Positive solutions for weighted singular $p$-Laplace equations via Nehari manifolds

TL;DR

This paper establishes the existence of multiple positive solutions for weighted singular p-Laplace equations with discontinuous weights using Nehari manifold techniques, addressing challenges posed by irregular weights.

Contribution

The paper introduces a novel approach employing Nehari manifolds to prove multiple positive solutions for weighted singular p-Laplace equations with discontinuous weights.

Findings

Existence of at least two positive bounded solutions

Application of Nehari manifold method to irregular weights

Handling of discontinuous weight functions in singular equations

Abstract

In this paper we study weighted singular $p$-Laplace equations involving a bounded weight function which can be discontinuous. Due to its discontinuity classical regularity results cannot be applied. Based on Nehari manifolds we prove the existence of at least two positive bounded solutions of such problems.

Equations

$$\displaystyle\begin{split}&-\operatorname*{div}(\xi(x)|\nabla u|^{p-2}\nabla u)=a(x)u^{-\gamma}+\lambda u^{r-1}\quad\text{in }\Omega\\ &u\big{|}_{\partial\Omega}=0,\quad 0<\gamma<1,\quad 1<p<r<p^{*},\quad u\geq 0,\quad\lambda>0.\end{split}$$

$$p^{\ast }=\{\begin{array}{ll}\frac{Np}{N-p}& \text{if }p<N,\\ +\infty & \text{if }N\le p.\end{array}$$

$$\Vert u\Vert =\Vert \nabla u{\Vert }_p\text{for all }u\in W_0^{1,p}(\Omega ),$$

$$p^{\ast }=\{\begin{array}{ll}\frac{Np}{N-p}& \text{if }p<N,\\ +\infty & \text{if }N\le p.\end{array}$$

$$⟨A(u),\phi ⟩={\int }_{\Omega }\xi (x)|\nabla u{|}^{p-2}\nabla u\cdot \nabla \phi dx\text{for all }u,\phi \in W_0^{1,p}(\Omega ).$$

$$u_n\stackrel{w}{\to }u\text{ in }{W}_0^{1,p}(\Omega )\text{and}\underset{n\to \infty }{\mathrm{limsup}}⟨A(u_n),u_n-u⟩\le 0,$$

$${\phi }_{\lambda }(u)=\frac{1}{p}{\int }_{\Omega }\xi (x)|\nabla u{|}^{p}dx-\frac{1}{1-\gamma }{\int }_{\Omega }a(x)|u{|}^{1-\gamma }dx-\frac{\lambda }{r}\Vert u{\Vert }_r^r.$$

$$N_{\lambda }=\left\{u\in W_0^{1,p}(\Omega ):{\int }_{\Omega }\xi (x)|\nabla u{|}^{p}dx={\int }_{\Omega }a(x)|u{|}^{1-\gamma }dx+\lambda \Vert u{\Vert }_r^r,u\ne 0\right\}.$$

$$N_{\lambda }^{+}$$

$$N_{\lambda }^0$$

$$N_{\lambda }^{-}$$

$$-\frac{1}{r}{\int }_{\Omega }\xi (x)|\nabla u{|}^{p}dx+\frac{1}{r}{\int }_{\Omega }a(x)|u{|}^{1-\gamma }dx=-\frac{\lambda }{r}\Vert u{\Vert }_r^r.$$

$$\begin{array}{rl}{\phi }_{\lambda }(u)& =\left[\frac{1}{p}-\frac{1}{r}\right]{\int }_{\Omega }\xi (x)|\nabla u{|}^{p}dx-\left[\frac{1}{1-\gamma }-\frac{1}{r}\right]{\int }_{\Omega }a(x)|u{|}^{1-\gamma }dx\\ & \ge \left[\frac{1}{p}-\frac{1}{r}\right]{\xi }_0\Vert u{\Vert }^p-\left[\frac{1}{1-\gamma }-\frac{1}{r}\right]{\int }_{\Omega }a(x)|u{|}^{1-\gamma }dx\\ & \ge c_1\Vert u{\Vert }^p-c_2\Vert u{\Vert }^{1-\gamma }\end{array}$$

