Nonlinear Eigenvalue Problems for the Dirichlet (p,2)-Laplacian

Abstract

We consider a nonlinear eigenvalue problem driven by the Dirichlet $(p,2)$-Laplacian. The parametric reaction is a Carathéodory function which exhibits $(p-1)$-sublinear growth as $x\to +\infty$ and as $x\to 0^{+}$. Using variational tools and truncation and comparison techniques, we prove a bifurcation-type theorem describing the “spectrum” as $\lambda >0$ varies. We also prove the existence of a smallest positive eigenfunction for every eigenvalue. Finally, we indicate how the result can be extended to $(p,q)$-equations ($q\ne 2$).

1. Introduction

Let $\mathsf{\Omega }\subseteq {\mathbb{R}}^N$ be a bounded domain with $C^2$-boundary $\partial \mathsf{\Omega }$. In this paper, we study the following nonlinear eigenvalue problem for the Dirichlet $(p,2)$-Laplacian

$$\left(P_{\lambda }\right)\left\{\begin{array}{c}-{\mathsf{\Delta }}_{p}u\left(z\right)-\mathsf{\Delta }u\left(z\right)=\lambda f(z,u\left(z\right))\mathrm{in}\mathsf{\Omega },\hfill \\ {u|}_{\partial \mathsf{\Omega }}=0,u⩾0,\lambda >0,2<p.\hfill \end{array}\right.$$

For every $r\in (1,\infty )$ by ${\mathsf{\Delta }}_r$ we denote the r-Laplacian differential operator defined by

$${\mathsf{\Delta }}_{r}u={\mathrm{div}\left(\right|Du|}^{r-2}Du)\forall u\in W_0^{1,p}\left(\mathsf{\Omega }\right)$$

($Du$ stands for the gradient of u). When $r=2$, we have the usual Laplacian denoted by $\mathsf{\Delta }$.

In the reaction, $\lambda >0$ is a parameter and $f(z,x)$ is a Carathéodory function. Such a function is jointly measurable. We assume that for almost all $z\in \mathsf{\Omega }$, $f(z,·)$ is $(p-1)$-sublinear as $x\to +\infty$. We are looking for positive solutions as the parameter $\lambda >0$ varies. Our work complements those by Gasiński and Papageorgiou [1] and Papageorgiou, Rădulescu and Repovš [2] where the reaction is $(p-1)$-superlinear in $x\in \mathbb{R}$. Moreover, in the aforementioned works, the equation is driven by the p-Laplacian differential operator which is homogeneous, a property used by the authors in the proof of their results. In contrast, here, the $(p,2)$-Laplace differential operator is not homogeneous.

We mention that equations driven by the sum of two differential operators of different structures (such as $(p,2)$-equations) arise in the mathematical models of many physical processes. We refer to the survey papers of Marano and Mosconi [3], Rădulescu [4] and the references therein.

2. Mathematical Background—Hypotheses

The main spaces in the analysis of problem $\left(P_{\lambda }\right)$ are the Sobolev space $W_0^{1,p}\left(\mathsf{\Omega }\right)$ and the Banach space

$$C_0^1\left(\overline{\mathsf{\Omega }}\right)=\{u\in C^1\left(\overline{\mathsf{\Omega }}\right):u{|}_{\partial \mathsf{\Omega }}=0\}.$$

By $\parallel ·\parallel$, we denote the norm of the Sobolev space $W_0^{1,p}\left(\mathsf{\Omega }\right)$. On account of the Poincaré inequality, we have

$$\parallel u\parallel ={\parallel Du\parallel }_p\forall u\in W_0^{1,p}\left(\mathsf{\Omega }\right).$$

The Banach space $C_0^1\left(\mathsf{\Omega }\right)$ is an ordered Banach space with positive (order) cone

$$C_{+}=\{u\in C_0^1\left(\mathsf{\Omega }\right):u\left(z\right)⩾0\mathrm{for}\mathrm{all}z\in \overline{\mathsf{\Omega }}\}.$$

This cone has a nonempty interior given by

$$\mathrm{int}{C}_{+}=\{u\in C_{+}:u\left(z\right)>0\mathrm{for}\mathrm{all}z\in \mathsf{\Omega },\frac{\partial u}{\partial n}{|}_{\partial \mathsf{\Omega }}<0\},$$

with n being the outward unit normal on $\partial \mathsf{\Omega }$ and $\frac{\partial u}{\partial n}={(Du,n)}_{{\mathbb{R}}^N}$.

We know that if $r\in (1,+\infty )$, then $W_0^{1,r}{\left(\mathsf{\Omega }\right)}^{*}=W^{-1,r^{\prime }}\left(\mathsf{\Omega }\right)$ ($\frac{1}{r}+\frac{1}{r^{\prime }}=1$). Let $A_r:W_0^{1,r}\left(\mathsf{\Omega }\right)⟶W^{-1,r^{\prime }}\left(\mathsf{\Omega }\right)$ by the operator defined by

$$\langle A_r\left(u\right),h\rangle ={\int }_{\mathsf{\Omega }}{\left|Du\right|}^{r-2}{(Du,Dh)}_{{\mathbb{R}}^N}dz\mathrm{for}\mathrm{all}u,h\in W_0^{1,r}\left(\mathsf{\Omega }\right).$$

The next proposition gathers the main properties of this operator (see Gasiński and Papageorgiou [5]).

Proposition 1.

The operator $A_r:W_0^{1,r}\left(\mathsf{\Omega }\right)⟶W^{-1,r^{\prime }}\left(\mathsf{\Omega }\right)$ is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (thus maximal monotone too) and of type ${\left(S\right)}_{+}$, that is, $A_r$ has the following property:

• if $u_n⟶u$ weakly in $W_0^{1,r}\left(\mathsf{\Omega }\right)$ and ${\displaystyle \underset{n\to \infty }{lim\; sup}\langle A_r\left(u_n\right),u_n-u\rangle ⩽0}$, then $u_n⟶u$ in $W_0^{1,r}\left(\mathsf{\Omega }\right)$.

If $r=2$, then we write $A_2=A\in \mathcal{L}(H_0^1\left(\mathsf{\Omega }\right),H^{-1}\left(\mathsf{\Omega }\right))$.

The Dirichlet r-Laplace differential operator has a principal eigenvalue denoted by ${\widehat{\lambda }}_1\left(r\right)$. Therefore, if we consider the nonlinear eigenvalue problem

$$\left\{\begin{array}{c}-{\mathsf{\Delta }}_{r}u\left(z\right)=\widehat{\lambda }{\left|u\left(z\right)\right|}^{r-2}u\left(z\right)\mathrm{in}\mathsf{\Omega },\hfill \\ {u|}_{\partial \mathsf{\Omega }}=0,\hfill \end{array}\right.$$

then this problem has a smallest eigenvalue ${\widehat{\lambda }}_1\left(r\right)>0$ which is isolated and simple. It has the following variational characterization:

$${\widehat{\lambda }}_1\left(r\right)=\underset{u\in W_0^{1,r}\left(\mathsf{\Omega }\right),u\ne 0}{inf}\frac{{\parallel Du\parallel }_r^r}{{\parallel u\parallel }_r^r}.$$

(1)

For $x\in \mathbb{R}$, we define $x^{\pm }=max\{\pm x,0\}$. Then, for $u\in W_0^{1,p}\left(\mathsf{\Omega }\right)$, we set $u^{\pm }\left(z\right)=u{\left(z\right)}^{\pm }$ for all $z\in \mathsf{\Omega }$. We know that

$$u^{\pm }\in W_0^{1,p}\left(\mathsf{\Omega }\right),u=u^{+}=u^{-},\left|u\right|=u^{+}+u^{-}.$$

A set $S\subseteq W_0^{1,p}\left(\mathsf{\Omega }\right)$ is said to be “downward directed”, if given $u_1,u_2\in S$, we can find $u\in S$ such that $u⩽u_1$, $u⩽u_2$.

