Positive solutions of an elliptic Neumann problem with a sublinear indefinite nonlinearity

TL;DR

This paper proves the existence of positive solutions for a nonlinear elliptic Neumann problem with a sign-changing weight, analyzes their asymptotic behavior as the nonlinearity approaches linearity, and explores explicit conditions and properties of solutions.

Contribution

It extends previous existence results for positive solutions of the elliptic Neumann problem with sublinear indefinite nonlinearity and provides detailed asymptotic and explicit solution conditions.

Findings

Existence of positive solutions for q between q_0 and 1.

Asymptotic behavior of solutions as q approaches 1 from below.

Conditions ensuring positive solutions in radial symmetric cases.

Abstract

Let $\Omega\subset\mathbb{R}^{N}$ ($N\geq1$) be a bounded and smooth domain and $a:\Omega\rightarrow\mathbb{R}$ be a sign-changing weight satisfying $\int_{\Omega}a<0$. We prove the existence of a positive solution $u_{q}$ for the problem $(P_{a,q})$: $-\Delta u=a(x)u^{q}$ in $\Omega$, $\frac{\partial u}{\partial\nu}=0$ on $\partial\Omega$, if $q_{0}<q<1$, for some $q_{0}=q_{0}(a)>0$. In doing so, we improve the existence result previously established in [16]. In addition, we provide the asymptotic behavior of $u_{q}$ as $q\rightarrow1^{-}$. When $\Omega$ is a ball and $a$ is radial, we give some explicit conditions on $q$ and $a$ ensuring the existence of a positive solution of $(P_{a,q})$. We also obtain some properties of the set of $q$'s such that $(P_{a,q})$ admits a solution which is positive on $\overline{\Omega}$. Finally, we present some results on nonnegative solutions having dead cores. Our approach combines bifurcation techniques, a priori bounds and the sub-supersolution method. Several methods and results apply as well to the Dirichlet counterpart of $(P_{a,q})$.

Equations

$$\left\{\begin{array}[c]{lll}-\Delta u=a(x)u^{q}&\mathrm{in}&\Omega,\\ \frac{\partial u}{\partial\nu}=0&\mathrm{on}&\partial\Omega,\end{array}\right.$$

$$a\text{ changes sign and }{\int }_{\Omega }a<0.$$

$${\mathcal{P}}^{\circ }:=\left\{u\in {\mathcal{C}}^1(\overline{\Omega }):u>0\text{ on }\overline{\Omega }\right\}.$$

$${\Omega }_{+}\text{ has finitely many connected components and }\left|(\text{supp }{a}^{+})\setminus {\Omega }_{+}\right|=0.$$

$$\left\{\begin{array}[c]{lll}-\Delta\phi=\mu a(x)\phi&\mathrm{in}&\Omega,\\ \frac{\partial\phi}{\partial\nu}=0&\mathrm{on}&\partial\Omega,\end{array}\right.$$

$${\Gamma }_1:=\{(q,u)=(1,t{\varphi }_1):t>0\}.$$

$$\left\{\begin{array}[c]{lll}-\Delta\varphi=qa(x)u^{q-1}\varphi+\gamma\varphi&\mathrm{in}&\Omega,\\ \frac{\partial\varphi}{\partial\nu}=0&\mathrm{on}&\partial\Omega.\end{array}\right.$$

$$t_{\ast }:=\exp \left[-\frac{{\int }_{\Omega }a\left(x\right){\varphi }_1^2\log {\varphi }_1}{{\int }_{\Omega }a\left(x\right){\varphi }_1^2}\right].$$

$$u_q\sim {\mu }_1(a{)}^{-\frac{1}{1-q}}{t}_{\ast }{\varphi }_1\text{mbox}asq\to 1^{-},$$

$$B_{R_0}$$

$$A_{R_0,R}$$

$${\omega }_{N-1}$$

$$\frac{1-q}{1+q}{\int }_{A_{R_0,R}}{a}^{-}\le {\int }_{B_{R_0}}{a}^{+}.$$

$$\frac{-{\int }_{\Omega }a}{{\int }_{\Omega }\left|a\right|}\le q<1.$$

$$a_{\delta }=a^{+}-\delta a^{-},\text{mbox}with\delta >{\delta }_0:=\frac{{\int }_{\Omega }{a}^{+}}{{\int }_{\Omega }{a}^{-}},$$

$$\frac{1-q}{2q+N\left(1-q\right)}{\omega }_{N-1}{R}_0^N{\Vert a^{-}\Vert }_{L^{\infty }(B_{R_0})}<{\int }_{A_{R_0,R}}{a}^{+}.$$