$$\lambda \Vert u{\Vert }_r^r<\frac{p+\gamma -1}{r+\gamma -1}{\int }_{\Omega }\xi (x)|\nabla u{|}^{p}dx.$$

$$-\frac{1}{1-\gamma }{\int }_{\Omega }a(x)|u{|}^{1-\gamma }dx=-\frac{1}{1-\gamma }{\int }_{\Omega }\xi (x)|\nabla u{|}^{p}dx+\frac{\lambda }{1-\gamma }\Vert u{\Vert }_r^r.$$

$${\phi }_{\lambda }(u)$$

$$<\left[\frac{-(p+\gamma -1)}{p(1-\gamma )}+\frac{r+\gamma -1}{r(1-\gamma )}\cdot \frac{p+\gamma -1}{r+\gamma -1}\right]{\int }_{\Omega }\xi (x)|\nabla u{|}^{p}dx$$

$$=\frac{p+\gamma -1}{1-\gamma }\left[\frac{1}{r}-\frac{1}{p}\right]{\int }_{\Omega }\xi (x)|\nabla u{|}^{p}dx$$

$$<0.$$

$$(p+\gamma -1){\int }_{\Omega }\xi (x)|\nabla u{|}^{p}dx=\lambda (r+\gamma -1)\Vert u{\Vert }_r^r.$$

$$\begin{array}{rl}& (r+\gamma -1){\int }_{\Omega }\xi (x)|\nabla u{|}^{p}dx-(r+\gamma -1){\int }_{\Omega }a(x)|u{|}^{1-\gamma }dx\\ & =\lambda (r+\gamma -1)\Vert u{\Vert }_r^r.\end{array}$$

$$(r-p){\int }_{\Omega }\xi (x)|\nabla u{|}^{p}dx=(r+\gamma -1){\int }_{\Omega }a(x)|u{|}^{1-\gamma }dx.$$

$$(r-p){\xi }_0\Vert u{\Vert }^p\le (r+\gamma -1)c_3\Vert u{\Vert }^{1-\gamma }$$

$$\Vert u{\Vert }^{p+\gamma -1}\le c_4$$

$$\Vert u{\Vert }^p\le \lambda c_5\Vert u{\Vert }^r$$

$${\left[\frac{1}{\lambda c_5}\right]}^{\frac{1}{r-p}}\le \Vert u\Vert .$$

$${\phi }_{\lambda }(u^{\ast })=m_{\lambda }^{+}=\underset{N_{\lambda }^{+}}{\inf }{\phi }_{\lambda }$$

$${\phi }_{\lambda }(u_n)\searrow m_{\lambda }^{+}<0\text{as }n\to \infty .$$

$$\{u_n{\}}_{n\ge 1}\subseteq W_0^{1,p}(\Omega )\text{ is bounded.}$$

$$u_n\stackrel{w}{\to }{u}^{\ast }\text{in }{W}_0^{1,p}(\Omega )\text{and}{u}_n\to u^{\ast }\text{in }{L}^r(\Omega ).$$

$${\phi }_{\lambda }(u^{\ast })\le \underset{n\to \infty }{\mathrm{liminf}}{\phi }_{\lambda }(u_n)<0={\phi }_{\lambda }(0).$$

$${\psi }_{u^{\ast }}(t)={\phi }_{\lambda }(t{u}^{\ast })\text{for all }t\ge 0.$$

$${\eta }_{u^{\ast }}(t)=t^{p-r}{\int }_{\Omega }\xi (x)|\nabla u^{\ast }{|}^{p}dx-t^{-\gamma -r+1}{\int }_{\Omega }a(x)|u^{\ast }{|}^{1-\gamma }dx\text{for all }t>0.$$

Full text

1. Introduction

Let $\Omega\subseteq\mathbb{R}^{N}$, $N\geq 1$, be a bounded domain with a Lipschitz boundary $\partial\Omega$. In this paper, we study the following nonlinear singular Dirichlet problem

