}}}{p_{-}}-\lambda A_{1} \int _{\Omega }\bigl(s^{q+1}\phi ^{q+1}-1 \bigr) \,dx \\ & \leq \frac{Ns^{p_{+}}}{p_{-}}-\lambda A_{1} \int _{\Omega }\bigl(s^{q+1}\phi ^{q+1}-1 \bigr) \,dx \\ & = \frac{Ns^{p_{+}}}{p_{-}}-A_{1}s^{q+1} \Vert \phi \Vert ^{q+1}_{q+1} \lambda +\lambda A_{1} \vert \Omega \vert \\ & = \biggl\{ \frac{Nc^{p_{+}}\lambda ^{-rp_{+}}}{p_{-}}-A_{1}c^{q+1} \lambda ^{-r(q+1)+1} \Vert \phi \Vert ^{q+1}_{q+1} \biggr\} +\lambda A_{1} \vert \Omega \vert \\ & \leq c^{p_{+}} \biggl\{ \frac{N\lambda ^{-rp_{+}}}{p_{-}}-A_{1}c^{q+1-p_{+}} \lambda ^{-r(q+1)+1} \Vert \phi \Vert ^{q+1}_{q+1} \biggr\} +\lambda A_{1} \vert \Omega \vert . \end{aligned} $$
(3.1)

Thus,

$$ \begin{aligned} J_{\lambda}(s\phi ) & \leq c^{p_{+}} \biggl\{ \frac{N\lambda ^{-rp_{+}}}{p_{-}}-\frac{N+1}{p_{-}}\lambda ^{-r(q+1)+1} \biggr\} +\lambda A_{1} \vert \Omega \vert \\ & = c^{p_{+}}\lambda ^{-rp_{+}} \biggl\{ \frac{N}{p_{-}}- \frac{N+1}{p_{-}}\lambda ^{-r(q+1)+1+rp_{+}} \biggr\} +\lambda A_{1} \vert \Omega \vert \\ & = -\frac{c^{p_{+}}\lambda ^{-rp_{+}}}{p_{-}}+\lambda A_{1} \vert \Omega \vert . \end{aligned} $$
(3.2)

Taking \(\lambda _{1}<\min \{1, (p_{-}A_{1}|\Omega |c^{-p_{+}})^{ \frac{-1}{(1+rp_{+})}}\}\), the lemma is proven. □

Lemma 3.2

Assume \(r=\frac{1}{q+1-p_{+}}>0\). There exists \(\tau >0\), \(c_{1}>0\) and \(\lambda _{2}\in (0,1)\) such that if \(\|u\|_{W_{0}^{1,p_{+}}(\Omega )}=\tau \lambda ^{-\tau}\), then \(J_{\lambda}(u)\geq c_{1}(\tau \lambda ^{-r})^{p_{+}}\) for all \(\lambda \in (0,\lambda _{2})\).

Proof

By the Sobolev embedding Theorem 2.1, there exists \(K_{1}>0\) such that if \(u\in W_{0}^{1,p_{+}}(\Omega )\), then \(\|u\|_{q+1}\leq K_{1}\|u\|_{W_{0}^{1,p_{+}}(\Omega )}\). Assume

$$ \tau =\min \bigl\{ \bigl(2p_{+}K_{1}^{q+1}B_{1} \bigr)^{-r}, c \Vert u \Vert _{W_{0}^{1,p_{+}}( \Omega )}\bigr\} . $$
(3.3)