$$\mathcal{A}={\mathcal{A}}_a:=\{q\in (0,1):\text{any nontrivial nonnegative solution of }(P_{a,q})\text{ lies in }{\mathcal{P}}^{\circ }\}$$

$$\mathcal{I}={\mathcal{I}}_a:=\{q\in (0,1):(P_{a,q})\text{ has a solution }u\in {\mathcal{P}}^{\circ }\}.$$

$${\Omega }_{+}\text{ is connected and }\left|(\text{supp }{a}^{+})\setminus {\Omega }_{+}\right|=0,$$

$$\partial {\Omega }_{+}\text{ satisfies the inner sphere condition with respect to }{\Omega }_{+}.$$

$$r:=\frac{2}{1-q_1}\in \left(2,\infty \right),\text{and}a\left(x\right):=r^{1-\frac{2}{r}}\left(1-r{\cos }^{2}x\right),\text{for }x\in \Omega .$$

$$\left\{\begin{array}[c]{lll}-u^{\prime\prime}=a(x)u^{q_{1}}&\mathrm{in}&\Omega,\\ u^{\prime}=0&\mathrm{on}&\partial\Omega.\end{array}\right.$$

$$b_1\in L^{\infty }(\Omega ),b_2\in \mathcal{C}(\overline{\Omega }),b_1,b_2\ge 0\text{ and }\text{supp }{b}_1\cap \{x\in \Omega :b_2(x)>0\}=\varnothing .$$

$$G_{\sigma }:=\{x\in G:\text{dist}(x,\partial G)>\sigma \}.$$

$$\left\{\begin{array}[c]{l}b\in L^{\infty}(\Omega),\ 0\not\equiv b\geq 0,\mbox{ and }\\ \emptyset\not=G\subset\Omega\mbox{ is an open subset such that }\mathrm{supp}\ b\cap G=\emptyset.\end{array}\right.$$

$$\text{supp }{b}_1\cup \{x\in \Omega :b_2(x)>0\}=\Omega .$$

$$\left\{\begin{array}[c]{lll}-\Delta u=a_{\delta}(x)u^{q}&\mathrm{in}&\Omega,\\ u=0&\mathrm{on}&\partial\Omega,\end{array}\right.$$

$$\left\{\begin{array}[c]{lll}-\Delta\phi=\lambda m\left(x\right)\phi&\mathrm{in}&\Omega,\\ \phi=0&\mathrm{on}&\partial\Omega,\end{array}\right.$$

$$v_n=c^{\frac{1}{1-q_n}}{u}_n\ge \psi \text{mbox}inB,$$

$${\int }_{\Omega }|\nabla v_n{|}^2=\left({\int }_{\Omega }a(x)v_n^{q_n+1}\right)\Vert u_n{\Vert }^{q_n-1}.$$

Full text

[TABLE]

if $q_{0}<q<1$, for some $q_{0}=q_{0}(a)>0$. In doing so, we improve the existence result previously established in [16]. In addition, we provide the asymptotic behavior of $u_{q}$ as $q\rightarrow 1^{-}$. When $\Omega$ is a ball and $a$ is radial, we give some explicit conditions on $q$ and $a$ ensuring the existence of a positive solution of $(P_{a,q})$. We also obtain some properties of the set of $q$’s such that $(P_{a,q})$ admits a solution which is positive on $\overline{\Omega}$. Finally, we present some results on nonnegative solutions having dead cores. Our approach combines bifurcation techniques, *a priori *bounds and the sub-supersolution method. Several methods and results apply as well to the Dirichlet counterpart of $(P_{a,q})$.

1 Introduction

Let $\Omega$ be a bounded and smooth domain of $\mathbb{R}^{N}$ with $N\geq 1$, and $0<q<1$. The purpose of this article is to discuss the existence of positive solutions for the problem

[TABLE]

where $\Delta$ is the usual Laplacian in $\mathbb{R}^{N}$, and $\nu$ is the outward unit normal to $\partial\Omega$. Throughout this article, unless otherwise stated, we assume that $r>N$ and $a\in L^{r}\left(\Omega\right)$ is such that

[TABLE]

We set $a^{\pm}:=\max(\pm a,0)$. Note that the change of sign in $a$ means that $|\text{supp }a^{\pm}|>0$, where $|A|$ stands for the Lebesgue measure of $A\subset\mathbb{R}^{N}$. By a nonnegative solution of $(P_{a,q})$ we mean a function $u\in W^{2,r}\left(\Omega\right)$ (and hence $u\in\mathcal{C}^{1}(\overline{\Omega})$) that satisfies the equation for the weak derivatives and the boundary condition in the usual sense, and such that $u\geq 0$ in $\Omega$. If, in addition, $u>0$ in $\Omega$, then we call it a positive solution of $(P_{a,q})$. In this case, we shall also say that $(q,u)$ is a positive solution of $(P_{a,q})$. Let us denote by $\mathcal{P}^{\circ}$ the interior of the positive cone of $\mathcal{C}^{1}(\overline{\Omega})$, i.e.,

[TABLE]

We observe that a positive solution of $(P_{a,q})$ need not belong to $\mathcal{P}^{\circ}$ (see e.g. Remark 1.9 (i) below).