If $u,v:\mathsf{\Omega }⟶\mathbb{R}$ are measurable functions, then we write $u\prec v$ if and only if for all compact sets $K\subseteq \mathsf{\Omega }$, we have

$$0<c_K⩽v\left(z\right)-u\left(z\right)\mathrm{for}\mathrm{a}.\mathrm{a}.z\in K.$$

Evidently if $u,v\in C\left(\overline{\mathsf{\Omega }}\right)$ and $u\left(z\right)<v\left(z\right)$ for all $z\in \mathsf{\Omega }$, then $u\prec v$.

Now, we introduce the hypotheses on the reaction $f(z,x)$.

H:$f:\mathsf{\Omega }\times \mathbb{R}⟶\mathbb{R}$ is a Carathéodory function such that for a.a. $z\in \mathsf{\Omega }$, $f(z,0)=0$, $f(z,x)>0$ for all $x>0$ and

(i)

For every $\varrho >0$, there exists $a_{\varrho }\in L^{\infty }\left(\mathsf{\Omega }\right)$ such that

$$f(z,x)⩽a_{\varrho }\left(z\right)\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega },\mathrm{all}0⩽x⩽\varrho ;$$

(ii)

${lim}_{x\to +\infty }\frac{f(z,x)}{x^{p-1}}=0$ uniformly for a.a. $z\in \mathsf{\Omega }$;

(iii)

${lim}_{x\to 0^{+}}\frac{f(z,x)}{x^{p-1}}=0$ uniformly for a.a. $z\in \mathsf{\Omega }$;

(iv)

for every $\varrho >0$, there exists s${\widehat{\xi }}_{\varrho }>0$ such that for a.a. $z\in \mathsf{\Omega }$, the function $x⟼f(z,x)+{\widehat{\xi }}_{\varrho }{x}^{p-1}$ is nondecreasing on $[0,\varrho ]$.

Remark 1.

Since we look for positive solutions and the above hypotheses concern the positive semiaxis ${\mathbb{R}}_{+}=[0,+\infty )$, without any loss of generality we may assume that

$$f(z,x)=0fora.a.z\in \mathsf{\Omega },allx⩽0.$$

(2)

Hypothesis $H\left(ii\right)$ implies that $f(z,·)$ is $(p-1)$-sublinear as $x\to +\infty$ while hypothesis $H\left(iii\right)$ says that $f(z,·)$ is sublinear near $0^{+}$. Hypothesis $H\left(iv\right)$ is essentially a one-sided local Lipschitz condition.

3. Positive Solutions

We introduce the following two sets:

$$\begin{array}{ccc}\hfill \mathcal{L}& =& \{\lambda >0:\mathrm{problem}\left(P_{\lambda }\right)\mathrm{admits}\mathrm{a}\mathrm{positive}\mathrm{solution}\};\hfill \\ \hfill S_{\lambda }& =& \mathrm{the}\mathrm{set}\mathrm{of}\mathrm{positive}\mathrm{solutions}\mathrm{for}\mathrm{problem}\left(P_{\lambda }\right).\hfill \end{array}$$

We also set

$${\lambda }_{*}=inf\mathcal{L}.$$

First, we establish the existence of admissible parameters (eigenvalues) and determine the regularity properties of the corresponding solutions (eigenfunctions).

Proposition 2.

If hypotheses H hold, then $\mathcal{L}\ne \varnothing$ and $S_{\lambda }\subseteq \mathrm{int}{C}_{+}$ for all $\lambda >0$.

Proof. 

For every $\lambda >0$, let ${\phi }_{\lambda }:W_0^{1,p}\left(\mathsf{\Omega }\right)⟶\mathbb{R}$ be the $C^1$-functional defined by

$${\phi }_{\lambda }\left(u\right)=\frac{1}{p}{\parallel Du\parallel }_p^p+\frac{1}{2}{\parallel Du\parallel }_2^2-{\int }_{\mathsf{\Omega }}F(z,u^{+})dz\forall u\in W_0^{1,p}\left(\mathsf{\Omega }\right),$$

with $F(z,x)={\int }_0^{x}f(z,s)ds$. From hypotheses $H\left(i\right),\left(ii\right)$, we see that given $\epsilon >0$, we can find $c_{\epsilon }>0$ such that

$$\begin{array}{c}\hfill 0⩽F(z,x)⩽\frac{\epsilon }{p}{x}^p+c_{\epsilon }\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega },\mathrm{all}x⩾0.\end{array}$$

(3)

For $u\in W_0^{1,p}\left(\mathsf{\Omega }\right)$, using (3) we have

$${\phi }_{\lambda }\left(u\right)⩾\frac{1}{p}\left({\parallel Du\parallel }_p^p-\lambda \epsilon {\parallel u\parallel }_p^p\right)+\frac{1}{2}{\parallel Du\parallel }_p^p-\lambda c_{\epsilon }{\left|\mathsf{\Omega }\right|}_N,$$

with ${|·|}_N$ being the Lebesgue measure on ${\mathbb{R}}^N$. Using (1) with $r=p$, we obtain

$${\phi }_{\lambda }\left(u\right)⩾\frac{1}{p}\left(1-\frac{\lambda \epsilon }{{\widehat{\lambda }}_p\left(p\right)}\right){\parallel Du\parallel }_p^p-\lambda c_{\epsilon }{\left|\mathsf{\Omega }\right|}_N.$$

Choosing $\epsilon \in (0,\frac{{\widehat{\lambda }}_1\left(p\right)}{\lambda })$, we infer that

$${\phi }_{\lambda }\left(u\right)⩾c_1{\parallel u\parallel }^p-\lambda c_{\epsilon }{\left|\mathsf{\Omega }\right|}_N,$$

for some $c_1>0$ and thus ${\phi }_{\lambda }$ is coercive.