If \(\|u\|_{W_{0}^{1,p_{+}}(\Omega )}=\tau \lambda ^{-r}\), then

$$ \begin{aligned} J_{\lambda}(u)& = \sum _{i=1}^{N} \frac{1}{p_{i}} \int _{\Omega } \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx -\lambda \int _{\Omega }F(u) \,dx \\ & \geq \frac{1}{p_{+}} \int _{ \Omega } \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{+}}\,dx - \lambda \int _{\Omega }F(u) \,dx \\ & = \frac{\tau \lambda ^{-r}}{p_{+}}-\lambda \int _{\Omega }F(u) \,dx \\ & \geq \frac{(\tau \lambda ^{-r})^{p_{+}}}{p_{+}}- \lambda \int _{\Omega }B_{1} \vert u \vert ^{q+1} \,dx - \lambda \vert \Omega \vert B_{1} \\ & \geq \frac{(\tau \lambda ^{-r})^{p_{+}}}{p_{+}}- \lambda B_{1}K_{1}^{q+1} \Vert u \Vert _{W_{0}^{1,p_{+}}(\Omega )}^{q+1} - \lambda \vert \Omega \vert B_{1} \\ & = \frac{(\tau \lambda ^{-r})^{p_{+}}}{p_{+}}-\lambda B_{1}K_{1}^{q+1}{ \tau ^{(q+1)} \lambda ^{-r(q+1)}} -\lambda \vert \Omega \vert B_{1} \\ & = \lambda ^{-rp_{+}} \biggl\{ \frac{\tau ^{p_{+}}}{2p_{+}}-\lambda ^{1+rp_{+}} \vert \Omega \vert B_{1} \biggr\} \\ & \geq \lambda ^{-rp_{+}}\frac{\tau ^{p_{+}}}{4p_{+}}, \end{aligned} $$
(3.4)

where we have used that \(\tau \leq (2p_{+}K_{1}^{q+1}B_{1})^{-r}\) (see (3.3)). Taking \(c_{1}=\frac{\tau ^{p_{+}}}{4p_{+}}\) and \(\lambda _{2}=\tau ^{\frac{p_{+}}{1+rp_{+}}}(4p_{+}B_{1}|\Omega |)^{- \frac{1}{1+rp_{+}}}\), the lemma is proven. □

Next, using the Mountain-Pass Theorem we prove that (1.2) has a solution \(u_{\lambda}\in W_{0}^{1,\overrightarrow{p}}(\Omega )\).

Lemma 3.3

Let \(\lambda _{3}= \min \{\lambda _{1}, \lambda _{2}\}\). There exists \(c_{2} > 0\) such that, for each \(\lambda \in (0,\lambda _{3})\), the functional \(J_{\lambda}\) has a critical point \(u_{\lambda}\) of mountain-pass type that satisfies \(J_{\lambda}(u_{\lambda})\leq c_{2}\lambda ^{-p_{+}r}\).

Proof

First, we show that \(J_{\lambda}\) satisfies the Palais–Smale condition.

Assume that \(\{u_{n}\}_{n}\) is a sequence in \(W_{0}^{1,\overrightarrow{p}} (\Omega )\) such that \(\{J_{\lambda}(u_{n})\}_{n}\) is bounded and \(J_{\lambda}^{\prime}(u_{n})\to 0\). Hence, there exists \(\nu >0\) such that

$$ \bigl\langle J^{\prime}_{\lambda}(u_{n}),u_{n} \bigr\rangle \leq \Vert u_{n} \Vert _{W_{0}^{1, \overrightarrow{p}} (\Omega )} $$

for \(n\geq \nu \). Thus,

$$ -\sum_{i=1}^{N} \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx -\sum_{i=1}^{N} \biggl( \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}} \biggr)^{ \frac{1}{p_{i}}}\,dx \leq -\lambda \int _{\Omega} f(u_{n})u_{n} \,dx \quad \text{for } n\geq \nu . $$

Let K be a constant such that \(|J_{\lambda}(u_{n})| \leq K\) for all \(n = 1, 2,\ldots \) . From (1.4), we obtain

$$ \begin{aligned} &\sum_{i=1}^{N} \frac{1}{p_{i}} \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx - \frac{\lambda}{\theta} \int _{\Omega} f(u_{n})u_{n} \,dx + \frac{\lambda}{\theta}M \vert \Omega \vert \\ &\quad \leq \sum_{i=1}^{N} \frac{1}{p_{i}} \int _{ \Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx - \lambda \int _{\Omega} F(u_{n}) \,dx \\ &\quad \leq K. \end{aligned} $$

From the last two inequalities we have

$$ \sum_{i=1}^{N}\biggl( \frac{1}{p_{i}}-\frac{1}{\theta}\biggr) \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx - \sum_{i=1}^{N} \frac{1}{\theta} \biggl( \int _{ \Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}} \biggr)^{\frac{1}{p_{i}}}\,dx \leq K-\frac{\lambda}{\theta}M \vert \Omega \vert . $$
(3.5)