Very few works have been devoted to $(P_{a,q})$, the first and main one being [4], where the following results were established (see Theorem 2.1 and Lemmas 2.1 and 3.1 therein):

Theorem 1.0** (Bandle-Pozio-Tesei [4]).**

Let $0<q<1$ and $a$ be a sign-changing Hölder continuous function on $\overline{\Omega}$. Then, the following three assertions hold:

1. (i)

If $(P_{a,q})$ has a positive solution then $\int_{\Omega}a<0$. 2. (ii)

If $\int_{\Omega}a<0$, then $(P_{a,q})$ has at least one nontrivial nonnegative solution. 3. (iii)

$(P_{a,q})$* has at most one solution in $\mathcal{P}^{\circ}$.*

Let us mention that some of the above results were extended in [1] to a problem that is a linear perturbation of $(P_{a,q})$. However, no sufficient conditions for the existence of positive solutions have been provided in [1, 4]. Let us point out that, due to the non-Lipschitzian character of $u^{q}$ at $u=0$ and the change of sign in $a$, the strong maximum principle does not apply to $(P_{a,q})$. As a consequence, one cannot derive the positivity of nontrivial nonnegative solutions of $(P_{a,q})$.

To the best of our knowledge, the first existence result on positive solutions of $(P_{a,q})$ has been proved in our recent work [16]. We recall it now. Denoting by $\Omega_{+}=\Omega_{+}(a)$ the largest open subset of $\Omega$ where $a>0$ a.e., let us consider the following condition:

[TABLE]

Under $(H_{1})$, we showed that every nontrivial nonnegative solution of $(P_{a,q})$ belongs to $\mathcal{P}^{\circ}$ if $q$ is close enough to $1$ (see [16, Theorem 1.7]). This positivity result was proved via a continuity argument inspired by [14, Theorem 4.1] (see also [15]), which is based on the fact that the strong maximum principle applies to $(P_{a,q})$ if $q=1$. As a consequence, assuming in addition $(H_{0})$, we deduced that, for $q$ close enough to $1$, $(P_{a,q})$ has a solution in $\mathcal{P}^{\circ}$, which is the unique nontrivial nonnegative solution of $(P_{a,q})$ (see [16, Corollary 1.8]). Let us mention that, in general, uniqueness of nonnegative solutions for $(P_{a,q})$ does not hold (see e.g. the proof of Theorem 1.8 (ii) below).

Regarding the Dirichlet counterpart of $(P_{a,q})$, we refer to [3, 18] for the existence of nontrivial nonnegative solutions, and to [10, 11, 13, 16] for the existence of a positive solution. Let us mention, as already pointed out in [1, 3, 4], that problems like $(P_{a,q})$ and its Dirichlet counterpart naturally arise in population dynamics models, cf. [12, 17].

Our purpose in this article is to carry on the investigation of $(P_{a,q})$, refining and extending the existence results on positive solutions established in [16]. In particular, following a different approach to the one in [16], we shall remove $(H_{1})$ and prove that under $(H_{0})$ the problem $(P_{a,q})$ has a unique solution $u_{q}\in\mathcal{P}^{\circ}$ for $q$ close to $1$. As a byproduct, we deduce that $(H_{0})$ is necessary and sufficient for the existence of a positive solution of $(P_{a,q})$ for some $q\in(0,1)$, see Corollary 1.2. Moreover, we shall provide the stability properties of $u_{q}$ and its asymptotic behavior as $q\rightarrow 1^{-}$ (see Theorem 1.1 below). Note that the stability analysis for solutions in $\mathcal{P}^{\circ}$ of $(P_{a,q})$ is not easily carried out for $q\in(0,1)$ in general (see Remark 2.6 (ii)).

Under $(H_{0})$, let us denote by $\mu_{1}\left(a\right)$ the first positive eigenvalue of the problem

[TABLE]

and by $\phi_{1}=\phi_{1}(a)$ the associated positive eigenfunction satisfying $\int_{\Omega}\phi_{1}^{2}=1$. It is well known that $\mu_{1}(a)$ is simple, and $\phi_{1}\in\mathcal{P}^{\circ}$.

