Positive Solutions for the Neumann p-Laplacian with Superdiffusive Reaction

Abstract

We consider a generalized logistic equation driven by the Neumann p-Laplacian and with a reaction that exhibits a superdiffusive kind of behavior. Using variational methods based on the critical point theory, together with truncation and comparison techniques, we show that there exists a critical value $\lambda _*>0$ of the parameter, such that if $\lambda >\lambda _*$, the problem has at least two positive solutions, if $\lambda =\lambda _*$, the problem has at least one positive solution and it has no positive solution if $\lambda \in (0,\lambda _*)$. Finally, we show that for all $\lambda \geqslant \lambda _*$, the problem has a smallest positive solution.

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 nonlinear parametric Neumann problem:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle -\Delta _p u(z)+\beta (z) u(z)^{p-1}\ =\ \lambda g\big (z,u(z)\big )-f\big (z,u(z)\big )\quad \text {in} \quad \Omega ,\\ \displaystyle \frac{\partial u}{\partial n} =0\quad \text {on} \quad \partial \Omega ,\ \lambda >0,\ u>0, \end{array} \right. \qquad \qquad (P)_{\lambda } \end{aligned}$$

with $\beta \in L^{\infty }(\Omega )_+$, $\beta \ne 0$. Here $\Delta _p$ denotes the p-Laplace differential operator, defined by

$$\begin{aligned} \Delta _p u \ =\ \text {div}\big (\Vert \nabla u\Vert ^{p-2}\nabla u\big ) \quad \forall u\in W^{1,p}(\Omega ), \end{aligned}$$

with $p\in (1,+\infty )$. Also $n(\cdot )$ denotes the outward unit normal on $\partial \Omega$. When the reaction in $(P)_{\lambda }$ has the particular form

$$\begin{aligned} \lambda \zeta ^{q-1}-\zeta ^{r-1}, \end{aligned}$$

with $q<r$, then the resulting equation is the p-logistic equation (or simply the logistic equation when $p=2$). The logistic equation is important in mathematical biology (see Gurtin and Mac Camy [21] and Afrouzi and Brown [1]) and describes the dynamics of biological populations whose mobility is density dependent.

There are three different types of the p-logistic equation, depending on the value of the exponent q with respect to p. More precisely, we have

• the “subdiffusive” type, when $q<p<r$;

• the “equidiffusive” type, when $q=p<r$;

• the “superdiffusive” type, when $p<q<r$.

The subdiffusive and equidiffusive cases are similar, but the superdiffusive case differs essentially and it exhibits bifurcation phenomena (see Takeuchi [29, 30] and Filippakis et al. [7], where the Dirichlet problem is studied).

The aim of this work, is to prove a bifurcation-type theorem for the positive solutions of $(P)_{\lambda }$ as the parameter $\lambda >0$ varies in $(0,+\infty )$ and the reaction $\zeta \longmapsto \lambda g(z,\zeta )-f(z,\zeta )$ (which is more general than the standard p-logistic equation; see Afrouzi and Brown [1]), exhibits a superdiffusive kind of behavior. To the best of our konwledge, the Neumann p-logistic equation has not been studied. There is only the recent work of Marano-Papageorgiou [25], where the equidiffusive case is examined.

Our approach is variational based on the critical point theory, combined with suitable truncation and comparison techniques. In the next section, for the convenience of the reader we recall main mathematical tools which we will use in the sequel.

This work is the outgrowth of a remark made by the referee of [19]. In that paper, the authors deal with the parametric equation

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle -\Delta _p u(z)\ =\ \lambda f\big (z,u(z)\big ) \quad \text {in} \quad \Omega ,\\ \displaystyle u|_{\partial \Omega } =0 \quad \text {on}\quad \partial \Omega \end{array} \right. \end{aligned}$$

and some analogous bifurcation-type results were proved. It was pointed out by the referee that in mathematical biology, the Neumann model is a more realistic one. For some other recent results on nonlinear Neumann boundary value problems involving p-Laplacian, we refer to Gasiński and Papageorgiou [11–17].

2 Mathematical Background

Let X be a Banach space and let $X^*$ be its topological dual. By $\langle \cdot ,\cdot \rangle$ we denote the duality brackets for the pair $(X,X^*)$. Let $\varphi \in C^1(X)$. We say that $\varphi$ satisfies the Palais–Smale condition, if the following holds:

“Every sequence $\{x_n\}_{n\geqslant 1}\subseteq X$, such that $\big \{\varphi (x_n)\big \}_{n\geqslant 1}\subseteq \mathbb {R}$ is bounded and

$$\begin{aligned} \varphi '(x_n) \ \longrightarrow \ 0 \quad \text {in} \quad X^*\ \text {as} \quad n\rightarrow +\infty , \end{aligned}$$

admits a strongly convergent subsequence.”

Using this compactness-type condition on $\varphi$, we can state the following theorem, known in the literature as the “mountain pass theorem”.

Theorem 2.1

If X is a Banach space, $\varphi \in C^1(X)$ satisfies the Palais–Smale condition, $x_0,x_1\in X$, $0<\varrho <\Vert x_0-x_1\Vert$,

$$\begin{aligned} \max \big \{\varphi (x_0),\varphi (x_1)\big \}< & {} \inf \big \{\varphi (x):\ \Vert x-x_0\Vert =\varrho \big \} \ =\ \eta _{\varrho },\\ c= & {} \inf _{\gamma \in \Gamma }\max _{0\leqslant t\leqslant 1} \varphi \big (\gamma (t)\big ), \end{aligned}$$

where

$$\begin{aligned} \Gamma \ =\ \big \{\gamma \in C\big ([0,1];X\big ):\ \gamma (0)=x_0,\ \gamma (1)=x_1 \big \}, \end{aligned}$$

then $c\geqslant \eta _{\varrho }$ and c is a critical value of $\varphi$ (i.e., there exists $\widehat{x}\in X$, such that $\varphi '(\widehat{x})=0$ and $\varphi (\widehat{x})=c$).

In the study of problem $(P)_{\lambda }$, we will use the Sobolev space $W^{1,p}(\Omega )$ and the ordered Banach space $C^1(\overline{\Omega })$. The positive cone of the latter is

$$\begin{aligned} C_+ \ =\ \big \{ u\in C^1(\overline{\Omega }):\ u(z)\geqslant 0\ \quad \text {for all} \quad \ z\in \overline{\Omega } \big \}. \end{aligned}$$

This cone has a nonempty interior, given by

$$\begin{aligned} \mathrm {int}\, C_+ \ =\ \bigg \{ u\in C_+:\ u(z)>0 \quad \text {for all} \quad z\in \overline{\Omega } \bigg \}. \end{aligned}$$

The next result relates local minimizers in $W^{1,p}(\Omega )$ with local minimizers in the smaller Banach space $C^1(\overline{\Omega })$. A result of this type was first proved for the Dirichlet Laplacian by Brézis and Nirenberg [5] and was later extended to the p-Laplacian by García Azorero et al. [8] and Guo and Zhang [20] (in the latter, for $p\geqslant 2$). Extensions to the Neumann p-Laplacian or Neumann p-Laplacian-like operators can be found in Motreanu et al. [26] and Motreanu and Papageorgiou [28].

So let $f_0:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}$ be a Carathéodory function (i.e., for all $\zeta \in \mathbb {R}$, the function $z\longmapsto f_0(z,\zeta )$ is measurable and for almost all $z\in \Omega$, the function $\zeta \longmapsto f_0(z,\zeta )$ is continuous), which exhibits subcritical growth in $\zeta \in \mathbb {R}$, i.e.,

$$\begin{aligned} \big | f_0(z,\zeta )\big |\ \leqslant \ a(z)+c|\zeta |^{r-1} \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \in \mathbb {R}, \end{aligned}$$

with $a\in L^{\infty }(\Omega )_+$, $c>0$ and $1<r<p^*$, where

$$\begin{aligned} p^* \ =\ \left\{ \begin{array}{lll} \frac{Np}{N-p} &{} \quad \text {if} &{} \quad p<N,\\ +\infty &{} \quad \text {if} &{} \quad p\geqslant N. \end{array} \right. \end{aligned}$$

We set

$$\begin{aligned} F_0(z,\zeta )\,ds \ =\ \int _0^{\zeta } f_0(z,s)\,ds \end{aligned}$$

and consider the $C^1$-functional $\psi _0:W^{1,p}(\Omega )\longrightarrow \mathbb {R}$, defined by

$$\begin{aligned} \psi _0(u) \ =\ \frac{1}{p}\Vert \nabla u\Vert _p^p -\int _{\Omega } F_0\big (z,u(z)\big )\,dz \quad \forall u\in W^{1,p}(\Omega ). \end{aligned}$$