Additionally, using the Sobolev imbedding theorem, we see that ${\phi }_{\lambda }$ is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find $u_0\in W_0^{1,p}\left(\mathsf{\Omega }\right)$ such that

$$\begin{array}{c}\hfill {\phi }_{\lambda }\left(u_0\right)=\underset{u\in W_0^{1,p}\left(\mathsf{\Omega }\right)}{min}{\phi }_{\lambda }\left(u\right).\end{array}$$

(4)

On account of the strict positivity of $f(z,·)$, if $\overline{u}\in \mathrm{int}{C}_{+}$, then

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}F(z,\overline{u})dz>0.\end{array}$$

(5)

Then, we have

$$\begin{array}{ccc}\hfill {\phi }_{\lambda }\left(\overline{u}\right)& =& \frac{1}{p}\parallel D\overline{u}{\parallel }_p^p+\frac{1}{2}{\parallel D\overline{u}\parallel }_2^2-\lambda {\int }_{\mathsf{\Omega }}F(z,\overline{u})dz\hfill \\ & =& c_2-\lambda {\int }_{\mathsf{\Omega }}F(z,\overline{u})dz,\hfill \end{array}$$

with $c_2=c_2\left(\overline{u}\right)>0$. From (5) and by choosing $\lambda >0$ big, we have

$${\phi }_{\lambda }\left(\overline{u}\right)<0,$$

so

$${\phi }_{\lambda }\left(u_0\right)<0={\phi }_{\lambda }\left(0\right)$$

(see (4)) and thus

$$u_0\ne 0.$$

From (4), we have

$${\phi }_{\lambda }^{\prime }\left(u_0\right)=0,$$

so

$$\begin{array}{c}\hfill \langle A_p\left(u_0\right),h\rangle +\langle A\left(u_0\right),h\rangle =\lambda {\int }_{\mathsf{\Omega }}f(z,u_0^{+})hdz\forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right).\end{array}$$

(6)

In (6), we choose $h=-u_0^{-}\in W_0^{1,p}\left(\mathsf{\Omega }\right)$. We obtain

$$\parallel D{u}_0^{-}{\parallel }_p⩽0,$$

thus $u_0⩾0$ and $u_0\ne 0$.

Then, from (6), we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-{\mathsf{\Delta }}_p{u}_0\left(z\right)-\mathsf{\Delta }{u}_0\left(z\right)=\lambda f(z,u_0\left(z\right))\mathrm{in}\mathsf{\Omega },\hfill \\ u_0{|}_{\partial \mathsf{\Omega }}=0,\hfill \end{array}\right.\end{array}$$

(7)

for $\lambda >0$ big and so $\mathcal{L}\ne \varnothing$.

From Theorem 7.1 of Ladyzhenskaya and Ural’tseva [6], we have that $u_0\in L^{\infty }\left(\mathsf{\Omega }\right)$. Then, the nonlinear regularity theory of Lieberman [7] implies that $u_0\in C_{+}\backslash \left\{0\right\}$. Let $\varrho =\parallel u_0{\parallel }_{\infty }$ and let ${\widehat{\xi }}_{\varrho }>0$ be as postulated by hypothesis $H\left(iv\right)$. From (7), we have

$$-{\mathsf{\Delta }}_p{u}_0\left(z\right)-\mathsf{\Delta }{u}_0\left(z\right)+\lambda {\widehat{\xi }}_{\varrho }{u}_0{\left(z\right)}^{p-1}⩾0\mathrm{in}\mathsf{\Omega },$$

so

$${\mathsf{\Delta }}_p{u}_0\left(z\right)+\mathsf{\Delta }{u}_0\left(z\right)⩽\lambda {\widehat{\xi }}_{\varrho }{u}_0{\left(z\right)}^{p-1}\mathrm{in}\mathsf{\Omega },$$

and thus $u_0\in \mathrm{int}{C}_{+}$ (see Pucci and Serrin [8] (pp. 111, 120)). Therefore, we conclude that $S_{\lambda }\subseteq \mathrm{int}{C}_{+}$ for all $\lambda >0$.  □

Next, we show that $\mathcal{L}$ is connected (more precisely, an upper half-line).

Proposition 3.

If hypotheses H hold, $\lambda \in \mathcal{L}$ and $\vartheta >\lambda$, then $\vartheta \in \mathcal{L}$.

Proof. 

Since $\lambda \in \mathcal{L}$, we can find $u_{\lambda }\in S_{\lambda }\in \mathrm{int}{C}_{+}$ (see Proposition 2). We introduce the Carathéodory function $k(z,x)$ defined by

$$k(z,x)=\left\{\begin{array}{ccc}f(z,u_{\lambda }\left(z\right))\hfill & \mathrm{if}\hfill & x⩽u_{\lambda }\left(z\right)\hfill \\ f(z,x)\hfill & \mathrm{if}\hfill & u_{\lambda }\left(z\right)<x.\hfill \end{array}\right.$$

(8)

We set

$$K(z,x)={\int }_0^{x}k(z,s)ds$$

and consider the $C^1$-functional ${\psi }_{\vartheta }:W_0^{1,p}\left(\mathsf{\Omega }\right)⟶\mathbb{R}$ defined by

$${\psi }_{\vartheta }\left(u\right)=\frac{1}{p}{\parallel Du\parallel }_p^p+\frac{1}{2}{\parallel Du\parallel }_2^2-{\int }_{\mathsf{\Omega }}\vartheta K(z,u)dz\forall u\in W_0^{1,p}\left(\mathsf{\Omega }\right).$$

Note that (8) and hypotheses $H\left(i\right),\left(ii\right)$ imply that, given $\epsilon >0$, we can find ${\widehat{c}}_{\epsilon }>0$ such that

$$K(z,x)⩽\frac{\epsilon }{p}{x}^p+{\widehat{c}}_{\epsilon }\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega },\mathrm{all}x\in \mathbb{R}.$$

(9)

Using (9) and choosing $\epsilon >0$ small, as in the proof of Proposition 2, we show that ${\psi }_{\vartheta }$ is coercive. In addition, it is sequentially weakly lower semicontinuous. Therefore, we can find $u_{\vartheta }\in W_0^{1,p}\left(\mathsf{\Omega }\right)$ such that

$${\psi }_{\vartheta }\left(u_{\vartheta }\right)=\underset{u\in W_0^{1,p}\left(\mathsf{\Omega }\right)}{min}{\psi }_{\vartheta }\left(u\right),$$

so ${\psi }_{\vartheta }^{\prime }\left(u_{\vartheta }\right)=0$ and thus

$$\langle A_p\left(u_{\vartheta }\right),h\rangle +\langle A\left(u_{\vartheta }\right),h\rangle ={\int }_{\mathsf{\Omega }}\vartheta k(z,u_{\vartheta })hdz\forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right).$$

(10)

In (10), we choose $h={(u_{\lambda }-u_{\vartheta })}^{+}\in W_0^{1,p}\left(\mathsf{\Omega }\right)$. Then, using (8), we have

$$\begin{array}{ccc}& & \langle A_p\left(u_{\vartheta }\right),{(u_{\lambda }-u_{\vartheta })}^{+}\rangle +\langle A\left(u_{\vartheta }\right),{(u_{\lambda }-u_{\vartheta })}^{+}\rangle \hfill \\ & =& {\int }_{\mathsf{\Omega }}\vartheta f(z,u_{\lambda }){(u_{\lambda }-u_{\vartheta })}^{+}dz\hfill \\ & ⩾& {\int }_{\mathsf{\Omega }}\lambda f(z,u_{\lambda }){(u_{\lambda }-u_{\vartheta })}^{+}dz\hfill \\ & =& \langle A_p\left(u_{\lambda }\right),{(u_{\lambda }-u_{\vartheta })}^{+}\rangle +\langle A\left(u_{\lambda }\right),{(u_{\lambda }-u_{\vartheta })}^{+}\rangle \hfill \end{array}$$

since $f⩾0$ and $u_{\lambda }\in S_{\lambda }$. Thus,

$$u_{\lambda }⩽u_{\vartheta }$$

(11)

(see Proposition 1).