We shall look at $q$ as a bifurcation parameter in $(P_{a,q})$. As a matter of fact, note that if $\mu_{1}(a)=1$, then $u=t\phi_{1}$ solves $(P_{a,1})$, i.e., $(P_{a,q})$ has the trivial line $\Gamma_{1}$ of solutions in $\mathcal{P}^{\circ}$, where

[TABLE]

We shall obtain, for $q$ close to $1$, a curve of solutions in $\mathcal{P}^{\circ}$ bifurcating from $\Gamma_{1}$ (see Figure 1).

Let us recall that a solution $u\in\mathcal{P}^{\circ}$ of $(P_{a,q})$ is said to be asymptotically stable (respect. unstable) if $\gamma_{1}(q,u)>0$ (respect. $<0$), where $\gamma_{1}(q,u)$ is the first eigenvalue of the linearized eigenvalue problem at $u$, namely,

[TABLE]

In addition, $u$ is said to be weakly stable if $\gamma_{1}(q,u)\geq 0$.

Set

[TABLE]

We are now in position to state our main results.

Theorem 1.1**.**

Assume $(H_{0})$. Then there exists $q_{0}=q_{0}\left(a\right)\in(0,1)$ such that $(P_{a,q})$ has a unique solution $u_{q}\in\mathcal{P}^{\circ}$ for $q_{0}<q<1$. Moreover, $u_{q}$ is asymptotically stable and satisfies the asymptotics

[TABLE]

i.e. $\mu_{1}(a)^{\frac{1}{1-q}}u_{q}\rightarrow t_{*}\phi_{1}$ in $\mathcal{C}^{1}(\overline{\Omega})$ as $q\to 1^{-}$. More specifically:

1. (i)

If $\mu_{1}(a)=1$, then $u_{q}\rightarrow t_{\ast}\phi_{1}$ in $\mathcal{C}^{1}(\overline{\Omega})$ as $q\rightarrow 1^{-}$. 2. (ii)

If $\mu_{1}(a)>1$, then $u_{q}\rightarrow 0$ in $\mathcal{C}^{1}(\overline{\Omega})$ as $q\rightarrow 1^{-}$. 3. (iii)

If $\mu_{1}(a)<1$, then $\displaystyle\min_{\overline{\Omega}}u_{q}\rightarrow\infty$ as $q\rightarrow 1^{-}$.

One may easily see that Theorem 1.0 (i) still holds if $a\in L^{r}(\Omega)$, with $r>N$, cf. the proof of [4, Lemma 2.1]. As a consequence of this result and Theorem 1.1, we derive the following:

Corollary 1.2**.**

$(P_{a,q})$* has a positive solution (or a solution in $\mathcal{P}^{\circ}$) for some $q\in(0,1)$ if and only if $(H_{0})$ holds.*

We shall prove Theorem 1.1 using a bifurcation technique based on the Lyapunov-Schmidt reduction, which yields the existence of bifurcating solutions in $\mathcal{P}^{\circ}$ from $\Gamma_{1}$ provided that $\mu_{1}(a)=1$. By a suitable rescaling, we deduce then the results for the case $\mu_{1}(a)\neq 1$. Let us also point out that, in general, it is hard to give a lower estimate for $q_{0}\left(a\right)$, see Remark 1.9 (i) below.

When $\Omega$ is a ball and $a$ is radial, we shall exhibit some explicit conditions on $q$ and $a$ so that $(P_{a,q})$ admits a positive solution. This will be done via the well known sub-supersolutions method. In Theorem 1.3 below we give a condition that guarantees the existence of a positive solution (not necessarily in $\mathcal{P}^{\circ}$), while Theorem 1.5 provides us with a solution in $\mathcal{P}^{\circ}$.

Given $0<R_{0}<R$, we write

[TABLE]

If $f$ is a radial function, we write (with a slight abuse of notation) $f\left(x\right):=f\left(\left|x\right|\right):=f\left(r\right)$. We first consider the case that $\text{supp }a^{+}$ is contained in $B_{R_{0}}$ for some $R_{0}\in(0,R)$.

Theorem 1.3**.**

Let $\Omega:=B_{R}$ and $a\in L^{\infty}\left(\Omega\right)$ be a radial function such that $\int_{\Omega}a<0$. Assume that there exists $R_{0}>0$ such that:

• •

$a\geq 0$* in $B_{R_{0}}$;*

• •

$a\leq 0$* in $A_{R_{0},R}$;*

• •

$r\rightarrow a(r)$* is differentiable and nonincreasing in $\left(R_{0},R\right)$, and*

[TABLE]

Then $(P_{a,q})$ has a positive solution, which is unique if $(H_{1})$ holds.