Theorem 2.2

If $u_0\in W^{1,p}(\Omega )$ is a local $C^1(\overline{\Omega })$-minimizer of $\psi _0$, i.e., there exists $\varrho _1>0$, such that

$$\begin{aligned} \psi _0(u_0)\ \leqslant \ \psi _0(u_0+h) \quad \forall h\in C^1(\overline{\Omega }),\ \Vert h\Vert _{C^1(\overline{\Omega })}\leqslant \varrho _1, \end{aligned}$$

then $u_0\in C^1(\overline{\Omega })$ and it is a local $W^{1,p}(\Omega )$-minimizer of $\psi _0$, i.e., there exists $\varrho _2>0$, such that

$$\begin{aligned} \psi _0(u_0)\ \leqslant \ \psi _0(u_0+h) \quad \forall h\in W^{1,p}(\Omega ),\ \Vert h\Vert \leqslant \varrho _2. \end{aligned}$$

Remark 2.3

In [26, 28], the result was stated in terms of $W^{1,p}_n(\Omega )=\overline{C^1_n(\overline{\Omega })}^{\Vert \cdot \Vert }$, where

$$\begin{aligned} C^1_n(\overline{\Omega }) \ =\ \bigg \{u\in C^1(\overline{\Omega }):\ \frac{\partial u}{\partial n}=0 \quad \text {on} \quad \partial \Omega \bigg \}. \end{aligned}$$

Actually, there is no need for this restriction.

Let $A:W^{1,p}(\Omega )\longrightarrow W^{1,p}(\Omega )^*$ be the nonlinear map defined by

$$\begin{aligned} \big \langle A(u),y\big \rangle \ =\ \int _{\Omega }\Vert \nabla u\Vert ^{p-2} (\nabla u,\nabla y)_{\mathbb {R}^N}\,dz \quad \forall u,y\in W^{1,p}(\Omega ). \end{aligned}$$

(2.1)

The next result can be found in Aizicovici et al. [3, Proposition 2].

Proposition 2.4

The map $A:W^{1,p}(\Omega )\longrightarrow W^{1,p}(\Omega )^*$ defined by (2.1) is continuous, strictly monotone (hence maximal monotone too) and of type $(S)_+$, i.e., if $\{u_n\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega )$ is a sequence, such that $u_n\longrightarrow u$ weakly in $W^{1,p}(\Omega )$ and

$$\begin{aligned} \limsup _{n\rightarrow +\infty } \big \langle A(u_n),\ u_n-u\big \rangle \ \leqslant \ 0, \end{aligned}$$

then $u_n\longrightarrow u$ in $W^{1,p}(\Omega )$.

The next simple lemma, will be useful in our estimations and can be found in Aizicovici et al. [4, Lemma 2]. Recall that by $\Vert \cdot \Vert$ we denote the norm of the Sobolev space $W^{1,p}(\Omega )$, i.e.,

$$\begin{aligned} \Vert u\Vert \ =\ \big ( \Vert u\Vert _p^p +\Vert \nabla u\Vert _p^p\big )^{\frac{1}{p}} \quad \forall u\in W^{1,p}(\Omega ). \end{aligned}$$

Lemma 2.5

If $\beta \in L^{\infty }(\Omega )$, $\beta (z)\geqslant 0$ for almost all $z\in \Omega$ and $\beta \ne 0$, then there exists $\xi _0>0$, such that

$$\begin{aligned} \Vert \nabla u\Vert _p^p +\int _{\Omega }\beta |u|^p\,dz \ \geqslant \ \xi _0\Vert u\Vert ^p \quad \forall u\in W^{1,p}(\Omega ). \end{aligned}$$

We conclude this section by fixing some notation. By $|\cdot |_N$ we denote the Lebesgue measure on $\mathbb {R}^N$. For every $u\in W^{1,p}(\Omega )$, we set $u^{\pm }=\max \{\pm u,0\}$. We know that

$$\begin{aligned} u^{\pm }\in W^{1,p}(\Omega ),\quad u\ =\ u^+-u^-,\quad |u|\ =\ u^+ + u^-. \end{aligned}$$

Finally for every measurable function $h:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}$, we define

$$\begin{aligned} N_h(u)(\cdot ) \ =\ h\big (\cdot ,u(\cdot )\big ) \quad \forall u\in W^{1,p}(\Omega ) \end{aligned}$$

(the Nemytskii map corresponding to h).

3 A Bifurcation-Type Theorem

The hypotheses on the data of problem $(P)_{\lambda }$ are the following:

$\underline{H_g}$ $g:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}$ is a Carathéodory function, such that $g(z,0)=0$ for almost all $z\in \Omega$ and

(i):

we have

$$\begin{aligned} \big |g(z,\zeta )\big |\ \leqslant a(z)+c|\zeta |^{r-1} \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \in \mathbb {R}, \end{aligned}$$

with $a\in L^{\infty }(\Omega )_+$, $c>0$ and $p<r<p^*$;

(ii):

there exist $\vartheta >q>p$, such that

$$\begin{aligned} 0 \ <\ \eta _g \ \leqslant \ \liminf _{\zeta \rightarrow +\infty }\frac{g(z,\zeta )}{\zeta ^{q-1}} \ \leqslant \ \limsup _{\zeta \rightarrow +\infty }\frac{g(z,\zeta )}{\zeta ^{q-1}} \ \leqslant \ \widehat{\eta }_g \end{aligned}$$

uniformly for almost all $z\in \Omega$ and for almost all $z\in \Omega$, the function $\zeta \longmapsto \frac{g(z,\zeta )}{\zeta ^{\vartheta -1}}$ is nonincreasing on $(0,+\infty )$;

(iii):

we have

$$\begin{aligned} \lim _{\zeta \rightarrow 0^+}\frac{g(z,\zeta )}{\zeta ^{q-1}} \ =\ 0 \end{aligned}$$

uniformly for almost all $z\in \Omega$;

(iv):

there exist two functions $\sigma _0,\sigma _1:(0,+\infty )\longrightarrow (0,+\infty )$, both upper semicontinuous, such that

$$\begin{aligned} \sigma _0(\zeta )\ \leqslant \ g(z,\zeta ) \ \leqslant \ \sigma _1(\zeta ) \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta >0. \end{aligned}$$

$\underline{H_f}$ $f:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}$ is a Carathéodory function, such that $f(z,0)=0$ for almost all $z\in \Omega$ and

(i):

we have

$$\begin{aligned} \big |f(z,\zeta )\big |\ \leqslant a(z)+c|\zeta |^{r-1} \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \in \mathbb {R}, \end{aligned}$$

with $a\in L^{\infty }(\Omega )_+$, $c>0$ and $p<r<p^*$;

(ii):

with $\vartheta >q>p$ as in hypothesis $H_g(ii)$, we have

$$\begin{aligned} 0 \ <\ \eta _f \ \leqslant \ \liminf _{\zeta \rightarrow +\infty }\frac{f(z,\zeta )}{\zeta ^{\vartheta -1}} \ \leqslant \ \limsup _{\zeta \rightarrow +\infty }\frac{f(z,\zeta )}{\zeta ^{\vartheta -1}} \ \leqslant \ \widehat{\eta }_f \end{aligned}$$

uniformly for almost all $z\in \Omega$ and for almost all $z\in \Omega$, the function $\zeta \longmapsto \frac{f(z,\zeta )}{\zeta ^{p-1}}$ is nondecreasing on $(0,+\infty )$;

(iii):

we have

$$\begin{aligned} 0 \ \leqslant \ \liminf _{\zeta \rightarrow 0^+}\frac{f(z,\zeta )}{\zeta ^{q-1}} \ \leqslant \ \limsup _{\zeta \rightarrow 0^+}\frac{f(z,\zeta )}{\zeta ^{q-1}} \ \leqslant \ \zeta ^* \end{aligned}$$

uniformly for almost all $z\in \Omega$;

(iv):

there exists a lower semicontinuous function $\sigma _2:(0,+\infty )\longrightarrow (0,+\infty )$, such that

$$\begin{aligned} \sigma _2(\zeta )\ \leqslant \ f(z,\zeta ) \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta >0. \end{aligned}$$

$\underline{H_0}$ For every $\lambda >0$ and $\varrho >0$, we can find $\gamma _{\varrho }=\gamma _{\varrho }(\lambda )>0$, such that for almost all $z\in \Omega$, the function $\zeta \longmapsto \lambda g(z,\zeta )-f(z,\zeta )+\gamma _{\varrho }\zeta ^{\vartheta -1}$ is nondecreasing on $[0,\varrho ]$ ($\vartheta >q>p$ as in the hypothesis $H_g(ii)$).