From (8), (10) and (11), we infer that

$$\left\{\begin{array}{c}-{\mathsf{\Delta }}_p{u}_{\vartheta }\left(z\right)-\mathsf{\Delta }{u}_{\vartheta }\left(z\right)=\vartheta f(z,u_{\vartheta }\left(z\right))\mathrm{in}\mathsf{\Omega },\hfill \\ u_{\vartheta }{|}_{\partial \mathsf{\Omega }}=0,\hfill \end{array}\right.$$

so $u_{\vartheta }\in S_{\vartheta }\subseteq C_{+}$ and thus $\vartheta \in \mathcal{L}$.  □

A byproduct of the above proof is the following corollary.

Corollary 1.

If hypotheses H hold, $\lambda \in \mathcal{L}$ and $u_{\lambda }\in S_{\lambda }\subseteq \mathrm{int}{C}_{+}$ and $\vartheta >\lambda$, then $\vartheta \in \mathcal{L}$ and we can find $u_{\vartheta }\in S_{\vartheta }\subseteq \mathrm{int}{C}_{+}$ such that $u_{\lambda }⩽u_{\vartheta }$.

We can improve this corollary using the strong comparison principle of Gasiński and Papageorgiou [1] (Proposition 3.2).

Proposition 4.

If hypotheses H hold, $\lambda \in \mathcal{L}$ and $u_{\lambda }\in S_{\lambda }\subseteq \mathrm{int}{C}_{+}$ and $\vartheta >\lambda$, then $\vartheta \in \mathcal{L}$ and we can find $u_{\vartheta }\in S_{\vartheta }\subseteq \mathrm{int}{C}_{+}$ such that $u_{\vartheta }-u_{\lambda }\in \mathrm{int}{C}_{+}$.

Proof. 

From Corollary 1, we already know that $\vartheta \in \mathcal{L}$ and there exists $u_{\vartheta }\in S_{\vartheta }\subseteq \mathrm{int}{C}_{+}$ such that

$$u_{\lambda }⩽u_{\vartheta },u_{\lambda }\ne u_{\vartheta }.$$

(12)

Consider the function $a:{\mathbb{R}}^N⟶{\mathbb{R}}^N$ defined by

$$a\left(y\right)={\left|y\right|}^{p-2}y+y\forall y\in {\mathbb{R}}^N.$$

Evidently, $a\in C^1({\mathbb{R}}^N;{\mathbb{R}}^N)$ (recall that $2<p$) and we have

$$\nabla a\left(y\right)={\left|y\right|}^{p-2}\left(\mathrm{id}+(p-2)\frac{y\otimes y}{{\left|y\right|}^2}\right)+\mathrm{id}\forall y\ne 0,$$

so

$${\left(\nabla a\left(y\right),\xi ,\xi \right)}_{{\mathbb{R}}^N}⩾{\left|\xi \right|}^2\forall y,\xi \in {\mathbb{R}}^N.$$

Then, the tangency principle of Pucci and Serrin [8] (Theorem 2.5.2, p. 35) implies that

$$u_{\lambda }\left(z\right)<u_{\vartheta }\left(z\right)\forall z\in \mathsf{\Omega }$$

(13)

(see (12)). Let $\varrho =\parallel u_{\vartheta }{\parallel }_{\infty }$ and let ${\widehat{\xi }}_{\varrho }>0$ be as postulated by hypothesis $H\left(iv\right)$. We pick ${\tilde{\xi }}_{\varrho }>{\widehat{\xi }}_{\varrho }$ and using (12), hypothesis $H\left(iv\right)$ and the facts that $f⩾0$ and $u_{\lambda }⩽u_{\vartheta }$, we have

$$\begin{array}{ccc}\hfill & & -{\mathsf{\Delta }}_p{u}_{\vartheta }-\mathsf{\Delta }{u}_{\vartheta }+\vartheta {\tilde{\xi }}_{\varrho }{u}_{\vartheta }^{p-1}\hfill \\ & =& \vartheta (f(z,u_{\vartheta })+{\widehat{\xi }}_{\varrho }{u}_{\vartheta }^{p-1})+\vartheta ({\tilde{\xi }}_{\varrho }-{\widehat{\xi }}_{\varrho })u_{\vartheta }^{p-1}\hfill \\ & ⩾& \vartheta (f(z,u_{\lambda })+{\widehat{\xi }}_{\varrho }{u}_{\lambda }^{p-1})+\vartheta ({\tilde{\xi }}_{\varrho }-{\widehat{\xi }}_{\varrho })u_{\vartheta }^{p-1}\hfill \\ & ⩾& \lambda f(z,u_{\lambda })+\vartheta {\tilde{\xi }}_{\varrho }{u}_{\lambda }^{p-1}\hfill \\ & =& -{\mathsf{\Delta }}_p{u}_{\lambda }-\mathsf{\Delta }{u}_{\lambda }+\vartheta {\tilde{\xi }}_{\varrho }{u}_{\lambda }^{p-1}\mathrm{in}\mathsf{\Omega }.\hfill \end{array}$$

(14)

Note that on account of (13), we have

$$0\prec \vartheta ({\tilde{\xi }}_{\varrho }-{\widehat{\xi }}_{\varrho })(u_{\vartheta }^{p-1}-u_{\lambda }^{p-1}).$$

(15)

Then, (14), (15) and Proposition 3.2 of Gasiński and Papageorgiou [1] imply that $u_{\vartheta }-u_{\lambda }\in \mathrm{int}{C}_{+}$.  □

Proposition 5.

If hypotheses H hold, then ${\lambda }_{*}>0$.

Proof. 

We argue by contradiction. Suppose that ${\lambda }_{*}=0$. Let ${\left\{{\lambda }_n\right\}}_{n\in \mathbb{N}}\subseteq \mathcal{L}$ be such that ${\lambda }_n\to 0^{+}$ and consider $u_n=u_{{\lambda }_n}\subseteq \mathrm{int}{C}_{+}$ for all $n\in \mathbb{N}$. We have

$$\langle A_p\left(u_n\right),h\rangle +\langle A\left(u_n\right),h\rangle ={\int }_{\mathsf{\Omega }}{\lambda }_{n}f(z,u_n)hdz\forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right),n\in \mathbb{N}.$$

(16)

On account of hypotheses $H\left(i\right),\left(ii\right)$, given $\epsilon >0$, we can find $c_{\epsilon }>0$ such that

$$0⩽f(z,u_n\left(z\right))⩽\epsilon u_n{\left(z\right)}^{p-1}+c_{\epsilon }\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega },n\in \mathbb{N}.$$

(17)