[TABLE]

In this problem the differential operator is a weighted $p$-Laplacian with a weight $\xi\in L^{\infty}(\Omega)$, $\xi\geq 0$ and $\xi$ is supposed to be bounded away from zero. Since $\xi$ is discontinuous in general, we cannot use the nonlinear global regularity theory of Lieberman [8] and the nonlinear strong maximum principle, see Pucci-Serrin [14, pp. 111 and 120]. The fact that these two basic tools are no longer available leads to a different approach in the analysis of problem (Pλ) which is based on the Nehari method. On the right-hand side of (Pλ) we have the competing effects of two different nonlinearities. One is the singular term $s\to a(x)s^{-\gamma}$ with $s>0$ and the other one is a parametric $(p-1)$-superlinear perturbation $s\to\lambda s^{r-1}$ with $s\geq 0$ and $p<r<p^{*}$ with $p^{*}$ being the critical Sobolev exponent corresponding to $p$ defined by

[TABLE]

We are looking for positive solutions of problem (Pλ) and we show that problem (Pλ) has at least two positive solutions for all $\lambda\geq 0$.

Singular problems with such competition phenomena were investigated by Sun-Wu-Long [15] and Haitao [4] for semilinear equations driven by the Laplacian and by Giacomoni-Schindler-Takáč [3], Papageorgiou-Smyrlis [10], Papageorgiou-Winkert [12] and Perera-Zhang [13] for equations driven by the $p$-Laplacian. We also refer to the works of Leonardi-Papageorgiou [6], [7]. In all the mentioned works the weight function $\xi$ is equal to one and so we can use the global elliptic regularity theory and the strong maximum principle. These tools are crucial in the proofs of the works above and are combined with variational methods and suitable truncation and comparison techniques. The regularity theory guarantees that the solutions are in $C^{1}_{0}(\overline{\Omega})$ and then the strong maximum principle, so-called Hopf theorem, implies that these solutions are in $\operatorname*{int}\left(C^{1}_{0}(\overline{\Omega})_{+}\right)$ which is the interior of the positive order cone of $C^{1}_{0}(\overline{\Omega})$.

Without these facts the proofs of the works above are no more valid. As we already indicated, in our setting, these results do not hold, so we need to employ a different approach.

2. Preliminaries

We denote by $W^{1,p}_{0}(\Omega)$ the usual Sobolev space with norm $\|\cdot\|$. By the Poincaré inequality we have

[TABLE]

where $\|\cdot\|_{p}$ denotes the norm of $L^{p}(\Omega)$ and $L^{p}\left(\Omega;\mathbb{R}^{N}\right)$, respectively. The norm of $\mathbb{R}^{N}$ is denoted by $|\cdot|$ and “$\cdot$” stands for the inner product in $\mathbb{R}^{N}$. By $p^{*}>1$ we denote the Sobolev critical exponent for $p$ defined by

[TABLE]

Let $\xi\in L^{\infty}(\Omega)$ with $0<\operatorname*{ess~{}inf}_{\Omega}\xi$ and let $A\colon W^{1,p}_{0}(\Omega)\to W^{-1.p^{\prime}}(\Omega)=W^{1,p}_{0}(\Omega)^{*}$ with $\frac{1}{p}+\frac{1}{p^{\prime}}=1$ be defined by

[TABLE]

The next proposition states the main properties of this map and it can be found in Gasiński-Papageorgiou [1, Problem 2.192, p. 279].

Proposition 2.1**.**

The map $A:W^{1,p}_{0}(\Omega)\to W^{-1,p^{\prime}}(\Omega)$ defined in (2.1) is bounded, that is, it maps bounded sets to bounded sets, continuous, strictly monotone, hence maximal monotone and it is of type $(\operatorname*{S})_{+}$, that is,

[TABLE]

imply $u_{n}\to u$ in $W^{1,p}_{0}(\Omega)$.