Remark 3.1

Since we are interested in positive solutions and hypotheses $H_g$, $H_f$ and $H_0$ concern the positive semiaxis $\mathbb {R}_+= [0,+\infty )$, we may (and will) assume that

$$\begin{aligned} g(z,\zeta ) \ =\ f(z,\zeta ) \ =\ 0 \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \leqslant 0. \end{aligned}$$

Example 3.2

The following functions satisfy hypotheses $H_g$, $H_f$ and $H_0$ (for the sake of simplicity we drop the z-dependence):

(a) $g(\zeta )= \left\{ \begin{array}{lll} \zeta ^{s-1} &{} \quad \text {if} &{} \quad \zeta \in [0,1],\\ \zeta ^{q-1} &{} \quad \text {if} &{} \quad \zeta >1 \end{array} \right.$ and $f(\zeta )=\zeta ^{\vartheta -1}$    for all $\zeta \geqslant 0$, with $p<q<s<\vartheta <p^*$.

(b) $g(\zeta )= \left\{ \begin{array}{lll} \zeta ^{s-1}-\zeta ^{\vartheta -1} &{} \quad \text {if} &{} \quad \zeta \in [0,1],\\ \zeta ^{q-1}-\zeta ^{p-1} &{} \quad \text {if} &{} \quad \zeta >1 \end{array} \right.$ and $f(\zeta )=\zeta ^{\vartheta -1}-\zeta ^{s-1}$ for all $\zeta \geqslant 0$, with $p<q<s<\vartheta <p^*$.

Example (a) corresponds to the standard superdiffusive p-logistic reaction (see Afrouzi and Brown [1]).

By a positive solution of problem $(P)_{\lambda }$, we understand a function $u\in W^{1,p}(\Omega )$, $u\ne 0$, which is a weak solution of $(P)_{\lambda }$. Then $u\in L^{\infty }(\Omega )$ (see e.g., Gasiński and Papageorgiou [9, 18] and Hu and Papageorgiou [23]). Invoking Theorem 2 of Lieberman [24], we have that $u\in C_+\setminus \{0\}$. Let $\varrho =\Vert u\Vert _{\infty }$ and let $\gamma _{\varrho }=\gamma _{\varrho }(\lambda )>0$ be as postulated by hypothesis $H_0$. We have

$$\begin{aligned}&-\Delta _p u(z) +\beta (z) u(z)^{p-1} +\gamma _{\varrho } u(z)^{\vartheta -1}\\&\quad = \lambda g\big (z,u(z)\big ) -f\big (z, u(z)\big ) +\gamma _{\varrho } u(z)^{\vartheta -1} \ \geqslant \ 0 \quad \text {for almost all}\ z\in \Omega \end{aligned}$$

(see Motreanu and Papageorgiou [27]), so

$$\begin{aligned} \Delta _p u(z) \ \leqslant \ \big ( \Vert \beta \Vert _{\infty }+\gamma _{\varrho }\varrho ^{\vartheta -p}\big ) u(z)^{p-1} \quad \text {for almost all}\ z\in \Omega \end{aligned}$$

and finally

$$\begin{aligned} u\ \in \ \mathrm {int}\, C_+ \end{aligned}$$

(see Vázquez [31]).

So, we see that the positive solutions of problem $(P)_{\lambda }$, if they exist, belong in $\mathrm {int}\, C_+$.

Let

$$\begin{aligned} \mathcal {Y} \ =\ \big \{ \lambda >0:\ \text {problem }(P)_{\lambda } \text { has a positive solution}. \big \} \end{aligned}$$

Proposition 3.3

If hypotheses $H_g$, $H_f$ and $H_0$ hold, then $\inf \mathcal {Y}>0$.

Proof

By virtue of hypothesis $H_g(ii)$, we can find $\eta _1>0$ and $M>0$, such that

$$\begin{aligned} g(z,\zeta )\ \leqslant \ \eta _1\zeta ^{q-1} \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \geqslant M. \end{aligned}$$

(3.1)

On the other hand, from hypothesis $H_g(iii)$, for a given $\varepsilon >0$, we can find $\delta \in (0,1)$ small, such that

$$\begin{aligned} g(z,\zeta )\ \leqslant \ \varepsilon \zeta ^{p-1} \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \in [0,\delta ]. \end{aligned}$$

(3.2)

The function $\zeta \longmapsto \frac{\sigma _1(\zeta )}{\zeta ^{q-1}}$ is upper semicontinuous on $[\delta ,M]$ and so, we can find $\widehat{\xi }\in [\delta ,M]$, such that

$$\begin{aligned} \frac{\sigma _1(\zeta )}{\zeta ^{q-1}} \ \leqslant \ \frac{\sigma _1(\widehat{\zeta })}{\widehat{\zeta }^{q-1}} \ =\ \eta _2(\varepsilon ) \quad \forall \zeta \in [\delta ,M], \end{aligned}$$

so

$$\begin{aligned} g(z,\zeta ) \ \leqslant \ \eta _2\zeta ^{q-1} \quad \text {for almost all}\ \zeta \in [\delta ,M] \end{aligned}$$

(3.3)

(see hypothesis $H_g(iv)$). From (3.1), (3.2) and (3.3), it follows that

$$\begin{aligned} g(z,\zeta ) \ \leqslant \ \varepsilon \zeta ^{p-1}+\widehat{\eta }\zeta ^{q-1} \quad \text {for almost all}\ \zeta \geqslant 0, \end{aligned}$$

(3.4)

with $\widehat{\eta }(\varepsilon )=\max \{\eta _1,\eta _2\}>0$.

In a similar fashion, using hypotheses $H_f\,({ ii})$, $({ iii})$ and $({ iv})$, for a given $\varepsilon >0$, we can find $\vartheta =\vartheta (\varepsilon )>0$, such that

$$\begin{aligned} f(z,\zeta ) \ \geqslant \ \vartheta \zeta ^{q-1}-\varepsilon \zeta ^{p-1} \quad \text {for almost all}\ \zeta \geqslant 0. \end{aligned}$$

(3.5)

Let us fix $\varepsilon \in \big (0,\frac{\xi _0}{2}\big )$ ($\xi _0>0$ as in Lemma 2.5) and let $\widehat{\lambda }\leqslant \min \big \{1,\frac{\vartheta }{\widehat{\eta }}\big \}$. From (3.4) and (3.5), we have

$$\begin{aligned} \widehat{\lambda }g(z,\zeta )-f(z,\zeta )\leqslant & {} (\widehat{\lambda }+1)\varepsilon \zeta ^{p-1}+(\widehat{\lambda }\widehat{\eta }-\vartheta )\zeta ^{q-1}\nonumber \\\leqslant & {} 2\varepsilon \zeta ^{p-1} \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \geqslant 0. \end{aligned}$$

(3.6)

Suppose that for $\lambda \in (0,\widehat{\lambda })$, problem $(P)_{\lambda }$ has a positive solution (i.e., $\lambda \in \mathcal {Y}$). Then we can find a positive solution $u_{\lambda }\in \mathrm {int}\, C_+$ of $(P)_{\lambda }$. Hence

$$\begin{aligned} A(u_{\lambda })+\beta u_{\lambda }^{p-1} \ =\ \lambda N_g(u_{\lambda })-N_f(u_{\lambda }) \end{aligned}$$

(3.7)

(see (2.1) for the definition of A). On (3.7) we act with $u_{\lambda }$ and obtain

$$\begin{aligned} \Vert \nabla u_{\lambda }\Vert _p^p +\int _{\Omega }\beta u_{\lambda }^p\,dz \ =\ \int _{\Omega }\big (\lambda g(z,u_{\lambda })-f(z,u_{\lambda })\big ) u_{\lambda }\,dz, \end{aligned}$$

so using Lemma 2.5 and (3.6), we have

$$\begin{aligned} \xi _0\Vert u_{\lambda }\Vert ^p \ \leqslant \ 2\varepsilon \Vert u_{\lambda }\Vert ^p. \end{aligned}$$

recalling that $\varepsilon \in \big (0,\frac{\xi _0}{2}\big )$, we conclude that $u_{\lambda }=0$, a contradiction. Therefore $\inf \mathcal {Y}\geqslant \widehat{\lambda }>0$. $\square$

If $\mathcal {Y}=\emptyset$, then $\inf \mathcal {Y}=+\infty$. In the next proposition, we establish the nonemptiness of $\mathcal {Y}$.

Proposition 3.4

If hypotheses $H_g, H_f$ and $H_0$ hold, then $\mathcal {Y}\ne \emptyset$ and if $\lambda \in \mathcal {Y}$ and $\tau >\lambda$, then $\tau \in \mathcal {Y}$.