Now, we consider two cases. Case (i): If \((\int _{\Omega} \vert \frac{\partial u_{n}}{\partial x_{i}} \vert ^{p_{i}} )^{ \frac{1}{p_{i}}}\leq 1\), for \(i=1,\ldots , N\), then \(\{u_{n}\}\) is a bounded sequence. Case (ii): If there exists \(1\leq j\leq N\) such that \((\int _{\Omega} \vert \frac{\partial u_{n}}{\partial x_{j}} \vert ^{p_{j}} )^{ \frac{1}{p_{j}}}> 1\), then

$$ \biggl( \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{j}} \biggr\vert ^{p_{j}}\,dx \biggr)^{ \frac{1}{p_{j}}} \leq \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{j}} \biggr\vert ^{p_{j}}\,dx . $$

This shows (3.5) can be written as

$$ \begin{aligned} \biggl(\frac{1}{p_{+}}-\frac{1}{\theta} \biggr) \biggl( \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{j}} \biggr\vert ^{p_{j}}\,dx \biggr)^{\frac{1}{p_{j}}}-\sum_{i=1}^{N} \frac{1}{\theta} \biggl( \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}}\leq K-\frac{\lambda}{\theta}M \vert \Omega \vert . \end{aligned} $$

This proves that \(\{u_{n}\}\) is a bounded sequence. Thus, without loss of generality, we may assume that \(\{u_{n}\}\) converges weakly. Let \(u\in W_{0}^{1,\overrightarrow{p}}(\Omega )\) be its weak limit. Since \(q< \frac{Np_{+}}{(N-p_{+})}\), by the Sobolev embedding theorem we may assume that \(\{u_{n}\}\) converges to u in \(L^{q}(\Omega )\). These assumptions and Hölder’s inequality imply

$$ \int _{\Omega}\lambda f(u_{n}) (u_{n}-u) \to 0. $$
(3.6)

From (3.6) and \(\lim_{n\to +\infty} J^{\prime}_{\lambda}(u_{n})=0\), we have

$$ \lim_{n\to +\infty}\sum _{i=1}^{N} \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}-2} \frac{\partial u_{n}}{\partial x_{i}}\biggl( \frac{\partial u_{n}}{\partial x_{i}}- \frac{\partial u}{\partial x_{i}}\biggr)\,dx =0. $$
(3.7)

Using again that u is the weak limit of \(\{u_{n}\}\) in \(W_{0}^{1,\overrightarrow{p}}(\Omega )\) we also have

$$ \lim_{n\to +\infty}\sum _{i=1}^{N} \int _{\Omega} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}-2} \frac{\partial u}{\partial x_{i}}\biggl( \frac{\partial u_{n}}{\partial x_{i}}- \frac{\partial u}{\partial x_{i}}\biggr)\,dx =0. $$
(3.8)

By Hölder’s inequality,

$$ \begin{aligned} &\sum_{i=1}^{N} \int _{\Omega} \biggl( \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}-2} \frac{\partial u_{n}}{\partial x_{i}}- \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}-2} \frac{\partial u}{\partial x_{i}} \biggr) \biggl( \frac{\partial u_{n}}{\partial x_{i}}- \frac{\partial u}{\partial x_{i}} \biggr)\,dx \\ &\quad \geq \sum_{i=1}^{N} \int _{\Omega} \biggl( \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}- \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}-1} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert - \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}-1} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert + \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}} \biggr)\,dx \\ &\quad = \sum_{i=1}^{N} \biggl( \int _{\Omega} \biggl( \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}+ \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}} \biggr)\,dx - \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}-1} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert \,dx \\ &\quad \quad {}- \int _{\Omega} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}-1} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert \,dx \biggr) \\ &\quad \geq \sum_{i=1}^{N} \biggl[ \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx - \biggl( \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{p_{i}-1}{p_{i}}} \biggl( \int _{\Omega} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}} \\ &\quad \quad - \biggl( \int _{\Omega} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{p_{i}-1}{p_{i}}} \biggl( \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}}+ \int _{\Omega} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr] \\ &\quad = \sum_{i=1}^{N} \biggl\{ \biggl[ \biggl( \int _{ \Omega} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{\frac{p_{i}-1}{p_{i}}}- \biggl( \int _{ \Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{\frac{p_{i}-1}{p_{i}}} \biggr] \\ &\quad \quad \times \biggl[ \biggl( \int _{\Omega} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{\frac{1}{p_{i}}}- \biggl( \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}} \biggr] \biggr\} \\ &\quad \geq 0. \end{aligned} $$
(3.9)