In (16), first, we choose $h=u_n\in W_0^{1,p}\left(\mathsf{\Omega }\right)$ and then on the right hand side we use (17). We obtain

$$\parallel D{u}_n{\parallel }_p^p⩽\epsilon \parallel u_n{\parallel }_p^p+c_3\parallel u_n\parallel \forall n\in \mathbb{N},$$

for some $c_3=c_3\left(\epsilon \right)>0$, so

$$\left(1-\frac{\epsilon }{{\widehat{\lambda }}_1\left(p\right)}\right){\parallel u_n\parallel }^{p-1}⩽c_3\forall n\in \mathbb{N}$$

(see (1) with $r=p$). Choosing $\epsilon \in (0,{\widehat{\lambda }}_1\left(p\right))$, we see that the sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}\subseteq W_0^{1,p}\left(\mathsf{\Omega }\right)$ is bounded. We may assume that

$$u_n⟶u_{*}\mathrm{weakly}\mathrm{in}{W}_0^{1,p}\left(\mathsf{\Omega }\right)\mathrm{and}{u}_n⟶u_{*}\mathrm{in}{L}^p\left(\mathsf{\Omega }\right).$$

(18)

In (16), we choose $h=u_n-u_{*}\in W_0^{1,p}\left(\mathsf{\Omega }\right)$, pass to the limit as $n\to +\infty$ and use (18). We obtain

$$\underset{n\to +\infty }{lim}\left(\langle A_p\left(u_n\right),u_n-u_{*}\rangle +\langle A\left(u_n\right),u_n-u_{*}\rangle \right)=0,$$

so, using the monotonicity of A, we obtain

$$\underset{n\to +\infty }{lim\; sup}\left(\langle A_p\left(u_n\right),u_n-u_{*}\rangle +\langle A\left(u\right),u_n-u_{*}\rangle \right)=0,$$

thus

$$\underset{n\to +\infty }{lim\; sup}(\langle A_p\left(u_n\right),u_n-u_{*}\rangle )⩽0$$

and hence

$$u_n⟶u_{*}\mathrm{in}{W}_0^{1,p}\left(\mathsf{\Omega }\right)$$

(19)

(see Proposition 1). Hypotheses $H\left(i\right),\left(ii\right),\left(iii\right)$ imply that given $\epsilon >0$, we can find $c_4=c_4\left(\epsilon \right)>0$ such that

$$0⩽f(z,x)⩽\epsilon x+c_4{x}^{p-1}\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega },x⩾0,$$

(20)

so

$$0⩽f(z,u_n\left(z\right))⩽\epsilon u_n\left(z\right)+c_4{u}_n{\left(z\right)}^{p-1}\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega },n\in \mathbb{N},$$

thus the sequence $\{f(·,u_n(·))\subseteq L^{p^{\prime }}\left(\mathsf{\Omega }\right)$ is bounded (see (19) and recall that $p^{\prime }<2<p$). Therefore, if in (16) we pass to the limit as $n\to +\infty$, we obtain

$$\langle A_p\left(u_{*}\right),h\rangle +\langle A\left(u_{*}\right),h\rangle =0\forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right).$$

Choosing $h=u_{*}\in W_0^{1,p}\left(\mathsf{\Omega }\right)$, we obtain

$$\parallel D{u}_{*}{\parallel }_p⩽0,$$

so

$$u_{*}=0.$$

(21)

From (19) and the nonlinear regularity theory of Lieberman [7], we know that there exist $\alpha \in (0,1)$ and $c_5>0$ such that

$$u_n\in C_0^{1,\alpha }\left(\overline{\mathsf{\Omega }}\right)\mathrm{and}{\parallel u_n\parallel }_{C_0^{1,\alpha }\left(\overline{\mathsf{\Omega }}\right)}⩽c_5\forall n\in \mathbb{N}.$$

(22)

Since the embedding $C_0^{1,\alpha }\left(\overline{\mathsf{\Omega }}\right)\subseteq C_0^1\left(\overline{\mathsf{\Omega }}\right)$ is compact, from (19), (21) and (22), we infer that

$$u_n⟶0\mathrm{in}{C}_0^1\left(\overline{\mathsf{\Omega }}\right)\mathrm{as}n\to +\infty .$$

(23)

Let $y_n=\frac{u_n}{\parallel u_n{\parallel }_{1,2}}$, for $n\in \mathbb{N}$, with ${\parallel ·\parallel }_{1,2}$ denoting the norm of $H_0^1\left(\mathsf{\Omega }\right)$. We have

$$\parallel y_n{\parallel }_{1,2}=0,y_n⩾0\forall n\in \mathbb{N}.$$

We may assume that

$$y_n⟶y\mathrm{weakly}\mathrm{in}{H}_0^1\left(\mathsf{\Omega }\right),y_n⟶y\mathrm{in}{L}^2\left(\mathsf{\Omega }\right),y⩾0.$$

(24)

From (16), we have

$$\parallel u_n{\parallel }_{1,2}^{p-2}\langle A_p\left(y_n\right),h\rangle +\langle A\left(y_n\right),h\rangle ={\lambda }_n{\int }_{\mathsf{\Omega }}\frac{f(z,u_n)}{\parallel u_n{\parallel }_{1,2}}hdz\forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right).$$

(25)

On account of (20), we have

$$0⩽\frac{f(z,u_n\left(z\right))}{\parallel u_n{\parallel }_{1,2}}⩽\epsilon y_n\left(z\right)+u_n{\left(z\right)}^{p-2}{y}_n\left(z\right)⩽c_6{y}_n\left(z\right)\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega },n\in \mathbb{N},$$

for some $c_6>0$ and thus

$$\mathrm{the}\mathrm{sequence}{\left\{\frac{f(·,u_n(·))}{\parallel u_n\parallel }\right\}}_{n\in \mathbb{N}}\subseteq L^p\left(\mathsf{\Omega }\right)\mathrm{is}\mathrm{bounded}$$

(26)

(recall that, if $2<p$, then $p^{\prime }<2$). Therefore, if in (25) we pass to the limit as $n\to +\infty$ and use (23), (24) and (26), we obtain

$$\langle A\left(y\right),h\rangle ⩽0\forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right),$$

so $y=0$ and hence $\parallel D{y}_n{\parallel }_2⟶0$ and $n\to +\infty$ (see (25)), a contradiction since $\parallel y_n{\parallel }_{1,2}=1$ for all $n\in \mathbb{N}$. Therefore, we conclude that ${\lambda }_{*}>0$.  □

Next, we prove a multiplicity result when $\lambda >{\lambda }_{*}$.

Proposition 6.

If hypotheses H hold and $\lambda >{\lambda }_{*}$, then problem $\left(P_{\lambda }\right)$ has at least two positive solutions

$$u_0,\widehat{u}\in \mathrm{int}{C}_{+},u_0\ne \widehat{u}.$$

Proof. 