Proof

Let $\varphi _{\lambda }:W^{1,p}(\Omega )\longrightarrow \mathbb {R}$ be the energy functional for problem $(P)_{\lambda }$, defined by

$$\begin{aligned} \varphi _{\lambda }(u) \ =\ \frac{1}{p}\Vert \nabla u\Vert _p^p +\frac{1}{p}\int _{\Omega }\beta |u|^p\,dz -\lambda \int _{\Omega } G(z,u)\,dz +\int _{\Omega } F(z,u)\,dz \end{aligned}$$

for all $u\in W^{1,p}(\Omega )$. Evidently $\varphi _{\lambda }\in C^1\big (W^{1,p}(\Omega )\big )$. By virtue of hypotheses $H_g(i),(ii)$, we can find $\xi _1>0$ and $c_1>0$, such that

$$\begin{aligned} G(z,\zeta ) \ \leqslant \ \xi _1(\zeta ^+)^q+c_1 \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \in \mathbb {R}. \end{aligned}$$

(3.8)

Since $q<\vartheta$, using Young inequality with $\varepsilon >0$, from (3.8) we see that for a given $\varepsilon >0$, we can find $c_2=c_2(\varepsilon )>0$, such that

$$\begin{aligned} G(z,\zeta ) \ \leqslant \ \varepsilon (\zeta ^+)^{\vartheta }+c_2 \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \in \mathbb {R}. \end{aligned}$$

(3.9)

Also, from hypotheses $H_f(i),(ii)$, we see that we can find $\xi _2>0$ and $c_3>0$, such that

$$\begin{aligned} F(z,\zeta ) \ \geqslant \ \xi _2(\zeta ^+)^{\vartheta }-c_3 \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \in \mathbb {R}. \end{aligned}$$

(3.10)

Then

$$\begin{aligned} \varphi _{\lambda }(u)= & {} \frac{1}{p}\Vert \nabla u\Vert _p^p +\frac{1}{p}\int _{\Omega }\beta |u|^p\,dz -\lambda \int _{\Omega } G(z,u)\,dz +\int _{\Omega } F(z,u)\,dz\nonumber \\\geqslant & {} \frac{\xi _0}{p}\Vert u\Vert ^p +(\xi _2-\lambda \varepsilon )\Vert u^+\Vert _{\vartheta }^{\vartheta } -c_4 \quad \forall u\in W^{1,p}(\Omega ), \end{aligned}$$

(3.11)

for some $c_4=c_4(\varepsilon )>0$ (see Lemma 2.5 and (3.9), (3.10)).

We choose $\varepsilon \in \big (0,\frac{\xi _2}{\lambda }\big ]$. Then, from (3.11), it follows that $\varphi _{\lambda }$ is coercive. Also, it is easy to see that $\varphi _{\lambda }$ is sequentially weakly lower semicontinuous. Therefore, by the Weierstrass theorem, we can find $u_{\lambda }\in W^{1,p}(\Omega )$, such that

$$\begin{aligned} \varphi _{\lambda }(u_{\lambda }) \ =\ \inf _{u\in W^{1,p}(\Omega )}\varphi _{\lambda }(u) \ =\ m_{\lambda }. \end{aligned}$$

(3.12)

Let $\overline{u}\in \text {int}\, C_+$. Then clearly for $\lambda >0$ big, we have $\varphi _{\lambda }(\overline{u})<0$. Hence

$$\begin{aligned} \varphi _{\lambda }(u_{\lambda }) \ =\ m_{\lambda } \ <\ 0 \ =\ \varphi _{\lambda }(0) \quad \forall \lambda >0,\ \text {big} \end{aligned}$$

(see (3.12)), so

$$\begin{aligned} u_{\lambda }\ \ne \ 0. \end{aligned}$$

(3.13)

From (3.12), we have

$$\begin{aligned} \varphi _{\lambda }'(u_{\lambda }) \ =\ 0 \quad \forall \lambda >0,\ \text {big} \end{aligned}$$

so

$$\begin{aligned} A(u_{\lambda })+\beta |u_{\lambda }|^{p-2}u_{\lambda } \ =\ \lambda N_g (u_{\lambda })-N_f(u_{\lambda }). \end{aligned}$$

(3.14)

On (3.14) we act with $-u_{\lambda }^-\in W^{1,p}(\Omega )$ and we obtain

$$\begin{aligned} \Vert \nabla u_{\lambda }^-\Vert _p^p +\int _{\Omega }\beta (u_{\lambda }^-)^p\,dz \ =\ 0, \end{aligned}$$

so

$$\begin{aligned} \xi _0\Vert u_{\lambda }^-\Vert ^p\ \leqslant \ 0 \end{aligned}$$

(see Lemma 2.5), i.e., $u_{\lambda }\geqslant 0$, $u_{\lambda }\ne 0$ (see (3.13)).

Then (3.14) becomes

$$\begin{aligned} A(u_{\lambda })+\beta u_{\lambda }^{p-1} \ =\ \lambda N_g(u_{\lambda })-N_f(u_{\lambda }), \end{aligned}$$

so

$$\begin{aligned} u_{\lambda }\ \text {solves problem}\ (P)_{\lambda }, \end{aligned}$$

i.e., $\mathcal {Y}\ne \emptyset$.

Now suppose that $\lambda \in \mathcal {Y}$ and $\tau >\lambda$. We choose $s\in (0,1)$, such that

$$\begin{aligned} \lambda \ =\ s^{\vartheta -1}\tau \end{aligned}$$

(3.15)

(recall that $\vartheta >p$ and $\lambda <\tau$). Since $\lambda \in \mathcal {Y}$, problem $(P)_{\lambda }$ has a solution $u_{\lambda }\in \mathrm {int}\,C_+$. We set $\underline{u}=s u_{\lambda }\in \text {int}\,C_+$. Then

$$\begin{aligned} -\Delta _p\underline{u}+\beta \underline{u}^{p-1} \ =\ s^{p-1}\big (-\Delta u_{\lambda }+\beta u_{\lambda }^{p-1}\big ) \ =\ s^{p-1} \big ( \lambda g(z,u_{\lambda })-f(z,u_{\lambda })\big ). \end{aligned}$$

(3.16)

By virtue of hypothesis $H_g(ii)$ and since $s\in (0,1)$, we have

$$\begin{aligned} \frac{g(z,u_{\lambda }(z))}{u_{\lambda }(z)^{\vartheta -1}} \ \leqslant \ \frac{g(z,\underline{u}(z))}{\underline{u}(z)^{\vartheta -1}} \ =\ \frac{g(z,\underline{u}(z))}{s^{\vartheta -1}u_{\lambda }(z)^{\vartheta -1}}, \end{aligned}$$

so

$$\begin{aligned} s^{\vartheta -1}g\big (z,u_{\lambda }(z)\big ) \ \leqslant \ g\big (z,su_{\lambda }(z)\big ) \ =\ g\big (z,\underline{u}(z)\big ) \quad \text {for almost all}\ z\in \Omega . \end{aligned}$$

(3.17)

Similarly, using hypothesis $H_f(ii)$, we have

$$\begin{aligned} \frac{f(z,u_{\lambda }(z))}{u_{\lambda }(z)^{p-1}} \ \geqslant \ \frac{f(z,\underline{u}(z))}{\underline{u}(z)^{p-1}} \ =\ \frac{f(z,\underline{u}(z))}{s^{p-1}u_{\lambda }(z)^{p-1}}, \end{aligned}$$

so

$$\begin{aligned} s^{p-1}f\big (z,u_{\lambda }(z)\big ) \ \geqslant \ f\big (z,su_{\lambda }(z)\big ) \ =\ f\big (z,\underline{u}(z)\big ) \quad \text {for almost all}\ z\in \Omega . \end{aligned}$$

(3.18)

Returning to (3.16) and using (3.15), (3.17) and (3.18), we have

$$\begin{aligned}&-\Delta _p \underline{u}(z)+\beta (z)\underline{u}(z)^{p-1}\nonumber \\&\quad = \lambda s^{p-1}g\big (z,u_{\lambda }(z)\big )-s^{p-1}f\big (z,u_{\lambda }(z)\big )\nonumber \\&\quad \leqslant s^{\vartheta -1}\tau g\big ( z,u_{\lambda }(z)\big )-f\big (z,\underline{u}(z)\big )\nonumber \\&\quad \leqslant \tau g\big (z,\underline{u}(z)\big )-f\big (z,\underline{u}(z)\big ) \quad \text {for almost all}\ z\in \Omega . \end{aligned}$$

(3.19)