The relations (3.7)–(3.9) imply that

$$ \begin{aligned} &\lim_{n\to \infty}\sum _{i=1}^{N} \biggl\{ \biggl[ \biggl( \int _{\Omega} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{p_{i}-1}{p_{i}}}- \biggl( \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{p_{i}-1}{p_{i}}} \biggr] \\ &\quad \times \biggl[ \biggl( \int _{\Omega} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}}- \biggl( \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}} \biggr] \biggr\} =0. \end{aligned} $$

This shows that for each \(i=1,\ldots , N\)

$$ \lim_{n\to \infty} \biggl( \int _{\Omega} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}}= \biggl( \int _{\Omega} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}}, $$

which implies that \(\lim_{n\to \infty} \|u_{n}\|_{W_{0}^{1,\overrightarrow{p}}(\Omega )} = \|u\|_{W_{0}^{1,\overrightarrow{p}}(\Omega )}\). Since \(u_{n}\rightharpoonup u\), \(u_{n}\to u\) in \(W_{0}^{1,\overrightarrow{p}}(\Omega )\). This proves that \(J_{\lambda}\) satisfies the Palais–Smale condition.

From (3.1) we obtain

$$ \begin{aligned} \max \bigl\{ J_{\lambda}(s\phi ):s\geq 0\bigr\} & \leq \biggl( \frac{Np_{+}}{p_{-}}\biggr)^{r(q+1)} \frac{C^{1+p_{+}r}((q+1)^{r(q-p_{+})+r}-p_{+})}{D^{p_{+}r}p_{+}(q+1)^{r(q+1)}} \lambda ^{-p_{+}r} +\lambda A_{1} \vert \Omega \vert \\ &: = c^{\prime}_{2}\lambda ^{-p_{+}r}+\lambda A_{1} \vert \Omega \vert \\ & \leq c^{\prime}_{2}\lambda ^{-p_{+}r}+\lambda ^{-p_{+}r} A_{1} \vert \Omega \vert \\ &: = c_{2}\lambda ^{-p_{+}r}, \end{aligned} $$
(3.10)

where \(C =\max \{\int _{\Omega} \vert \frac{\partial{u}}{\partial{x_{i}}} \vert ^{p_{i}}\,dx : \text{for } 1 \leq i\leq N \}\) and \(D=A_{1}\|\phi \|_{q+1}^{q+1}\). With this estimate and Lemma 3.2, the existence of \(u_{\lambda}\in W^{1,\overrightarrow{p}}_{0} (\Omega )\) such that \(\nabla J_{\lambda}(u_{\lambda})=0\) and

$$ c_{1}\bigl(\tau \lambda ^{-r} \bigr)^{p_{+}}\leq J_{\lambda}(u_{\lambda})\leq c_{2} \lambda ^{-p_{+}r} $$
(3.11)

follows by the Mountain-Pass Theorem. □

Remark 3.4

The solution \(u_{\lambda}\in W_{0}^{1,\overrightarrow{p}}(\Omega )\) is indeed in \(L^{\infty}(\Omega )\) (see [12, Lemma 2.4]) and [3, Sect. 4]).

Lemma 3.5

Let \(u_{\lambda}\) be as in Lemma 3.3. Then, there is a positive constant \(M_{0}\) such that

$$ M_{0}\lambda ^{-r}\leq \Vert u_{\lambda} \Vert _{\infty}. $$
(3.12)

Proof

Note that there exists \(c_{1} > 0\) such that \(J(u_{\lambda}) \geq c_{1}\lambda ^{-rp_{+}}\). On the other hand, \(F(s)\geq \min F >-\infty \) and \(f(s)s\leq B_{1}(|s|^{q+1} + |s|)\) for all \(s\in \mathbb{R}\). Then, there is a constant \(C_{1}>0\) such that

$$ \begin{aligned} \lambda \int _{\Omega }f(u_{\lambda})u_{ \lambda }\,dx & = \sum _{i=1}^{N} \int _{\Omega } \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \\ & \geq p_{-}\sum_{i=1}^{N} \frac{1}{p_{i}} \int _{\Omega } \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \\ & \geq p_{-}J(u_{\lambda})+p_{-}\lambda \int _{\Omega }F(u_{ \lambda})\,dx \\ & \geq p_{-}C_{1}\lambda ^{-rp_{+}}+p_{-} \vert \Omega \vert \lambda \min F \\ & \geq c_{1}\lambda ^{-rp_{+}}. \end{aligned} $$