Let $\mu \in ({\lambda }_{*},\lambda )$. We have $\mu ,\lambda \in \mathcal{L}$ and then, according to Proposition 4, we can find $u_0\in S_{\lambda }\subseteq \mathrm{int}{C}_{+}$ and $u_{\mu }\in S_{\mu }\subseteq \mathrm{int}{C}_{+}$ such that

$$u_0-u_{\mu }\in \mathrm{int}{C}_{+}.$$

(27)

We truncate $f(z,·)$ from below at $u_{\mu }\left(z\right)$ and introduce the Carathéodory function $e(z,x)$ defined by

$$e(z,x)=\left\{\begin{array}{ccc}f(z,u_{\mu }\left(z\right))\hfill & \mathrm{if}\hfill & x⩽u_{\mu }\left(z\right),\hfill \\ f(z,x)\hfill & \mathrm{if}\hfill & u_{\mu }\left(z\right)<x.\hfill \end{array}\right.$$

(28)

We set

$$E(z,x)={\int }_0^{x}e(z,s)ds$$

and consider the $C^1$-functional ${\widehat{\phi }}_{\lambda }:W_0^{1,p}\left(\mathsf{\Omega }\right)⟶\mathbb{R}$ defined by

$${\widehat{\phi }}_{\lambda }\left(u\right)=\frac{1}{p}{\parallel Du\parallel }_p^p+\frac{1}{2}{\parallel Du\parallel }_2^2-{\int }_{\mathsf{\Omega }}\lambda E(z,u)dz\forall u\in W_0^{1,p}\left(\mathsf{\Omega }\right).$$

Let

$$\left[u_{\mu }\right)=\{u\in W_0^{1,p}\left(\mathsf{\Omega }\right):u_{\mu }\left(z\right)⩽u\left(z\right)\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega }\}.$$

Then, from (28), we see that

$${\widehat{\phi }}_{\lambda }{{|}_{\left[u_{\mu }\right)}={\phi }_{\lambda }|}_{\left[u_{\mu }\right)}+\xi ,$$

(29)

with $\xi \in \mathbb{R}$. From the proof of Proposition 2, we know that ${\phi }_{\lambda }$ is coercive. Hence ${\phi }_{\lambda }$ is coercive. Additionally, ${\phi }_{\lambda }$ is sequentially weakly lower semicontinuous. Therefore, we can find ${\widehat{u}}_0\in W_0^{1,p}\left(\mathsf{\Omega }\right)$ such that

$${\widehat{\phi }}_{\lambda }\left({\widehat{u}}_0\right)=\underset{u\in W_0^{1,p}\left(\mathsf{\Omega }\right)}{min}{\widehat{\phi }}_{\lambda }\left(u\right),$$

(30)

so

$${\widehat{\phi }}_{\lambda }^{\prime }\left({\widehat{u}}_0\right)=0,$$

and hence

$$\langle A_p\left({\widehat{u}}_0\right),h\rangle +\langle A\left({\widehat{u}}_0\right),h\rangle ={\int }_{\mathsf{\Omega }}\lambda e(z,{\widehat{u}}_0)hdz\forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right).$$

(31)

Choose $h\in {(u_{\mu }-{\widehat{u}}_0)}^{+}\in W_0^{1,p}\left(\mathsf{\Omega }\right)$. Using (28), we have

$$\begin{array}{ccc}& & \langle A_p\left({\widehat{u}}_0\right),{(u_{\mu }-{\widehat{u}}_0)}^{+}\rangle +\langle A\left({\widehat{u}}_0\right),{(u_{\mu }-{\widehat{u}}_0)}^{+}\rangle \hfill \\ & =& {\int }_{\mathsf{\Omega }}\lambda f(z,u_{\mu }){(u_{\mu }-{\widehat{u}}_0)}^{+}dz\hfill \\ & ⩾& {\int }_{\mathsf{\Omega }}\mu f(z,u_{\mu }){(u_{\mu }-{\widehat{u}}_0)}^{+}dz\hfill \\ & =& \langle A_p\left(u_{\mu }\right),{(u_{\mu }-{\widehat{u}}_0)}^{+}\rangle +\langle A\left(u_{\mu }\right),{(u_{\mu }-{\widehat{u}}_0)}^{+}\rangle \hfill \end{array}$$

(since $f⩾0$, $\mu <\lambda$ and $u_{\mu }\in S_{\mu }$), so

$$u_{\mu }⩽{\widehat{u}}_0$$

(see Proposition 1).

Then, from (28) and (31), we infer that ${\widehat{u}}_0\in S_{\lambda }\subseteq \mathrm{int}{C}_{+}$.

If ${\widehat{u}}_0\ne u_0$, then this is the second positive solution of $\left(P_{\lambda }\right)$. Therefore, we assume that

$${\widehat{u}}_0=u_0.$$

From (27), (29) and (30), it follows that

$$u_0\in \mathrm{int}{C}_{+}\mathrm{is}\mathrm{a}\mathrm{local}{C}_0^1\left(\overline{\mathsf{\Omega }}\right)-\mathrm{minimizer}\mathrm{of}{\phi }_{\lambda }$$

and so

$$u_0\in \mathrm{int}{C}_{+}\mathrm{is}\mathrm{a}\mathrm{local}{W}_0^{1,p}\left(\mathsf{\Omega }\right)-\mathrm{minimizer}\mathrm{of}{\phi }_{\lambda }$$

(32)

(see Gasiński and Papageorgiou [9]).

Hypothesis $H\left(iii\right)$ implies that given $\epsilon >0$, we can find $\delta =\delta \left(\epsilon \right)>0$ such that

$$F(z,x)⩽\frac{\epsilon }{2}{x}^2\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega },\mathrm{all}\left|x\right|⩽\delta$$

(33)

(see (2)). Let $u\in C_0^1\left(\overline{\mathsf{\Omega }}\right)$ with ${\parallel u\parallel }_{C_0^1\left(\overline{\mathsf{\Omega }}\right)}⩽\delta$. We have

$$\begin{array}{ccc}\hfill {\phi }_{\lambda }\left(u\right)& ⩾& \frac{1}{p}{\parallel Du\parallel }_p^p+\frac{1}{2}{\parallel Du\parallel }_2^2-\frac{\lambda \epsilon }{2}{\parallel u\parallel }_2^2\hfill \\ & ⩾& \frac{1}{p}{\parallel Du\parallel }_p^p+\frac{1}{2}\left(1-\frac{\lambda \epsilon }{{\widehat{\lambda }}_1\left(2\right)}\right){\parallel Du\parallel }_2^2\hfill \end{array}$$

(see (1) with $r=2$). Choosing $\epsilon \in (0,\frac{{\widehat{\lambda }}_1\left(2\right)}{\lambda })$, we obtain

$${\phi }_{\lambda }\left(u\right)⩾\frac{1}{p}{\parallel u\parallel }^p\forall u\in C_0^1\left(\overline{\mathsf{\Omega }}\right),{\parallel u\parallel }_{C_0^1\left(\overline{\mathsf{\Omega }}\right)}⩽\delta ,$$

so

$$u=0\mathrm{is}\mathrm{a}\mathrm{local}{C}_0^1\left(\overline{\mathsf{\Omega }}\right)-\mathrm{minimizer}\mathrm{of}{\phi }_{\lambda }$$

and thus

$$u=0\mathrm{is}\mathrm{a}\mathrm{local}{W}_0^{1,p}\left(\mathsf{\Omega }\right)-\mathrm{minimizer}\mathrm{of}{\phi }_{\lambda }$$

(34)

(see Gasiński and Papageorgiou [9]).

We assume that ${\phi }_{\lambda }\left(0\right)=0⩽{\phi }_{\lambda }\left(u_0\right)$. The reasoning is similar if the opposite inequality holds, using (34) instead of (32).