3. Positive Solutions

We suppose the following hypotheses related to problem (Pλ) throughout this paper.

1. H0:

$\xi,a\in L^{\infty}(\Omega)$,   $0<\xi_{0}\leq\operatorname*{ess~{}inf}_{\Omega}\xi$,  $a(x)>0$ for a. a. $x\in\Omega$.

This hypothesis implies that the natural function space for the analysis of problem (Pλ) is the Sobolev space $W^{1,p}_{0}(\Omega)$.

Let $\varphi_{\lambda}\colon W^{1,p}_{0}(\Omega)\to\mathbb{R}$ be the energy functional for problem (Pλ) defined by

[TABLE]

It is clear that $\varphi_{\lambda}$ is not $C^{1}$. The corresponding Nehari manifold for this functional is given by

[TABLE]

We decompose $N_{\lambda}$ into three disjoint parts

[TABLE]

Note that $N_{\lambda}$ is much smaller than $W^{1,p}_{0}(\Omega)$ and contains the nontrivial weak solutions of (Pλ). It is possible for \varphi_{\lambda}\big{|}_{N_{\lambda}} to exhibit properties which fail globally. One such property is identified in the next proposition.

Proposition 3.1**.**

If hypotheses H0 hold, then \varphi_{\lambda}\big{|}_{N_{\lambda}} is coercive.

Proof.

Let $u\in N_{\lambda}$. From the definition of the Nehari manifold we have

[TABLE]

From (3.1) and hypotheses H0 we obtain

[TABLE]

for some $c_{1},c_{2}>0$, where we have used Theorem 13.17 of Hewitt-Stromberg [5, p. 196], the fact that $1-\gamma<1<p$ and the Sobolev embedding theorem. From (3.2) it is clear that \varphi_{\lambda}\big{|}_{N_{\lambda}} is coercive. ∎

Let $m^{+}_{\lambda}=\inf_{N_{\lambda}^{+}}\varphi_{\lambda}$.

Proposition 3.2**.**

If hypotheses H0 hold, then $m^{+}_{\lambda}<0$.

Proof.

From the definition of $N^{+}_{\lambda}$, we have, for $u\in N_{\lambda}^{+}$,

[TABLE]

Moreover, since $u\in N^{+}_{\lambda}\subseteq N_{\lambda}$, it holds

[TABLE]

Applying (3.3), (3.4), hypotheses H0 and recalling $0<\gamma<1<p<r$, we get for $u\in N^{+}_{\lambda}$

[TABLE]

Therefore, \varphi_{\lambda}\big{|}_{N_{\lambda}^{+}}<0 and so $m^{+}_{\lambda}<0$. ∎

Proposition 3.3**.**

If hypotheses H0 hold, then there exists $\lambda^{*}>0$ such that for all $\lambda\in(0,\lambda^{*})$ we have $N^{0}_{\lambda}=\emptyset$.

Proof.

We argue indirectly. So, suppose that for every $\lambda^{*}>0$ there exists $\lambda\in(0,\lambda^{*})$ such that $N^{0}_{\lambda}\neq\emptyset$. Hence, given $\lambda>0$, we can find $u\in N_{\lambda}$ such that

[TABLE]

Moreover, since $u\in N_{\lambda}$, one has

[TABLE]

Subtracting (3.5) from (3.6) results in

[TABLE]

Hence, by hypotheses H0,

[TABLE]

for some $c_{3}>0$. This implies

[TABLE]

for some $c_{4}>0$.

On the other hand, from (3.5), hypotheses H0 and the Sobolev embedding theorem, we obtain

[TABLE]

for some $c_{5}>0$ and thus,

[TABLE]

We let $\lambda\to 0^{+}$ and see that $\|u\|\to\infty$, contradicting (3.7). Therefore, we can find $\lambda^{*}>0$ such that $N^{0}_{\lambda}=\emptyset$ for all $\lambda\in(0,\lambda^{*})$. ∎

Proposition 3.4**.**

If hypotheses H0 hold, then there exists $\hat{\lambda}^{*}\in(0,\lambda^{*}]$ such that for every $\lambda\in(0,\hat{\lambda}^{*})$, there exists $u^{*}\in N^{+}_{\lambda}$ such that

[TABLE]

and $u^{*}(x)\geq 0$ for a. a. $x\in\Omega$.