Remark 1.4**.**

1. (i)

The condition (1.2) can also be formulated as

[TABLE]

In particular, we see that (1.2) is satisfied if $q$ is close enough to $1$. Note that if we replace $a$ by

[TABLE]

then the left-hand side in (1.3) approaches $1$ as $\delta\to\infty$, so that this condition becomes very restrictive for $a_{\delta}$ as $\delta\to\infty$. On the other side, we have that $\int_{\Omega}a_{\delta}\to 0^{-}$ as $\delta\to\delta_{0}^{+}$, so that (1.3) becomes much less constraining for $a_{\delta}$ as $\delta\to\delta_{0}^{+}$. A similar argument will be used in Remark 4.4. 2. (ii)

As one can see from the proof of Theorem 1.3, the condition (1.2) guarantees the existence of a positive subsolution for the corresponding Dirichlet problem. Thus, since arbitrarily large supersolutions can be easily obtained in the Dirichlet case (see e.g. [10, Remark 1.1]), it follows that (1.2) ensures the existence of a positive solution for the analogous Dirichlet problem. Moreover, we point out that this condition substantially improves some of the results known in that case (see [10, Section 3]).

Next we consider the case that $\text{supp }a^{-}$ is contained in $B_{R_{0}}$ for some $R_{0}\in(0,R)$.

Theorem 1.5**.**

Let $\Omega:=B_{R}$ and $a\in L^{\infty}\left(\Omega\right)$ be a radial function such that $\int_{\Omega}a<0$. Assume that there exists $R_{0}\in\left(0,R\right)$ such that $a\geq 0$ in $A_{R_{0},R}$, and

[TABLE]

Then $(P_{a,q})$ has a unique solution $u\in\mathcal{P}^{\circ}$.

Remark 1.6**.**

Observe that unlike Theorem 1.3, no differentiability nor monoto-nicity condition is imposed on $a^{-}$ in Theorem 1.5. Note again that (1.4) is also clearly satisfied if $q$ is close enough to $1$.

Our next results concern the sets

[TABLE]

and

[TABLE]

We observe that if $(H_{0})$ holds then $(P_{a,q})$ has a nontrivial nonnegative solution for any $0<q<1$ (see e.g. the proof of [16, Corollary 1.8]), so that $\mathcal{A}\subseteq\mathcal{I}$. In [16, Theorem 1.9], we proved, under $(H_{0})$ and $(H_{1})$, that $\mathcal{A}=(q_{a},1)$ for some $q_{a}\in[0,1)$.

Let us now introduce the following assumptions:

[TABLE]

[TABLE]

Note that $(H_{1}^{\prime})$ corresponds to $(H_{1})$ with $\Omega_{+}$ consisting of a single connected component.

Remark 1.7**.**

If $a$ is Hölder continuous, $\Omega_{+}$ has finitely many connected components and $(H_{+})$ holds, then [4, Theorem 3.1] shows, in particular, that $(P_{a,q})$ has at most one nonnegative solution $u$ such that $u>0$ in $\Omega_{+}$. Let us observe that their proof is still valid for $a\in L^{r}(\Omega)$ with $r>N$, assuming now $(H_{1})$ and $(H_{+})$.

We shall complement [16, Theorem 1.9] as follows:

Theorem 1.8**.**

1. (i)

Assume $(H_{0})$. Then $\mathcal{I}\not=\emptyset$. Moreover:

1. (i1)

Either $\mathcal{I}=[q_{i},1)$ with $q_{i}\in\left(0,1\right)$ or $\mathcal{I}=\left(q_{i},1\right)$ with $q_{i}\in[0,1)$. 2. (i2)

If $(H_{1}^{\prime})$ and $(H_{+})$ hold, then for all $q\in\left(0,1\right)$, there exists a unique nontrivial nonnegative solution of $(P_{a,q})$. In particular, $\mathcal{I}=\mathcal{A}$. 2. (ii)

Let $\Omega:=(x_{0},x_{1})\subset\mathbb{R}$. Given $q\in\left(0,1\right)$, there exists $a\in\mathcal{C}(\overline{\Omega})$ such that $q\in\mathcal{I}\setminus\mathcal{A}$.