We also assume that

$$K_{{\phi }_{\lambda }}=\{u\in W_0^{1,p}\left(\mathsf{\Omega }\right):{\phi }_{\lambda }^{\prime }\left(u\right)=0\}$$

(the critical set of ${\phi }_{\lambda }$) is finite. Otherwise, we already have an infinity of distinct positive solutions of $\left(P_{\lambda }\right)$. On account of (32) and using Theorem 5.7.6 of Papageorgiou, Rădulescu and Repovš [2] (p. 449), we can find $\varrho \in (0,1)$ small such that

$${\phi }_{\lambda }\left(0\right)=0⩽{\phi }_{\lambda }\left(u_0\right)<\underset{\parallel u-u_0\parallel =\varrho }{inf}{\phi }_{\lambda }\left(u\right)=m_{\lambda },0<\phi <\parallel u_0\parallel .$$

(35)

Recall that ${\phi }_{\lambda }$ is coercive (see the proof of Proposition 2). Therefore, from Proposition 5.1.15 of Papageorgiou, Rădulescu and Repovš [2] (p. 449), we have that

$${\phi }_{\lambda }\mathrm{satisfies}\mathrm{the}\mathrm{PS}-\mathrm{condition}.$$

(36)

Then, (35) and (36) permit the use of the mountain pass theorem. Therefore, we can find $\widehat{u}\in W_0^{1,p}\left(\mathsf{\Omega }\right)$ such that

$${\phi }_{\lambda }^{\prime }\left(\widehat{u}\right)=0\mathrm{and}{m}_{\lambda }⩽{\phi }_{\lambda }\left(\widehat{u}\right).$$

(37)

From (35) and (37), we conclude that

$$\widehat{u}\in S_{\lambda }\subseteq \mathrm{int}{C}_{+}\mathrm{and}\widehat{u}\ne u_0.$$

□

It remains to be decided what we can say for the critical parameter value ${\lambda }_{*}$. We show that ${\lambda }_{*}>0$ is admissible too.

Proposition 7.

If hypotheses H hold, then ${\lambda }_{*}\in \mathcal{L}$.

Proof. 

Let ${\left\{{\lambda }_n\right\}}_{n\in \mathbb{N}}\subseteq \mathcal{L}$ be such that ${\lambda }_n⟶{\lambda }_{*}^{+}$. We can find $u_n\in S_{{\lambda }_n}\subseteq \mathrm{int}{C}_{+}$ such that

$$\langle A_p\left(u_n\right),h\rangle +\langle A\left(u_n\right),h\rangle ={\lambda }_n{\int }_{\mathsf{\Omega }}f(z,u_n)hdz\forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right),n\in \mathbb{N}.$$

(38)

In (38), we use $h=u_n\in W_0^{1,p}\left(\mathsf{\Omega }\right)$. Then,

$$\parallel u_n{\parallel }^p⩽{\lambda }_1{\int }_{\mathsf{\Omega }}f(z,u_n)u_{n}dz\forall n\in \mathbb{N}.$$

(39)

On account of hypotheses $H\left(i\right),\left(ii\right)$, given $\epsilon >0$, we can find $c_{\epsilon }>0$ such that

$$0⩽f(z,x)x⩽\epsilon x^p+c_{\epsilon }\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega },\mathrm{all}x⩾0.$$

(40)

We use (40) in (39) and have

$$\parallel u_n{\parallel }^p⩽{\lambda }_1\frac{\epsilon }{{\widehat{\lambda }}_1\left(p\right)}\parallel u_n{\parallel }^p+c_{\epsilon }{\left|\mathsf{\Omega }\right|}_N$$

(see (1) with $r=p$ and recall that ${|·|}_N$ is the Lebesgue measure on ${\mathbb{R}}^N$), so

$$\left(1-\frac{{\lambda }_1}{{\widehat{\lambda }}_1\left(p\right)}\epsilon \right)\parallel u_n{\parallel }^p⩽c_{\epsilon }{\left|\mathsf{\Omega }\right|}_N\forall n\in \mathbb{N}.$$

We choose $\epsilon \in (0,\frac{{\widehat{\lambda }}_1\left(p\right)}{{\lambda }_1})$ and infer that the sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}\subseteq W_0^{1,p}\left(\mathsf{\Omega }\right)$ is bounded. Therefore, we may assume that

$$u_n⟶u_{*}\mathrm{weakly}\mathrm{in}{W}_0^{1,p}\left(\mathsf{\Omega }\right)\mathrm{and}{u}_n⟶u_{*}\mathrm{in}{L}^p\left(\mathsf{\Omega }\right).$$

Then, reasoning as in the proof of Proposition 5 (see the part of the proof after (18)), we show that

$$u_n⟶u_{*}\mathrm{in}{W}_0^{1,p}\left(\mathsf{\Omega }\right),u_{*}\ne 0.$$

Therefore, if in (38) we pass to the limit as $n\to +\infty$, then

$$\langle A_p\left(u_{*}\right),h\rangle +\langle A\left(u_{*}\right),h\rangle ={\lambda }_{*}{\int }_{\mathsf{\Omega }}f(f,u_{*})hdz\forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right),$$

so $u_{*}\in S_{{\lambda }_{*}}\subseteq \mathrm{int}{C}_{+}$ and so ${\lambda }_{*}\in \mathcal{L}$.  □

We have proved that

$$\mathcal{L}=[{\lambda }_{*},\infty ).$$

Next, we show that for every $\lambda \in \mathcal{L}$, problem $\left(P_{\lambda }\right)$ admits a smallest positive solution (minimal positive solution).

Proposition 8.

If hypotheses H hold and $\lambda \in \mathcal{L}$, then problem $\left(P_{\lambda }\right)$ admits a smallest solution $u_{\lambda }^{*}\in S_{\lambda }\subseteq \mathrm{int}{C}_{+}$ (that is, $u_{\lambda }^{*}⩽u$ for all $u\in S_{\lambda }$).

Proof. 

From Proposition 7 of Papageorgiou, Rădulescu and Repovš [10], we know that $S_{\lambda }$ is downward directed. Using Lemma 3.10 of Hu and Papageorgiou [11] (p. 178), we can find a decreasing sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}\subseteq S_{\lambda }$ such that

$$\underset{n\in \mathbb{N}}{inf}{u}_n=inf{S}_{\lambda }.$$

We have

$$\langle A_p\left(u_n\right),h\rangle +\langle A\left(u_n\right),h\rangle ={\int }_{\mathsf{\Omega }}\lambda f(z,u_n)hdz\forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right),n\in \mathbb{N}$$

(41)

and

$$0⩽u_n⩽u_1\forall n\in \mathbb{N}.$$

(42)

In (41), we choose $h=u_n\in W_0^{1,p}\left(\mathsf{\Omega }\right)$ and then use (42) and hypothesis $H\left(i\right)$ to establish that ${\left\{u_n\right\}}_{n\in \mathbb{N}}\subseteq W_0^{1,p}\left(\mathsf{\Omega }\right)$ is bounded. Therefore, we may assume that

$$u_n⟶u_{\lambda }^{*}\mathrm{weakly}\mathrm{in}{W}_0^{1,p}\left(\mathsf{\Omega }\right)\mathrm{and}{u}_n⟶u_{\lambda }^{*}\mathrm{in}{L}^p\left(\mathsf{\Omega }\right).$$