We consider the following truncation of the reaction in problem $(P)_{\tau }$:

$$\begin{aligned} h_{\tau }(z,\zeta ) \ =\ \left\{ \begin{array}{lll} \tau g\big (z,\underline{u}(z)\big )-f\big (z,\underline{u}(z)\big ) &{} \quad \text {if} &{} \quad \zeta \leqslant \underline{u}(z),\\ \tau g(z,\zeta )-f(z,\zeta ) &{} \quad \text {if} &{} \quad \underline{u}(z)<\zeta . \end{array} \right. \end{aligned}$$

(3.20)

This is a Carathéodory function. We set

$$\begin{aligned} H_{\tau }(z,\zeta ) \ =\ \int _0^{\zeta } h_{\tau }(z,s)\,ds \end{aligned}$$

and consider the $C^1$-functional $\psi _{\tau }:W^{1,p}(\Omega )\longrightarrow \mathbb {R}$, defined by

$$\begin{aligned} \psi _{\tau }(u) \ =\ \frac{1}{p}\Vert \nabla u\Vert _p^p +\frac{1}{p}\int _{\Omega }\beta |u|^p\,dz -\int _{\Omega }H_{\tau }(z,u)\,dz \quad \forall u\in W^{1,p}(\Omega ). \end{aligned}$$

As we did for $\varphi _{\lambda }$ earlier in this proof, we can check that $\psi _{\tau }$ is coercive and sequentially weakly lower semicontinuous. So, we can find $u_{\tau }\in W^{1,p}(\Omega )$, such that

$$\begin{aligned} \psi _{\tau }(u_{\tau }) \ =\ \inf _{u\in W^{1,p}(\Omega )}\psi _{\tau }(u), \end{aligned}$$

so

$$\begin{aligned} \psi _{\tau }'(u_{\tau }) \ =\ 0, \end{aligned}$$

thus

$$\begin{aligned} A(u_{\tau })+\beta |u_{\tau }|^{p-2}u_{\tau } \ =\ N_{h_{\tau }}(u_{\tau }). \end{aligned}$$

(3.21)

On (3.21) we act with $(\underline{u}-u_{\tau })^+\in W^{1,p}(\Omega )$ and obtain

$$\begin{aligned}&\big \langle A(u_{\tau }),\ (\underline{u}-u_{\tau })^+\big \rangle +\int _{\Omega }\beta |u_{\tau }|^{p-2}u_{\tau }(\underline{u}-u_{\tau })^+\,dz\\&\quad = \int _{\Omega } h_{\tau }(z,u_{\tau })(\underline{u}-u_{\tau })^+\,dz\\&\quad = \int _{\Omega }\big (\tau g(z,\underline{u})-f(z,\underline{u})\big )(\underline{u}-u_{\tau })^+\,dz\\&\quad \geqslant \big \langle A(\underline{u}),\ (\underline{u}-u_{\tau })^+\big \rangle +\int _{\Omega }\beta \underline{u}^{p-1}(\underline{u}-u_{\tau })^+\,dz \end{aligned}$$

(see (3.20) and (3.19)), so

$$\begin{aligned}&\int _{\{\underline{u}>u_{\tau }\}}\big (\Vert \nabla u_{\tau }\Vert ^{p-2}\nabla u_{\tau }-\Vert \nabla \underline{u}\Vert ^{p-2}\nabla \underline{u},\ \nabla u_{\tau }-\nabla \underline{u}\big )_{\mathbb {R}}dz \nonumber \\&\quad +\int _{\{\underline{u}>u_{\tau }\}}\beta \big ( |u_{\tau }|^{p-2}u_{\tau }-\underline{u}^{p-1}\big )(u_{\tau }-\underline{u})\,dz \leqslant 0. \end{aligned}$$

(3.22)

We recall the following elementary inequalities (see e.g., Gasiński and Papageorgiou [10, Lemma 6.2.13, p. 740]). If $1<p\leqslant 2$, then

$$\begin{aligned}&(p-1)|y-v|^2(1+|y|+|v|)^{p-2}\nonumber \\&\quad \le \big (|y|^{p-2}y-|v|^{p-2}v,\ y-v\big )_{\mathbb {R}^N} \quad \forall y,v\in \mathbb {R}^N \end{aligned}$$

(3.23)

and if $2<p$, then

$$\begin{aligned} \frac{1}{2^{p-2}}|y-v|^p\ \le \ \big (|y|^{p-2}y-|v|^{p-2}v,\ y-v\big )_{\mathbb {R}^N} \quad \forall y,v\in \mathbb {R}^N. \end{aligned}$$

(3.24)

If $1<p\leqslant 2$, then from (3.22), (3.23) and since $u_{\tau },\underline{u}\in \mathrm {int}\, C_+$, we have

$$\begin{aligned} \frac{p-1}{c_5}\int _{\{\underline{u}>u_{\tau }\}}\Vert \nabla u_{\tau }-\nabla \underline{u}\Vert ^2\,dz \ \leqslant \ 0 \end{aligned}$$

for some $c_5>0$, so

$$\begin{aligned} \big |\{\underline{u}>u_{\tau }\}\big |_N \ =\ 0, \end{aligned}$$

i.e., $\underline{u}\leqslant u_{\tau }$.

If $2<p$, then from (3.22) and (3.24), we have

$$\begin{aligned} \frac{1}{2^{p-2}}\int _{\{\underline{u}>u_{\tau }\}}\Vert \nabla u_{\tau }-\nabla \underline{u}\Vert ^p\,dz \ \leqslant \ 0, \end{aligned}$$

so

$$\begin{aligned} \big |\{\underline{u}>u_{\tau }\}\big |_N \ =\ 0, \end{aligned}$$

i.e., $\underline{u}\leqslant u_{\tau }$.

So, finally $\underline{u}\leqslant u_{\tau }$ and then (3.21) becomes

$$\begin{aligned} A(u_{\tau })+\beta u_{\tau }^{p-1} \ =\ \tau N_g(u_{\tau })-N_f(u_{\tau }) \end{aligned}$$

(see (3.20)), so $u_{\tau }\in \mathrm {int}\,C_+$ is a positive solution of $(P)_{\lambda }$, i.e., $\tau \in \mathcal {Y}$. $\square$

Proposition 3.5

If hypotheses $H_f$, $H_g$ and $H_0$ hold and $\lambda >\lambda _*$, then problem $(P)_{\lambda }$ has at least two positive solutions.

Proof

Let $\tau \in (\lambda _*,\lambda )\cap \mathcal {Y}$. Then, we can find $u_{\tau }\in \mathrm {int}\,C_+$, such that

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle -\Delta _p u_{\tau }(z)+\beta (z) u_{\tau }(z)^{p-1}\ =\ \tau g\big (z,u_{\tau }(z)\big )-f\big (z,u_{\tau }(z)\big ) \quad \text {in} \quad \Omega ,\\ \displaystyle \frac{\partial u_{\tau }}{\partial n} =0 \quad \text {on} \quad \partial \Omega . \end{array} \right. \end{aligned}$$

(3.25)

Proceeding as in the proof of Proposition 3.4, we introduce the following truncation of the reaction:

$$\begin{aligned} \widehat{h}_{\lambda }(z,\zeta ) \ =\ \left\{ \begin{array}{lll} \lambda g\big (z,u_{\lambda }(z)\big )-f\big (z,u_{\lambda }(z)\big ) &{} \quad \text {if} &{} \quad \zeta \leqslant u_{\tau }(z),\\ \lambda g(z,\zeta )-f(z,\zeta ) &{} \quad \text {if} &{} \quad u_{\tau }(z)<\zeta . \end{array} \right. \end{aligned}$$

(3.26)

This is a Carathéodory function. We set

$$\begin{aligned} \widehat{H}_{\lambda }(z,\zeta ) \ =\ \int _0^{\zeta }\widehat{h}_{\lambda }(z,s)\,ds \end{aligned}$$

and consider the $C^1$-functional $\widehat{\psi }:W^{1,p}(\Omega )\longrightarrow \mathbb {R}$, defined by

$$\begin{aligned} \widehat{\psi }_{\lambda }(u) \ =\ \frac{1}{p}\Vert \nabla u\Vert _p^p +\frac{1}{p}\int _{\Omega }\beta |u|^p\,dz -\int _{\Omega }\widehat{H}_{\lambda }(z,u)\,dz \quad \forall u\in W^{1,p}(\Omega ). \end{aligned}$$