Proof.

Let $\{u_{n}\}_{n\geq 1}\subseteq N^{+}_{\lambda}$ be a minimizing sequence, that is,

[TABLE]

Since $N^{+}_{\lambda}\subseteq N_{\lambda}$, from Proposition 3.1, we infer that

[TABLE]

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

[TABLE]

From (3.8) and $u_{n}\overset{\operatorname*{w}}{\to}u^{*}$ in $W^{1,p}_{0}(\Omega)$ we have

[TABLE]

Hence, $u^{*}\neq 0$.

We consider the fibering function $\psi_{u^{*}}\colon[0,\infty)\to\mathbb{R}$ defined by

[TABLE]

Moreover, let $\eta_{u^{*}}\colon(0,\infty)\to\mathbb{R}$ be the function defined by

[TABLE]

Note that as $t\to 0^{+}$, then $\eta_{u^{*}}(t)\to-\infty$, since $r-p<r+\gamma-1$ and $a(x)>0$ for a. a. $x\in\Omega$, see H0. Also, $\eta_{u^{*}}(t)\to 0$ as $t\to+\infty$ and $\eta_{u^{*}}(t)>0$ for

[TABLE]

Therefore, we can find $t_{0}>\hat{t}$ such that

[TABLE]

This maximizer is unique and it is given by the solution of

[TABLE]

Hence,

[TABLE]

We see that

[TABLE]

Let $\hat{\lambda}^{*}\in(0,\lambda^{*}]$ such that

[TABLE]

We can find $t_{1}<t_{0}<t_{2}$ such that

[TABLE]

In this proof we will only use $t_{1}$, we mention the existence of $t_{2}$ as above since it will be needed in the sequel when we will minimize over $N^{-}_{\lambda}$.

Note that $\psi_{u^{*}}\in C^{2}(0,\infty)$. Therefore,

[TABLE]

and

[TABLE]

From (3.10) we have

[TABLE]

which implies that

[TABLE]

We will now apply (3.12) in (3.11) and obtain

[TABLE]

But using (3.12) in (3.11) gives

[TABLE]

because of (3.10).

From (3.13) and (3.14) it follows that

[TABLE]

which implies

[TABLE]

Suppose that

[TABLE]

Applying (3.9), (3.10) and (3.16) we get

[TABLE]

From (3.17) we see that there exists $n_{0}\in\mathbb{N}$ such that

[TABLE]

Recall that $u_{n}\in N^{+}_{\lambda}\subseteq N_{\lambda}$ and $\psi^{\prime}_{u_{n}}(t)=t^{r}\eta_{u_{n}}(t)$. Hence

[TABLE]

Then, by (3.18), it follows $t_{1}>1$.

Since $\psi_{u^{*}}$ is decreasing on $(0,t_{1}]$, we have

[TABLE]

But recall that $t_{1}u^{*}\in N^{+}_{\lambda}$ because of (3.15). So, by (3.19), we obtain

[TABLE]

a contradiction. This proves that $u_{n}\to u^{*}$ in $W^{1,p}_{0}(\Omega)$, see Papageorgiou-Winkert [11, p. 225], and so, with regards to (3.8),

[TABLE]

We know that $u_{n}\in N^{+}_{\lambda}$ for all $n\in\mathbb{N}$. This implies

[TABLE]

Therefore

[TABLE]

On account of Proposition 3.3, since $\lambda\in(0,\hat{\lambda}^{*}]$, we cannot have equality in (3.20). Therefore $u^{*}\in N^{+}_{\lambda}$ and finally we have

[TABLE]

Since we can always replace $u^{*}$ by $|u^{*}|$, we may assume that $u^{*}\geq 0$ with $u^{*}\neq 0$. ∎

The next lemma is inspired by Lemma 3 of Sun-Wu-Long [15]. In what follows we denote by $B_{\varepsilon}(0)$ the open $\varepsilon$-ball in $W^{1,p}_{0}(\Omega)$ centered at the origin, that is,

[TABLE]

Lemma 3.5**.**

If hypotheses H0 hold and $u\in N^{+}_{\lambda}$, then there exist $\varepsilon>0$ and a continuous function $\vartheta\colon B_{\varepsilon}(0)\to\mathbb{R}_{+}$ such that

[TABLE]

Proof.