Remark 1.9**.**

1. (i)

Let $q_{1}\in\left(0,1\right)$, and define $\Omega:=\left(0,\pi\right)$,

[TABLE]

One can check that u\left(x\right):=\frac{\sin^{r}x}{r}\is a (strictly positive in $\Omega$) solution of

[TABLE]

It follows that $q_{1}\not\in\mathcal{A}$ because $u\not\in\mathcal{P}^{\circ}$. Now, since $a$ satisfies $(H_{1}^{\prime})$ and $(H_{+})$, we deduce from Theorem 1.8 (i2) that $u$ is the unique nontrivial nonnegative solution of $(P_{a,q_{1}})$, and $\mathcal{I}=\mathcal{A}$. Consequently, we have $q_{1}\not\in\mathcal{I}$. In particular, if $q_{0}\left(a\right)$ is given by Theorem 1.1, then $q_{0}\left(a\right)\geq q_{1}$. In the same way, if $q_{i}$ is provided by Theorem 1.8 (i1), then $q_{i}\geq q_{1}$. 2. (ii)

After the corresponding modifications, Theorem 1.8 (i) holds also for the Dirichlet counterpart of $(P_{a,q})$, replacing $(H_{0})$ by the condition that $a^{+}\not\equiv 0$. As a matter of fact, in this case one can check that the proof of Theorem 1.8 (i) can be carried out, using now [3, Theorems 2.1, 2.2, and Lemma 2.3].

Finally, we shall investigate the existence of nonnegative dead core solutions of $(P_{a,q})$. Following [3, 4], the set $\{x\in\Omega:u(x)=0\}$ is called the dead core of a nontrivial nonnegative solution $u$ of $(P_{a,q})$. Let us mention that in the proof of Theorem 1.8 (ii) we shall see that, when $N=1$, for any $q\in(0,1)$ there exists $a$ with $(P_{a,q})$ admitting a solution in $\mathcal{P}^{\circ}$ and also nonnegative solutions with nonempty dead cores.

Next we give some sufficient conditions for the existence of dead core solutions of $(P_{a,q})$. We introduce the following condition:

[TABLE]

Given a nonempty open subset $G\subseteq\Omega$ and $\sigma>0$, we set

[TABLE]

We call the set $\Omega\setminus\overline{\Omega_{\sigma}}$ a tubular neighborhood of $\partial\Omega$.

Theorem 1.10**.**

1. (i)

Let $q\in(0,1)$, and assume that $(H_{1}^{\prime})$ holds and $\Omega_{+}$ contains a tubular neighborhood of $\partial\Omega$. Then, every nontrivial nonnegative solution of $(P_{a,q})$ is positive on $\partial\Omega$. In particular, if $u$ is a nontrivial nonnegative solution of $(P_{a,q})$, then either $u\in\mathcal{P}^{\circ}$ or $u$ has a nonempty dead core. 2. (ii)

Let $a_{\delta}:=b_{1}-\delta b_{2}$, with $b_{1},b_{2}\not\equiv 0$ satisfying $(H_{2})$, and $\delta>0$. If we set $G:=\{x\in\Omega:b_{2}(x)>0\}$ then, given $0<\overline{q}<1$ and $\sigma>0$, there exists $\delta_{0}=\delta_{0}(\sigma,\overline{q})>0$ such that any nontrivial nonnegative solution of $(P_{a_{\delta},q})$ with $q\in(0,\overline{q}]$ vanishes in $G_{\sigma}$ if $\delta\geq\delta_{0}$.

Remark 1.11**.**

1. (i)

The conclusion of Theorem 1.10 (ii) still holds if $a_{\delta}:=b-\delta\chi_{G}$, with $b,G$ satisfying

[TABLE]

Here $\delta>0$ and $\chi_{G}$ is the characteristic function of $G$. 2. (ii)

Let $a_{\delta}:=b_{1}-\delta b_{2}$ with $b_{1},b_{2}\not\equiv 0$ satisfying $(H_{2})$, and $\delta>0$.

1. (ii1)

In addition to $(H_{1}^{\prime})$, let us assume that

[TABLE]

Let $q\in(0,1)$. Theorem 1.10 (ii) then shows that the support of any nontrivial nonnegative solution of $(P_{a_{\delta},q})$ approaches $\Omega_{+}$ (in some sense) as $\delta\rightarrow\infty$. 2. (ii2)

Combining Theorem 1.1 and Theorem 1.10 (ii), we find $\delta_{1}>0$ and $0<q_{1}\leq q_{0}<1$ such that any nontrivial nonnegative solution of $(P_{a_{\delta},q})$ with $\delta=\delta_{1}$ has a nonempty dead core for $q\in(0,q_{1}]$, whereas this problem has a unique solution in $\mathcal{P}^{\circ}$ and no other nontrivial nonnegative solutions for $q\in(q_{0},1)$. Furthermore, according to Theorem 1.8 (i2) and Theorem 1.10 (i), we see that if $(H_{1}^{\prime})$ and $(H_{+})$ hold and $\Omega_{+}$ contains a tubular neighborhood of $\partial\Omega$, then $q_{1}=q_{0}$, and the nontrivial nonnegative solution for $q\in(0,q_{0}]$ is also unique (see Figure 2). 3. (ii3)