As we did for $\varphi _{\lambda }$ in the proof of Proposition 3.4, we can check that $\widehat{\psi }_{\lambda }$ is coercive and sequentially weakly lower semicontinuous. So, we can find $u_{\lambda }^0\in W^{1,p}(\Omega )$, such that

$$\begin{aligned} \widehat{\psi }_{\lambda }(u_{\lambda }^0) \ =\ \inf _{u\in W^{1,p}(\Omega )}\widehat{\psi }_{\lambda }(u), \end{aligned}$$

and

$$\begin{aligned} \widehat{\psi }_{\lambda }'(u_{\lambda }^0) \ =\ 0, \end{aligned}$$

so

$$\begin{aligned} A(u_{\lambda }^0)+\beta |u_{\lambda }^0|^{p-2}u_{\lambda }^0 \ =\ N_{\widehat{h}_{\lambda }}(u_{\lambda }^0). \end{aligned}$$

From this, as before, acting with $(u_{\tau }-u_{\lambda }^0)^+\in W^{1,p}(\Omega )$ and using (3.25) and (3.26), we show that $u_{\tau }\leqslant u_{\lambda }^0$. Hence, we have

$$\begin{aligned} A(u_{\lambda }^0)+\beta (u_{\lambda }^0)^{p-1} \ =\ \lambda N_g (u_{\lambda }^0)-N_f(u_{\lambda }^0) \end{aligned}$$

(see (3.26)), so $u_{\lambda }^0\in \mathrm {int}\,C_+$ is a solution of $(P)_{\lambda }$ and $u_{\lambda }^0\geqslant u_{\tau }$.

Claim 1

$u_{\lambda }^0-u_{\tau }\in \mathrm {int}\,C_+$.

Let $\varrho =\Vert u_{\lambda }^0\Vert _{\infty }$. By hypothesis $H_0$, we can find $\gamma _{\varrho }=\gamma _{\varrho }(\lambda )>0$, such that for all $z\in \Omega$, the function $\zeta \longmapsto \lambda g(z,\zeta )-f(z,\zeta )+\gamma _{\varrho }\zeta ^{\vartheta -1}$ is nondecreasing on $[0,\varrho ]$. For $\delta >0$, we set

$$\begin{aligned} \overline{u}_{\tau } \ =\ u_{\tau }+\delta \ \in \ \mathrm {int}\, C_+. \end{aligned}$$

Then

$$\begin{aligned}&-\Delta _p\overline{u}_{\tau }+\beta \overline{u}_{\tau }^{p-1} +\gamma _{\varrho }\overline{u}_{\tau }^{\vartheta -1}\nonumber \\&\quad \leqslant -\Delta _p u_{\tau }+\beta u_{\tau }^{p-1}+\gamma _{\varrho }u_{\tau }^{\vartheta -1}+\xi (\delta )\nonumber \\&\quad = \tau g(z,u_{\tau })-f(z,u_{\tau })+\gamma _{\varrho }u_{\tau }^{\vartheta -1}+\xi (\delta )\nonumber \\&\quad = \lambda g(z,u_{\tau })-f(z,u_{\tau })+(\tau -\lambda )g(z,u_{\tau }) +\gamma _{\varrho }u_{\tau }^{\vartheta -1}+\xi (\delta )\nonumber \\&\quad \leqslant \lambda g(z,u_{\tau })-f(z,u_{\tau })-(\lambda -\tau )\sigma _0(u_{\tau }) +\gamma _{\varrho }u_{\tau }^{\vartheta -1}+\xi (\delta ) \end{aligned}$$

(3.27)

with $\xi (\delta )\rightarrow 0$ as $\delta \searrow 0$ (see hypothesis $H_g(iv)$ and recall that $\tau <\lambda$).

Since $u_{\tau }\in \mathrm {int}\, C_+$, the function $z\longmapsto \sigma _0\big (u_{\tau }(z)\big )$ is upper semicontinuous on $\overline{\Omega }$ (see hypothesis $H_g(iv)$). So, we can find $z_0\in \overline{\Omega }$, such that

$$\begin{aligned} \sigma _0\big (u_{\tau }(z_0)\big ) \ =\ \max _{z\in \overline{\Omega }}\sigma _0\big (u_{\tau }(z)\big ) \ >\ 0. \end{aligned}$$

(3.28)

We use (3.28) in (3.27). Since $\xi (\delta )\searrow 0$ and $\delta \searrow 0$ and $\lambda >\tau$, we infer that

$$\begin{aligned}&-\Delta _p\overline{u}_{\tau }+\beta \overline{u}_{\tau }^{p-1} +\gamma _{\varrho }\overline{u}_{\tau }^{\vartheta -1}\\&\quad \leqslant \lambda g(z,u_{\tau })-f(z,u_{\tau })+\gamma _{\varrho }u_{\tau }^{\vartheta -1}\\&\quad \leqslant \lambda g(z,u_{\lambda }^0)-f(z,u_{\lambda }^0)+\gamma _{\varrho }(u_{\lambda }^0)^{\vartheta -1}\\&\quad = -\Delta _p u_{\lambda }^0+\beta (u_{\lambda }^0)^{p-1}+\gamma _{\varrho }(u_{\lambda }^0)^{\vartheta -1} \quad \text {for almost all}\ z\in \Omega . \end{aligned}$$

for $\delta >0$ small (see $H_0$ and recall that $u_{\tau }\leqslant u_{\lambda }^0$). Acting on this inequality with $(\overline{u}_{\tau }-u_{\lambda }^0)^+\in W^{1,p}(\Omega )$ and using the nonlinear Green’s identity (see e.g., Gasiński and Papageorgiou [9]) as above, we obtain

$$\begin{aligned} \overline{u}_{\tau } \ =\ u_{\tau }+\delta \ \leqslant \ u_{\lambda }^0 \quad \forall \delta >0\ \text {small}, \end{aligned}$$

so

$$\begin{aligned} u_{\lambda }^0-u_{\tau }\ \in \ \mathrm {int}\, C_+. \end{aligned}$$

This proves Claim 1.

Let

$$\begin{aligned}{}[u_{\tau }) \ =\ \big \{ u\in W^{1,p}(\Omega ):\ u_{\tau }(z)\leqslant u(z) \quad \text {for almost all}\ z\in \Omega \big \}. \end{aligned}$$

From (3.26), we see that

$$\begin{aligned} \widehat{\psi }_{\lambda }|_{[u_{\tau })} \ =\ \varphi _{\lambda }|_{[u_{\tau })}+\widehat{c}, \end{aligned}$$

(3.29)

for some $\widehat{c}\in \mathbb {R}$. Then Claim 1 and (3.29) imply that $u_{\lambda }^0$ is a local $C^1(\overline{\Omega })$-minimizer of $\varphi _{\lambda }$. From Theorem 2.2, it follows that $u_{\lambda }^0$ is a local $W^{1,p}(\Omega )$-minimizer of $\varphi (\lambda )$.

By virtue of hypotheses $H_g$(iii) and $H_f$(iii), for a given $\varepsilon >0$ we can find $\delta =\delta (\varepsilon )>0$, such that

$$\begin{aligned} G(z,\zeta )\ \leqslant \ \frac{\varepsilon }{p}\zeta ^p \quad \text {and}\quad F(z,\zeta )\ \geqslant \ -\frac{\varepsilon }{p}\zeta ^p \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \in (0,\delta ]. \end{aligned}$$

(3.30)

So, if $u\in C^1(\overline{\Omega })$ with $\Vert u\Vert _{C^1(\overline{\Omega })}\leqslant \delta$, then

$$\begin{aligned} \varphi _{\lambda }(u)= & {} \frac{1}{p}\Vert \nabla u\Vert _p^p +\frac{1}{p}\int _{\Omega }\beta |u|^p\,dz -\lambda \int _{\Omega } G(z,u)\,dz +\int _{\Omega } F(z,u)\,dz\\\geqslant & {} \frac{\xi _0}{p}\Vert u\Vert ^p -\frac{\lambda +1}{p}\varepsilon \Vert u^+\Vert ^p\\\geqslant & {} \frac{\xi _0-(\lambda +1)\varepsilon }{p}\Vert u\Vert ^p \end{aligned}$$

(see Lemma 2.5) and (3.30). Choosing $\varepsilon \in \big (0,\frac{\xi _0}{\lambda +1}\big )$, we infer that

$$\begin{aligned} \varphi _{\lambda }(u)\ \geqslant \ 0 \ =\ \varphi _{\lambda }(0) \quad \forall u\in C^1(\overline{\Omega }),\ \Vert u\Vert _{C^1(\Omega )}\leqslant \delta , \end{aligned}$$

so

$$\begin{aligned} u\ =\ 0 \quad \text {is a local } C^1(\overline{\Omega })\text {-minimizer of} \ \varphi _{\lambda } \end{aligned}$$

and thus

$$\begin{aligned} u\ =\ 0 \quad \text {is a local }W^{1,p}(\Omega )\text {-minimizer of }\varphi _{\lambda } \end{aligned}$$

(see Theorem 2.2).