Thus, \(\lim_{\lambda \to 0}\|u_{\lambda}\|_{\infty}=+\infty \). On the other hand, by (1.5),

$$ \begin{aligned} \lambda \int _{\Omega }f(u_{\lambda})u_{ \lambda }\,dx & \leq B_{1}\lambda \int _{\Omega}\bigl( \vert u_{ \lambda} \vert ^{q+1}+ \vert u_{\lambda} \vert \bigr)\,dx \\ & \leq B_{1}\lambda \int _{\Omega}\bigl( \Vert u_{\lambda} \Vert _{ \infty}^{q+1}+ \Vert u_{\lambda} \Vert _{\infty}\bigr)\,dx \\ & \leq 2B_{1} \vert \Omega \vert \lambda \Vert u_{\lambda} \Vert _{\infty}^{q+1}, \end{aligned} $$

where we have used the fact that \(0 < \lambda < 1\). Finally, taking \(M_{0}= \frac{C_{1}}{2B_{1}|\Omega |}\), the lemma is proven. □

Lemma 3.6

Let \(u_{\lambda}\) be as in Lemma 3.3. Then, there exists \(c_{3}>0\) such that

$$ \sum_{i=1}^{N} \int _{\Omega} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}} \,dx \leq c_{3} \lambda ^{-rp_{+}} $$
(3.13)

for all \(\lambda \in (0,\lambda _{3})\).

Proof

By (1.4) and the definition of \(u_{\lambda}\),

$$ \begin{aligned} \lambda \int _{\Omega} \frac{\theta -p_{+}}{\theta}u_{\lambda }f(u_{\lambda})\,dx & \leq \lambda \int _{\Omega } \bigl(u_{\lambda }f(u_{\lambda})-p_{+}F(u_{ \lambda}) \bigr)\,dx -\frac{\lambda p_{+}M \vert \Omega \vert }{\theta} \\ & = \sum_{i=1}^{N} \int _{\Omega } \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx -p_{+} \int _{\Omega }F(u_{\lambda})\,dx - \frac{\lambda p_{+}M \vert \Omega \vert }{\theta} \\ & \leq p_{+} \Biggl( \sum_{i=1}^{N} \frac{1}{p_{i}} \int _{ \Omega } \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx - \int _{\Omega }F(u_{\lambda})\,dx \Biggr)- \frac{p_{+}\lambda M \vert \Omega \vert }{\theta} \\ & \leq c_{2}\lambda ^{-rp_{+}}+\frac{\lambda p_{+}M \vert \Omega \vert }{\theta} \\ & \leq 2 c_{2}\lambda ^{-rp_{+}}, \end{aligned} $$
(3.14)

where we have used \(0 < \lambda < 1\). Now, the result follows from (3.14) and the fact that \(u_{\lambda}\) is a weak solution of (1.2). □

4 Existence of a positive solution

Now, we can prove Theorem 1.2 as follows.

Proof

Suppose there exists a sequence \(\{\lambda _{j}\}_{j}, 1 >\lambda _{j} > 0\) for all j, converging to 0 such that the measure \(m(\{x\in \Omega ; u_{\lambda _{j}} (x)\leq 0\}) > 0\).