As we shall see from its proof, Theorem 1.10 (ii) holds also for the Dirichlet counterpart of $(P_{a_{\delta},q})$. In particular, it complements [16, Theorem 1.1] as follows: given $q\in(0,1)$ there exist $0<\delta_{1}<\delta_{0}$ such that every nontrivial nonnegative solution $u$ of

[TABLE]

satisfies $u>0$ in $\Omega$ and $\frac{\partial u}{\partial\nu}<0$ on $\partial\Omega$ for $\delta<\delta_{1}$, whereas $u$ has a nonempty dead core for $\delta>\delta_{0}$.

The rest of the paper is organized as follows: in Section $2$ we establish some bifurcation results and stability properties for solutions in $\mathcal{P}^{\circ}$ of $(P_{a,q})$, whereas Section $3$ is devoted to the proof of Theorems 1.1, 1.3 and 1.5. In Section $4$ we prove Theorem 1.8 and some corollaries of it. Finally, Section 5 is concerned with the existence of dead core solutions and the proof of Theorem 1.10.

2 Bifurcation analysis

This section is devoted to the bifurcation analysis of $(P_{a,q})$, where $q$ is the bifurcation parameter. First we establish, under $(H_{0})$ and $(H_{1})$, some a priori bounds for nontrivial nonnegative solutions of $(P_{a,q})$, which imply that no nontrivial nonnegative solutions bifurcate from zero or from infinity at any $q\in[0,1)$. More precisely, we shall see that given $q\in[0,1)$ there exists no sequence $\left\{q_{n}\right\}\subset(0,1)$ such that $q_{n}\to q$ and $(P_{a,q_{n}})$ has a nontrivial nonnegative solution $u_{n}$ satisfying $u_{n}\rightarrow 0$ in $\mathcal{C}(\overline{\Omega})$ or $\|u_{n}\|_{\infty}\rightarrow\infty$.

Proposition 2.1**.**

1. (i)

Assume $(H_{1})$. Then, given $q_{1}\in(0,1)$, there exists $C>1$ such that $\|u\|_{L^{\infty}(\Omega)}>C^{-1}$ for all nontrivial nonnegative solutions of $(P_{a,q})$ with $q\in(0,q_{1}]$. 2. (ii)

*Assume $(H_{0})$. Then, given $q_{1}\in(0,1)$, there exists $C>1$ such that $\|u\|_{L^{\infty}(\Omega)}<C$ for all nontrivial nonnegative solutions of $(P_{a,q})$ with $q\in(0,q_{1}]$. *

Proof.

1. (i)

First we obtain an a priori bound from below. Assume by contradiction that there exist $0<q_{n}\leq\overline{q}<1$ and $u_{n}$ nontrivial nonnegative solutions of $(P_{a,q_{n}})$ such that $u_{n}\rightarrow 0$ in $\mathcal{C}(\overline{\Omega})$. Then, thanks to $(H_{1})$, we may assume that $u_{n}\not\equiv 0$ in some fixed subdomain $\Omega^{\prime}\subset\Omega_{+}$. By the strong maximum principle, we deduce that $u_{n}>0$ in $\Omega^{\prime}$.

We fix $c>0$ sufficiently large such that $\lambda_{1}(ca,\Omega^{\prime})<1$, where $\lambda_{1}(m,\Omega)$ denotes the first positive eigenvalue of the Dirichlet problem

[TABLE]

and observe that $v_{n}:=c^{\frac{1}{1-q_{n}}}u_{n}$ are nontrivial nonnegative solutions of $(P_{ca,q_{n}})$. We now apply [16, Lemma 2.5] to get a ball $B\subset\Omega^{\prime}$ and a positive function $\psi$ in $B$ such that

[TABLE]

where $\psi$ and $B$ do not depend on $n$. It follows that $u_{n}\geq c^{-\frac{1}{1-q_{n}}}\psi$ in $B$, which provides a contradiction, since $q_{n}\leq\overline{q}<1$. 2. (ii)

We obtain now an a priori bound from above. Assume to the contrary that there exist $0<q_{n}\leq\overline{q}<1$ and $u_{n}$ nontrivial nonnegative solutions of $(P_{a,q_{n}})$ such that $\|u_{n}\|:=\|u_{n}\|_{H^{1}(\Omega)}\rightarrow\infty$. We set $v_{n}:=\frac{u_{n}}{\|u_{n}\|}$, so that we may assume that $v_{n}\rightharpoonup v_{0}$ in $H^{1}(\Omega)$ and $v_{n}\rightarrow v_{0}$ in $L^{s}(\Omega)$ for $s\in[1,2^{\ast})$. From $(P_{a,q_{n}})$ we have that