Without any loss of generality, we may assume that

$$\begin{aligned} \varphi _{\lambda }(0) \ =\ 0 \ \leqslant \ \varphi _{\lambda }(u_{\lambda }^0) \end{aligned}$$

(the analysis is similar if the opposite inequality is true). Moreover, we may assume that both local minimizers $u=0$ and $u=u_{\lambda }^0$ are isolated (otherwise it is clear that we have a whole sequence of positive solutions of $(P)_{\lambda }$ and so we are done). Reasoning as in Aizicovici et al. [2, Proposition 29], we can find $\varrho \in \big (0,\Vert u_{\lambda }^0\Vert \big )$ small, such that

$$\begin{aligned} \varphi _{\lambda }(0) \ =\ 0 \ \leqslant \ \varphi _{\lambda }(u_{\lambda }^0) \ <\ \inf \big \{\varphi _{\lambda }(u):\ \Vert u-u_{\lambda }^0\Vert =\varrho \big \} \ =\ \eta _0^{\lambda }. \end{aligned}$$

(3.31)

Recall that $\varphi _{\lambda }$ is coercive (see the proof of Proposition 3.4). Hence it satisfies the Palais–Smale condition. This fact and (3.1) permit the use of the mountain pass theorem (see Theorem 2.1) and so, we obtain $\widehat{u}_{\lambda }\in W^{1,p}(\Omega )$, such that

$$\begin{aligned} \eta _{\varrho }\ \leqslant \ \varphi _{\lambda }(\widehat{u}_{\lambda }) \end{aligned}$$

(3.32)

and

$$\begin{aligned} \varphi _{\lambda }'(\widehat{u}_{\lambda }) \ =\ 0. \end{aligned}$$

(3.33)

From (3.31) and (3.32), it follows that $\widehat{u}_{\lambda }\not \in \{0, u_{\lambda }^0\}$. From (3.33), we have

$$\begin{aligned} A(\widehat{u}_{\lambda }) +\beta \widehat{u}_{\lambda }^{p-1} \ =\ \lambda N_g(\widehat{u}_{\lambda })-N_f(\widehat{u}_{\lambda }), \end{aligned}$$

so $\widehat{u}_{\lambda }\in \mathrm {int}\, C_+$ is a solution of $(P)_{\lambda }$.

So, we conclude that $(P)_{\lambda }$ ($\lambda >\lambda _*$) has at least two positive solutions $u_{\lambda }^0,\widehat{u}_{\lambda }\in \mathrm {int}\,C_+$. $\square$

Next we examine what happens in the critical case $\lambda =\lambda _*$.

Proposition 3.6

If hypotheses $H_f$, $H_g$ and $H_0$ hold, then $\lambda _*\in \mathcal {Y}$.

Proof

Let $\lambda _n>\lambda _*$ for $n\geqslant 1$ be such that $\lambda _n\searrow \lambda _*$ and let $u_n=u_{\lambda _n}\in \mathrm {int}\,C_+$ be positive solutions for problem $(P)_{\lambda }$ for $n\geqslant 1$ (see Proposition 3.4). We have

$$\begin{aligned} A(u_n)+\beta u_n^{p-1} \ =\ \lambda _n N_g (u_n)-N_f(u_n) \quad \forall n\geqslant 1. \end{aligned}$$

(3.34)

By virtue of hypothesis $H_g(ii)$ and since $\vartheta >q$, we have

$$\begin{aligned} \lim _{\zeta \rightarrow +\infty } \frac{g(z,\zeta )}{\zeta ^{\vartheta -1}} \ =\ 0 \quad \text {uniformly for almost all}\ z\in \Omega . \end{aligned}$$

This fact combined with hypothesis $H_g(i)$, implies that for a given $\varepsilon >0$, we can find $c_6=c_6(\varepsilon )>0$, such that

$$\begin{aligned} g(z,\zeta )\zeta \ \leqslant \ \frac{\varepsilon }{\vartheta }(\zeta ^+)^{\vartheta }+c_6 \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \in \mathbb {R}. \end{aligned}$$

(3.35)

In a similar fashion, using hypotheses $H_f(i)$ and (ii), we see that we can find $\eta >0$ and $c_7>0$, such that

$$\begin{aligned} f(z,\zeta )\zeta \ \geqslant \ \frac{\eta }{\vartheta }(\zeta ^+)^{\vartheta }-c_7 \quad \text {for almost all}\ z\in \Omega ,\ \text {all}\ \zeta \in \mathbb {R}. \end{aligned}$$

(3.36)

On (3.34) we act with $u_n\in W^{1,p}(\Omega )$ and obtain

$$\begin{aligned} \Vert \nabla u_n\Vert _p^p +\int _{\Omega }\beta u_n^p\,dz= & {} \lambda _n\int _{\Omega }g(z,u_n)u_n\,dz -\int _{\Omega }f(z,u_n)u_n\,dz\nonumber \\\leqslant & {} \frac{\lambda _n\varepsilon -\eta }{\vartheta }\Vert u_n\Vert _{\vartheta }^{\vartheta }+c_8 \quad \forall n\geqslant 1, \end{aligned}$$

(3.37)

for some $c_8>0$ (see (3.35) and (3.36)).

We choose $\varepsilon \in \big (0,\frac{\eta }{\lambda _1}\big )$ (recall that $\lambda _n\leqslant \lambda _1$ for all $n\geqslant 1$). Then from (3.37) and Lemma 2.5, it follows that

$$\begin{aligned} \xi _0\Vert u_n\Vert ^p\ \leqslant \ c_8 \quad \forall n\geqslant 1 \end{aligned}$$

and so the sequence $\{u_n\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega )$ is bounded.

So, passing to a subsequence if necessary, we may assume that

$$\begin{aligned} u_n\longrightarrow & {} u_*\quad \text {weakly in}\ W^{1,p}(\Omega ), \end{aligned}$$

(3.38)

$$\begin{aligned} u_n\longrightarrow & {} u_*\quad \text {in}\ L^{\theta }(\Omega ), \end{aligned}$$

(3.39)

with $\theta <p^*$. On (3.34) we act with $u_n-u_*$, pass to the limit as $n\rightarrow +\infty$ and use (3.38). We obtain

$$\begin{aligned} \lim _{n\rightarrow +\infty }\big \langle A(u_n),\ u_n-u_*\big \rangle \ =\ 0, \end{aligned}$$

so

$$\begin{aligned} u_n\ \longrightarrow \ u_* \quad \text {in}\ W^{1,p}(\Omega ) \end{aligned}$$

(3.40)

(see Proposition 2.4).

So, if in (3.34) we pass to the limit as $n\rightarrow +\infty$ and use (3.40), we obtain

$$\begin{aligned} A(u_*)+\beta u_*^{p-1} \ =\ \lambda _* N_g (u_*)-N_f(u_*), \end{aligned}$$

so $u_*\in C_+$ and it solves problem $(P)_{\lambda _*}$.

It remains to show that $u_*\ne 0$. Arguing by contradiction, suppose that $u_*=0$. From (3.34), we have

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle -\Delta _p u_n(z)+\beta (z) u_n(z)^{p-1}\ =\ \lambda g\big (z,u_n(z)\big )-f\big (z,u_n(z)\big ) \quad \text {in} \quad \Omega ,\\ \displaystyle \frac{\partial u_n}{\partial n} =0 \quad \text {on} \quad \partial \Omega . \end{array} \right. \end{aligned}$$

(3.41)

From (3.41) and Theorem 2 of Lieberman [24], we know that we can find $\alpha \in (0,1)$ and $M>0$, such that

$$\begin{aligned} u_n\ \in \ C^{1,\alpha }(\overline{\Omega }) \quad \text {and}\quad \Vert u_n\Vert _{C^{1,\alpha }(\overline{\Omega })}\ \leqslant \ M \quad \forall n\geqslant 1. \end{aligned}$$

From the compactness of the embedding $C^{1,\alpha }(\overline{\Omega })\subseteq C^1(\overline{\Omega })$, we have

$$\begin{aligned} u_n\ \longrightarrow \ u_* \quad \text {in}\ C^1(\overline{\Omega }). \end{aligned}$$

Let

$$\begin{aligned} y_n \ =\ \frac{u_n}{\Vert u_n\Vert } \quad \forall n\geqslant 1. \end{aligned}$$

Then

$$\begin{aligned} y_n\ \geqslant \ 0,\quad \Vert y_n\Vert \ =\ 1 \quad \forall n\geqslant 1. \end{aligned}$$