Letting \(w_{j}=\frac{u_{\lambda _{j}}}{\|u_{\lambda _{j}}\|_{\infty}}\), we see that

$$ -\sum_{i=1}^{N} \Vert u_{\lambda _{j}} \Vert _{\infty}^{p_{i}-1} \frac{\partial}{\partial x_{i}}\biggl( \biggl\vert \frac{\partial w_{j}}{\partial x_{i}} \biggr\vert ^{p_{i}-2} \frac{\partial w_{j}}{\partial x_{i}}\biggr)=\lambda _{j}f(u_{\lambda _{j}}). $$
(4.1)

From Lemmas 3.5 and 3.6 there is a constant \(C_{3}\) such that

$$ \begin{aligned} \sum_{i=1}^{N} \int _{\Omega } \biggl\vert \frac{\partial w_{j}}{\partial x_{i}} \biggr\vert ^{p_{i}} \,dx & = \sum_{i=1}^{N} \biggl(\frac{1}{ \Vert u_{\lambda _{j}} \Vert _{\infty}} \biggr)^{p_{i}} \int _{\Omega } \biggl\vert \frac{\partial u_{\lambda _{j}}}{\partial x_{i}} \biggr\vert ^{p_{i}} \,dx \\ & \leq \sum_{i=1}^{N} \frac{1}{(M_{0}\lambda ^{-r})^{p_{i}}} \int _{\Omega } \biggl\vert \frac{\partial u_{\lambda _{j}}}{\partial x_{i}} \biggr\vert ^{p_{i}} \,dx \\ & \leq M_{1}\frac{1}{\lambda ^{-rp_{+}}}\sum_{i=1}^{N} \int _{\Omega } \biggl\vert \frac{\partial u_{\lambda _{j}}}{\partial x_{i}} \biggr\vert ^{p_{i}} \,dx \\ & \leq C_{3}. \end{aligned} $$
(4.2)

This shows that for each \(i=1,\ldots ,N\)

$$ \begin{aligned} \int _{\Omega } \biggl\vert \frac{\partial w_{j}}{\partial x_{i}} \biggr\vert ^{p_{i}} \,dx \leq c_{3} \end{aligned} $$
(4.3)

and therefore

$$ \begin{aligned} \Vert w_{j} \Vert _{W_{0}^{1,\overrightarrow{p}}(\Omega )}=\sum_{i=1}^{N} \biggl( \int _{\Omega } \biggl\vert \frac{\partial w_{j}}{\partial x_{i}} \biggr\vert ^{p_{i}} \,dx \biggr)^{ \frac{1}{p_{i}}}\leq D_{3}. \end{aligned} $$
(4.4)

By [3, Proposition 4.1] (or [13, Theorem 2]) the sequence \(w_{j}\) is uniformly bounded in \(L^{\infty}(\Omega )\). Therefore, one may denote its limit by ω.

Next, using comparison principles [12, Lemma 2.5], we prove that \(w(x)\geq 0\).

Let \(v_{0}\in W_{0}^{1, p_{+}}(\Omega )\) be the solution of

$$ \textstyle\begin{cases} -\Delta _{p_{+}}v_{0}=1&\text{in } \Omega , \\ v_{0}=0,&\text{on } \partial \Omega . \end{cases} $$
(4.5)

Let \(K_{j}:=\lambda _{j}\min \{f(t); t\in \mathbb{R}\}\|u_{\lambda _{j}} \|_{\infty}^{1-p_{+}}\). The solution \(v_{j}\) of the equation

$$ \textstyle\begin{cases} -\Delta _{p_{+}}v_{j}=K_{j}&\text{in } \Omega , \\ v_{0}=0,&\text{on } \partial \Omega , \end{cases} $$
(4.6)

is given by \(v_{j}= (-K_{j} )^{\frac{1}{p_{+}-1}}v_{0}\).

Since \(\lambda _{j}f(u_{\lambda _{j}})\|u_{\lambda _{j}}\|_{\infty}^{1-p_{+}} \geq K_{j}\), it follows by the comparison principle in [12, Lemma 2.5] that \(w_{j}\geq v_{j}\). Then, the fact that \(v_{j}(x)\to 0\) as \(j\to 0\) implies that \(w(x)\geq 0\) for all \(x\in \Omega \).