We do the proof only for $N^{+}_{\lambda}$, the proof for $N^{-}_{\lambda}$ works in the same way. So, let $L\colon W^{1,p}_{0}(\Omega)\times(0,\infty)\to\mathbb{R}$ be defined by

[TABLE]

Since $u\in N^{+}_{\lambda}\subseteq N_{\lambda}$, one has $L(0,1)=0$. Moreover, because $u\in N^{+}_{\lambda}$, it holds

[TABLE]

Then, by the implicit function theorem, see Gasiński-Papageorgiou [2, p. 481], we can find $\varepsilon>0$ and a continuous map $\vartheta\colon B_{\varepsilon}(0)\to\mathbb{R}_{+}$ such that

[TABLE]

Choosing $\varepsilon>0$ even smaller if necessary, we can have

[TABLE]

∎

Proposition 3.6**.**

If hypotheses H0 hold, $\lambda\in(0,\hat{\lambda}^{*}]$ and $h\in W^{1,p}_{0}(\Omega)$, then we can find $b>0$ such that

[TABLE]

Proof.

We consider the function $\mu_{h}\colon[0,\infty)\to\mathbb{R}$ defined by

[TABLE]

Recall that $u^{*}\in N^{+}_{\lambda}\subseteq N_{\lambda}$, see Proposition 3.4. Thus, we have

[TABLE]

and

[TABLE]

Combining (3.21), (3.22) and (3.23) we obtain that

[TABLE]

The function $\mu_{h}$ is continuous. So, we can find $b_{0}>0$ such that

[TABLE]

see (3.24). Lemma 3.5 implies that for every $t\in[0,b_{0})$, we can find $\hat{\vartheta}(t)>0$ such that

[TABLE]

Taking (3.25) into account we finally reach that

[TABLE]

∎

The next proposition shows that $N^{+}_{\lambda}$ is a natural constraint for the functional $\varphi_{\lambda}$, see Papageorgiou-Rădulescu-Repovš [9, p. 425].

Proposition 3.7**.**

If hypotheses H0 hold and $\lambda\in(0,\hat{\lambda}^{*})$, then $u^{*}$ is a weak solution of problem (Pλ).

Proof.

Let $h\in W^{1,p}_{0}(\Omega)$. From Proposition 3.6 we know that

[TABLE]

This means

[TABLE]

Multiplying by $\frac{1}{t}$ and letting $t\to 0^{+}$ gives

[TABLE]

for all $h\in W^{1,p}_{0}(\Omega)$. Hence,

[TABLE]

for all $h\in W^{1,p}_{0}(\Omega)$. Thus, $u^{*}$ is a weak solution of (Pλ). ∎

Now we are ready to generate the first positive solution of problem (Pλ).

Proposition 3.8**.**

If hypotheses H0 hold and $\lambda\in(0,\hat{\lambda}^{*})$, then problem (Pλ) admits a positive solution $u^{*}\in W^{1,p}_{0}(\Omega)$ such that $u\in L^{\infty}(\Omega)$, $u^{*}(x)>0$ for a. a. $x\in\Omega$ and $\varphi_{\lambda}(u^{*})<0$.

Proof.

According to Proposition 3.4 there exists $u^{*}\in W^{1,p}_{0}(\Omega)$ such that

[TABLE]

From Proposition 3.7 we know that $u^{*}$ is a weak solution of problem (Pλ).