So, passing to a subsequence if necessary, we may assume that

$$\begin{aligned} y_n\longrightarrow & {} y_*\quad \text {weakly in}\ W^{1,p}(\Omega ), \end{aligned}$$

(3.42)

$$\begin{aligned} y_n\longrightarrow & {} y_*\quad \text {in}\ L^{\vartheta }(\Omega ). \end{aligned}$$

(3.43)

From (3.34), we have

$$\begin{aligned} A(y_n)+\beta y_n^{p-1} \ =\ \lambda _n\frac{N_g(u_n)}{\Vert u_n\Vert ^{p-1}} -\frac{N_f(u_n)}{\Vert u_n\Vert ^{p-1}} \quad \forall n\geqslant 1. \end{aligned}$$

(3.44)

From hypotheses $H_g(i)$, (iii) and $H_f(i)$, (iii), it follows that

$$\begin{aligned} \text {the sequences}\ \bigg \{\frac{N_g(u_n)}{\Vert u_n\Vert ^{p-1}}\bigg \}_{n\geqslant 1}, \bigg \{\frac{N_f(u_n)}{\Vert u_n\Vert ^{p-1}}\bigg \}_{n\geqslant 1} \subseteq L^{p'}(\Omega )\ \text {are bounded} \end{aligned}$$

(where $\frac{1}{p}+\frac{1}{p'}=1$).

Acting on (3.44) with $y_n-y_*$, passing to the limit as $n\rightarrow +\infty$ and using (3.42), we obtain

$$\begin{aligned} \lim _{n\rightarrow +\infty }\big \langle A(y_n),\ y_n-y_*\big \rangle \ =\ 0, \end{aligned}$$

so

$$\begin{aligned} y_n\ \longrightarrow \ y_* \quad \text {in}\ W^{1,p}(\Omega ) \end{aligned}$$

(3.45)

(see Proposition 2.4) and so $\Vert y_*\Vert =1$.

Note that by virtue of hypotheses $H_g$(iii) and $H_f$(iii), we have

$$\begin{aligned} \frac{N_f(u_n)}{\Vert u_n\Vert ^{p-1}}\ \longrightarrow \ 0 \quad \text {and}\quad \frac{N_g(u_n)}{\Vert u_n\Vert ^{p-1}}\ \longrightarrow \ \widehat{\zeta }y_*^{p-1} \quad \text {weakly in}\ L^p(\Omega ), \end{aligned}$$

(3.46)

with $0\leqslant \widehat{\zeta }(z)\leqslant \zeta ^*$ for almost all $z\in \Omega$. So, if in (3.44) we pass to the limit as $n\rightarrow +\infty$ and we use (3.45) and (3.46), we obtain

$$\begin{aligned} A(y_*)+\beta y_*^{p-1} \ =\ -\widehat{\zeta }y_*^{p-1}, \end{aligned}$$

so

$$\begin{aligned} \Vert \nabla y_*\Vert _p^p +\int _{\Omega }\beta y_*^p\,dz \ \leqslant \ -\int _{\Omega }\widehat{\zeta }y_*^p\,dz \ \leqslant \ 0, \end{aligned}$$

thus

$$\begin{aligned} \xi _0\Vert y_*\Vert ^p \ \leqslant \ 0 \end{aligned}$$

(see Lemma 2.5) and finally, we have that $y_*=0$, which contradicts to (3.45).

This proves that $u_*\ne 0$. Hence $u_*\in \mathrm {int}\,C_+$ is a solution of problem $(P)_{\lambda _*}$. Therefore $\lambda _*\in \mathcal {Y}$. $\square$

We show that for every $\lambda \geqslant \lambda _*$, problem $(P)_{\lambda }$ has an extremal (smallest) positive solution.

Proposition 3.7

If hypotheses $H_f$, $H_g$, and $H_0$ hold and $\lambda \geqslant \lambda _*$, then problem $(P)_{\lambda }$ has a smallest positive solution $u_{\lambda }^*\in \mathrm {int}\, C_+$.

Proof

Let $S(\lambda )$ be the set of positive solutions for problem $(P)_{\lambda }$. Since $\lambda \geqslant \lambda _*$, $S(\lambda )\ne 0$ and $S(\lambda )\subseteq \mathrm {int}\,C_+$. Let $C\subseteq S(\lambda )$ be a chain (i.e., a nonempty linearly ordered subset of $S(\lambda )$). From Dunford and Schwartz [6, p.336], we know that we can find a sequence $\{u_n\}_{n\geqslant 1}\subseteq C$, such that

$$\begin{aligned} \inf _{n\geqslant 1}u_n \ =\ \inf C. \end{aligned}$$

Moreover, from Lemma 11.5(a) of Heikkilä and Lakshmikantham [22, p. 15], we know that we may assume that the sequence $\{u_n\}_{n\geqslant 1}$ is decreasing. We have

$$\begin{aligned} A(u_n)+\beta u_n^{p-1} \ =\ \lambda N_g(u_n) -N_f(u_n) \quad \forall n\geqslant 1, \end{aligned}$$

(3.47)

so

$$\begin{aligned} \Vert \nabla u_n\Vert _p^p +\int _{\Omega }\beta u_n^p\,dz \ =\ \int _{\Omega }\big (\lambda g(z,u_n)-f(z,u_n)\big )u_n\,dz \ \leqslant \ M_1 \quad \forall n\geqslant 1, \end{aligned}$$

for some $M_1>0$ (see hypotheses $H_g(i)$, $H_f(i)$ and recall that $u_n\leqslant u_1$ for all $n\geqslant 1$). So,

$$\begin{aligned} \xi _0\Vert u_n\Vert ^p \ \leqslant \ M_1 \quad \forall n\geqslant 1 \end{aligned}$$

(see Lemma 2.5) and thus the sequence $\{u_n\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega )$ is bounded.

So, passing to a subsequence if necessary, we may assume that

$$\begin{aligned} u_n\longrightarrow & {} u_*\quad \text {weakly in}\ W^{1,p}(\Omega ), \end{aligned}$$

(3.48)

$$\begin{aligned} u_n\longrightarrow & {} u_*\quad \text {in}\ L^{\theta }(\Omega ), \end{aligned}$$

(3.49)

with $\theta <p^*$. On (3.47) we act with $u_n-u_*$, pass to the limit as $n\rightarrow +\infty$ and use (3.48). Then

$$\begin{aligned} \lim _{n\rightarrow +\infty }\big \langle A(u_n),\ u_n-u_*\big \rangle \ =\ 0, \end{aligned}$$

so

$$\begin{aligned} u_n\ \longrightarrow \ u_* \quad \text {in}\ W^{1,p}(\Omega ) \end{aligned}$$

(see Proposition 2.4).

Reasoning as in the proof of Proposition 3.6, we show that $u_*\ne 0$ and so $u_*\in \mathrm {int}\,C_+$ is a positive solution of $(P)_{\lambda }$. Hence $u_*=\inf C\in S(\lambda )$ and since C was an arbitrary chain, from the Kuratowski-Zorn lemma, we infer that $S(\lambda )$ has a minimum element $u_{\lambda }^*\in \mathrm {int}\,C_+$. But $S(\lambda )$ is downward directed (i.e., if $u,v\in S(\lambda )$, then there exists $y\in S(\lambda )$, such that $y\leqslant \min \{u,v\}$; see Aizicovici et al. [3]). So, it follows that $u_{\lambda }^*\leqslant u$ for all $u\in S(\lambda )$, i.e., $u_{\lambda }^*\in \mathrm {int}\,C_+$ is the smallest positive solution of problem $(P)_{\lambda }$. $\square$

Summarizing the situation, we have the following bifurcation-type theorem describing the dependence of positive solutions of $(P)_{\lambda }$ on the parameter $\lambda >0$.

Theorem 3.8

If hypotheses $H_f$, $H_g$ and $H_0$ hold, then there exists $\lambda _*>0$, such that:

(a) :

for all $\lambda >\lambda _*$, problem $(P)_{\lambda }$ has at least two positive solutions

$$\begin{aligned} u_0,\ \widehat{u}\ \in \ \mathrm {int}\,C_+; \end{aligned}$$

(b) :

for $\lambda =\lambda _*$, problem $(P)_{\lambda }$ has at least one positive solution $u_*\in \mathrm {int}\,C_+$;

(c) :

for all $\lambda \in (0,\lambda _*)$, problem $(P)_{\lambda }$ has no positive solution.

Moreover, if $\lambda \geqslant \lambda _*$, then problem $(P)_{\lambda }$ has a smallest positive solution $u_{\lambda }^*\in \mathrm {int}\,C_+$.