Since, by hypothesis, \(q > p_{+}-1\), we have \(s = \frac{Np_{+}r}{(N -p_{+})} > 1\). This result, together with the Sobolev embedding Theorem, (1.3) and Lemma 3.6, gives

$$ \begin{aligned} \int _{\Omega } \bigl\vert f(u_{\lambda _{j}}) \bigr\vert ^{s} \Vert u_{\lambda _{j}} \Vert _{\infty}^{s(1-p_{+})}\,dx & \leq B^{s}2^{s-1} \int _{\Omega}\bigl( \vert u_{\lambda _{j}} \vert ^{(q+1-p_{+})s}+1\bigr)\,dx \\ & \leq C \bigl( \Vert u_{\lambda _{j}} \Vert _{W_{0}^{1,p_{+}}(\Omega )}^{ \frac{Np_{+}}{N-p_{+}}}+1 \bigr) \\ & \leq C \bigl(c_{3}\lambda _{j}^{-r\frac{Np_{+}}{N-p_{+}}}+1 \bigr), \end{aligned} $$
(4.7)

where \(C > 0\) is a constant independent of j and, without loss of generality, we have assumed \(\|u_{\lambda _{j}}\|_{\infty}\geq 1\). From (4.7) and the fact that \(\frac{rNp_{+}}{(sN-sp_{+})} = 1\) we see that \(\{\lambda _{j}f(u_{\lambda _{j}}) \|u_{\lambda _{j}}\|_{\infty}^{(1-p_{+})} \}\) is bounded in \(L^{s}(\Omega )\), so we may assume that it converges weakly. Let \(z\in L^{s}(\Omega )\) be the weak limit of such a sequence. Since \(\lambda _{j} \|u_{\lambda _{j}}\|_{\infty}^{(1-p_{+})}\to 0\) as \(j\to +\infty \) and f is bounded from below, \(z\geq 0\). Now, if \(\phi \in C_{0}^{\infty}(\Omega )\), then

$$ \begin{aligned} \sum_{i=1}^{N} \int _{\Omega } \biggl\vert \frac{\partial w}{\partial x_{i}} \biggr\vert ^{p_{i}-2}\biggl\langle \frac{\partial w}{\partial x_{i}}, \frac{\partial \phi}{\partial x_{i}}\biggr\rangle \,dx & = \lim_{j\to \infty} \sum_{i=1}^{N} \int _{\Omega } \biggl\vert \frac{\partial w_{j}}{\partial x_{i}} \biggr\vert ^{p_{i}-2}\biggl\langle \frac{\partial w_{j}}{\partial x_{i}}, \frac{\partial \phi}{\partial x_{i}}\biggr\rangle \,dx \\ & = \lim_{j\to \infty} \sum_{i=1}^{N} \int _{\Omega } \Vert u_{ \lambda _{j}} \Vert _{\infty}^{1-p_{i}} \biggl\vert \frac{\partial u_{\lambda _{j}}}{\partial x_{i}} \biggr\vert ^{p_{i}-2} \biggl\langle \frac{\partial u_{\lambda _{j}}}{\partial x_{i}}, \frac{\partial \phi}{\partial x_{i}}\biggr\rangle \,dx \\ & \geq \lim_{j\to \infty} \Vert u_{\lambda _{j}} \Vert _{\infty}^{1-p_{+}} \sum_{i=1}^{N} \int _{\Omega } \biggl\vert \frac{\partial u_{\lambda _{j}}}{\partial x_{i}} \biggr\vert ^{p_{i}-2} \biggl\langle \frac{\partial u_{\lambda _{j}}}{\partial x_{i}}, \frac{\partial \phi}{\partial x_{i}} \biggr\rangle \,dx \\ & = \lim_{j\to \infty} \int _{\Omega } \Vert u_{\lambda _{j}} \Vert _{\infty}^{1-p_{+}}\lambda _{j}f(u_{\lambda _{j}}) \phi \,dx \\ & = \int _{\Omega }z\phi \,dx . \end{aligned} $$
(4.8)

Therefore, \(-\Delta _{\overrightarrow{p}}w\geq z\). Since \(\|w_{j}\|_{\infty}= 1\), \(w\neq 0\). By [12, Lemma 2.5], \(w>0\) in Ω.

Therefore, since \(\{w_{j}\}_{j}\) converges w in \(L^{\infty}(\Omega )\), for sufficiently large \(j, w_{j}(x) > 0\) for all \(x\in \Omega \). Hence, \(u_{\lambda _{j}} (x) > 0\) for all \(x\in \Omega \), which contradicts the assumption that

$$ m\bigl(\bigl\{ x; u_{\lambda _{j}} (x) < 0\bigr\} \bigr) > 0. $$

This contradiction proves Theorem 1.2. □