(43)

Then, as before (see the proof of Proposition 5 after (18)), using (43) we obtain

$$u_n⟶u_{\lambda }^{*}\mathrm{in}{W}_0^{1,p}\left(\mathsf{\Omega }\right)\mathrm{and}{u}_{\lambda }^{*}\ne 0.$$

(44)

If in (41) we pass to the limit as $n\to +\infty$ and use (44), then

$$\langle A_p\left(u_{\lambda }^{*}\right),h\rangle +\langle A\left(u_{\lambda }^{*}\right),h\rangle ={\int }_{\mathsf{\Omega }}\lambda f(z,u_{\lambda }^{*})hdz\forall h\in W_0^{1,p}\left(\mathsf{\Omega }\right),$$

so $u_{\lambda }^{*}\in S_{\lambda }\subseteq \mathrm{int}{C}_{+}$, $u_{\lambda }^{*}=inf{S}_{\lambda }$.  □

The theorem that follows summarizes our findings concerning the changes in the set of positive solutions of $\left(P_{\lambda }\right)$ as $\lambda >0$ moves.

Theorem 1.

If hypotheses H hold, then there exists ${\lambda }_{*}>0$ such that

(a) 

for all $\lambda >{\lambda }_{*}$ problem $\left(P_{\lambda }\right)$ has at least two positive solutions $u_0,\widehat{u}\in \mathrm{int}{C}_{+}$, $u_0\ne \widehat{u}$;

(b) 

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

(c) 

for every $\lambda \in (0,{\lambda }_{*})$ problem $\left(P_{\lambda }\right)$ has no positive solution;

(d) 

for every $\lambda \in \mathcal{L}=[{\lambda }_{*},\infty )$, problem $\left(P_{\lambda }\right)$ has a smallest positive solution $u_{\lambda }^{*}\in \mathrm{int}{C}_{+}$.

Remark 2.

From Proposition 4, we know that the minimal solution map $\widehat{k}:\mathcal{L}⟶C_0^1\left(\overline{\mathsf{\Omega }}\right)$ defined by $\widehat{k}\left(\lambda \right)=u_{\lambda }^{*}$ is strictly increasing in the sense that

$$\mathrm{if}{\lambda }_{*}⩽\mu ⩽\lambda ,\mathrm{then}{u}_{\lambda }^{*}-u_{\mu }^{*}\in \mathrm{int}{C}_{+}.$$

It is worth mentioning that when the reaction $f(z,·)$ is $(p-1)$-superlinear, then we have the “bifurcation” in $\lambda >0$, for small values of the parameter (see [1,2]). Here, $f(z,·)$ is $(p-1)$-sublinear, and the “bifurcation” in $\lambda >0$ occurs for large values of the parameter.

4. $(\mathit{p},\mathit{q})$-Equations

In this section, we briefly mention the situation for the more general $(p,q)$-equations, $q\ne 2$. We now deal with the following nonlinear Dirichlet eigenvalue problem:

$${\left(P_{\lambda }\right)}^{\prime }\left\{\begin{array}{c}-{\mathsf{\Delta }}_{p}u\left(z\right)-{\mathsf{\Delta }}_{q}u\left(z\right)=\lambda f(z,u\left(z\right))\mathrm{in}\mathsf{\Omega },\hfill \\ {u|}_{\partial \mathsf{\Omega }}=0,u⩾0,\lambda >0,1<q<p.\hfill \end{array}\right.$$

If we strengthen the conditions on $f(z,·)$, we can have a similar “bifurcation-type” result for problem ${\left(P_{\lambda }\right)}^{\prime }$.

The new conditions on $f(z,x)$ are the following:

H’:$f:\mathsf{\Omega }\times \mathbb{R}⟶\mathbb{R}$ is a Carathéodory function, $f(z,0)=0$ for a.a. $z\in \mathsf{\Omega }$, hypotheses $H^{\prime }\left(i\right),\left(ii\right),\left(iii\right)$ are the same as the corresponding hypotheses $H\left(i\right),\left(ii\right),\left(iii\right)$ and $\left(iv\right)$ for a.a. $z\in \mathsf{\Omega }$, $f(z,·)$ is strictly increasing on ${\mathbb{R}}^{+}$.

Remark 3.

According to hypothesis $H^{\prime }\left(iv\right)$, we have

$$0<f(z,x)\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega },\mathrm{all}x>0.$$

The function $f(z,x)=a\left(z\right)x^{\tau -1}$ for a.a. $z\in \mathsf{\Omega }$, all $x⩾0$ with $a\in L^{\infty }\left(\mathsf{\Omega }\right)$ and $1<\tau <q<p$ satisfies hypotheses H’.

For the $(p,q)$-equation ($q\ne 2$), we cannot use the tangency principle of Pucci and Serrin [8] (p. 35) (see the proof of Proposition 4). Instead, on account of the stronger condition $H^{\prime }\left(iv\right)$, we can use Proposition 3.4 of Gasiński and Papageorgiou [1] (strong comparison principle) and have that $u_{\vartheta }-u_{\lambda }\in \mathrm{int}{C}_{+}$. Then, all the other results remain valid and so we can have the following bifurcation-type result for problem ${\left(P_{\lambda }\right)}^{\prime }$.

Theorem 2.

If hypotheses $H^{\prime }$ hold, then there exists ${\lambda }_{*}^{\prime }>0$ such that

(a) 

for all $\lambda >{\lambda }_{*}^{\prime }$, problem ${\left(P_{\lambda }\right)}^{\prime }$ has at least two positive solutions $u_0,\widehat{u}\in \mathrm{int}{C}_{+}$, $u_0\ne \widehat{u}$;

(b) 

for $\lambda ={\lambda }_{*}^{\prime }$, problem ${\left(P_{\lambda }\right)}^{\prime }$ has at least one positive solution $u_{*}\in \mathrm{int}{C}_{+}$;

(c) 

for every $\lambda \in {(0,{\lambda }_{*})}^{\prime }$, problem ${\left(P_{\lambda }\right)}^{\prime }$ has no positive solution;

(d) 

for every $\lambda \in {\mathcal{L}}^{\prime }=[{\lambda }_{*}^{\prime },\infty )$, problem ${\left(P_{\lambda }\right)}^{\prime }$ has a smallest positive solution $u_{\lambda }^{*}\in \mathrm{int}{C}_{+}$.

Remark 4.

The function $f(z,x)$ defined by

$$f(z,x)=\left\{\begin{array}{ccc}a\left(z\right)({\left(x^{+}\right)}^{r-1}+{\left(x^{+}\right)}^{\eta -1})\hfill & if\hfill & \left|x\right|⩽1,\hfill \\ a\left(z\right)ln\left(x^{+}\right)\hfill & if\hfill & 1<\left|x\right|,\hfill \end{array}\right.$$

with $a\in L^{\infty }\left(\mathsf{\Omega }\right)$, $p<r<\eta$ satisfies hypotheses H but not hypotheses $H^{\prime }$.