From Giacomoni-Schindler-Takáč [3, Lemma A.6, p. 142] we have that $u^{*}\in L^{\infty}(\Omega)$. Furthermore, the Harnack inequality, see Pucci-Serrin [14, p. 163] implies that

[TABLE]

∎

Now we start looking for a second positive solution. To this end, we will use the manifold $N^{-}_{\lambda}$.

Proposition 3.9**.**

If hypotheses H0 hold, then there exists $\hat{\lambda}^{*}_{0}\in(0,\hat{\lambda}^{*}]$ such that \varphi_{\lambda}\big{|}_{N^{-}_{\lambda}}\geq 0 for all $0<\lambda\leq\hat{\lambda}_{0}^{*}$.

Proof.

Let $u\in N_{\lambda}$. From the definition of $N^{-}_{\lambda}$ we have

[TABLE]

which implies

[TABLE]

Then, by the embedding $W^{1,p}_{0}(\Omega)\hookrightarrow L^{r}(\Omega)$, it follows

[TABLE]

for some $c_{5}>0$. Therefore

[TABLE]

Suppose that the result of the proposition is not true. This means that for every $\lambda>0$ there exists $u\in N^{-}_{\lambda}$ such that $\varphi_{\lambda}(u)<0$, that is,

[TABLE]

On the other hand, since $u\in N^{-}_{\lambda}\subseteq N_{\lambda}$, we have

[TABLE]

Using (3.28) in (3.27) yields

[TABLE]

which implies

[TABLE]

for some $c_{6}>0$. Hence

[TABLE]

and so

[TABLE]

for some $c_{7}>0$.

Now we use (3.29) in (3.26) and obtain

[TABLE]

This implies

[TABLE]

Letting $\lambda\to 0^{+}$ leads to a contradiction. So, we can find $0<\hat{\lambda}_{0}^{*}\leq\hat{\lambda}^{*}$ such that \varphi_{\lambda}\big{|}_{N^{-}_{\lambda}}\geq 0 for all $\lambda\in(0,\hat{\lambda}_{0}^{*}]$. ∎

Now we minimize $\varphi_{\lambda}$ on the manifold $N^{-}_{\lambda}$.

Proposition 3.10**.**

If hypotheses H0 hold and $\lambda\in(0,\lambda^{*}_{0})$, then we can find $v^{*}\in N^{-}_{\lambda}$ with $v^{*}\geq 0$ such that

[TABLE]

Proof.

The proof of the proposition is the same as that of Proposition 3.4. Only now as we already hinted in that proof, we use the point $t_{2}>t_{0}$ for which we have

[TABLE]

see (3.10). Then we conclude that

[TABLE]

∎

Applying Lemma 3.5 and reasoning as in the proofs of Propositions 3.6 and 3.7 we show that $N^{-}_{\lambda}$ is a natural constraint for the energy functional $\varphi_{\lambda}$ as well.

Proposition 3.11**.**

If hypotheses H0 hold and $\lambda\in(0,\hat{\lambda}_{0}^{*})$, then $v^{*}$ is a weak solution of problem (Pλ).

Therefore, we have a second positive solution $v^{*}\in W^{1,p}_{0}(\Omega)\cap L^{\infty}(\Omega)$ and by Harnack’s inequality we have $v^{*}(x)>0$ for a. a. $x\in\Omega$.

Finally, we can state the following multiplicity theorem for problem (Pλ).

Theorem 3.12**.**

If hypotheses H0 hold, then there exists $\hat{\lambda}^{*}_{0}>0$ such that for all $\lambda\in(0,\hat{\lambda}^{*}_{0})$, problem (Pλ) has at least two positive solutions

[TABLE]

and

[TABLE]

Remark 3.13**.**

It is an interesting open problem whether the multiplicity theorem above holds if we assume that

[TABLE]

but not necessarily bounded away from zero.

Acknowledgment

The authors wish to thank a knowledgeable referee for his/her corrections and remarks.

The second author thanks the National Technical University of Athens for the kind hospitality during a research stay in June 2019.