[TABLE]

Since $q_{n}\leq\overline{q}<1$, it follows that $\int_{\Omega}|\nabla v_{n}|^{2}\rightarrow 0$. Hence, we deduce that $v_{n}\rightarrow v_{0}$ in $H^{1}(\Omega)$, and $v_{0}$ is a positive constant. Finally, since $\int_{\Omega}a(x)v_{n}^{q_{n}+1}>0$ we derive that $\int_{\Omega}a\geq 0$, which contradicts $(H_{0})$. By elliptic regularity, we have the desired conclusion. $\blacksquare$

In view of Proposition 2.1, we see that, under $(H_{0})$ and $(H_{1})$, bifurcation from zero or from infinity can only occur at $q=1$. As already mentioned, we shall look at $q$ as a bifurcation parameter in $(P_{a,q})$, and then seek for bifurcating solutions in $\mathcal{P}^{\circ}$ from the trivial line $\Gamma_{1}=\{(1,t\phi_{1}):t>0\}$ when $\mu_{1}(a)=1$. To this end, we employ the Lyapunov-Schmidt reduction for $(P_{a,q})$, based on the positive eigenfunction $\phi_{1}$. We set

[TABLE]

The usual decomposition of $D(A)$ is given by the formula

[TABLE]

where $t=\int_{\Omega}u\phi_{1}$, and $w=u-(\int_{\Omega}u\phi_{1})\phi_{1}$. So, $X_{2}$ is characterized as

[TABLE]

On the other hand, put $Y:=L^{r}(\Omega)=Y_{1}+R(A)$, where

[TABLE]

and

[TABLE]

Let $Q$ be the projection of $Y$ to $R(A)$, given by

[TABLE]

We reduce $(P_{a,q})$ to the following coupled equations:

[TABLE]

The first equation yields

[TABLE]

where we have used the fact that $\int_{\Omega}Au\phi_{1}=\int_{\Omega}uA\phi_{1}=0$. The second equation implies that

[TABLE]

and thus, that

[TABLE]

Now, we see that $(q,t,w)=(1,t,0)$ satisfies (2.1) and (2.2) for any $t>0$. So, first we solve (2.1) with respect to $w$, around $(q,t,w)=(1,t_{0},0)$ for a fixed $t_{0}>0$. To this end, we introduce the mapping $F:(1-\delta,1+\delta)\times(t_{0}-d,t_{0}+d)\times B_{\rho}(0)\rightarrow R(A)$ given by

[TABLE]

where $B_{\rho}(w)$ is the ball in $X_{2}$ centered at $w$ and with radius $\rho>0$. It is clear that $F(1,t_{0},0)=0$. Moreover, the Fréchet derivative $F_{w}(q,t,w):X_{2}\rightarrow R(A)$ is given by

[TABLE]

We see that $F_{w}(1,t_{0},0)\varphi=-\Delta\varphi-a\left(x\right)\varphi$. Hence,

[TABLE]

Since $\varphi\in X_{2}$, it follows that $\int_{\Omega}(c\phi_{1})\phi_{1}=0$, and thus $c=0$. This means that $F_{w}(1,t_{0},0)$ is injective. It is also surjective from the fact that $\int_{\Omega}f\phi_{1}=0$ if and only if there exists $\varphi$ such that

[TABLE]

Since $F_{w}(1,t_{0},0)$ is continuous, from the Bounded Inverse Theorem we infer that $F_{w}(1,t_{0},0)$ is an isomorphism. Hence, the implicit function theorem applies, and consequently, we have

[TABLE]

We plug $w(q,t)$ into (2.2) to get the following bifurcation equation in $\mathbb{R}^{2}$:

[TABLE]

We are now in position to prove the following result:

Theorem 2.2**.**

Assume $(H_{0})$. If $\mu_{1}(a)=1$, then the following assertions hold:

1. (i)

Assume that $(q_{n},u_{n})\in(0,1)\times\mathcal{P}^{\circ}$ are solutions of $(P_{a,q_{n}})$ such that $(q_{n},u_{n})\rightarrow(1,t\phi_{1})\in\Gamma_{1}$ in $\mathbb{R}\times W^{2,r}(\Omega)$ for some $t>0$. Then, we have $t=t_{*}$, where $t_{*}$ is given by (1.1). 2. (ii)

The set of positive solutions of $(P_{a,q})$ consists of $\Gamma_{1}\cup\Gamma_{2}$ in a neighborhood of $(q,u)=(1,t_{\ast}\phi_{1})$ in $\mathbb{R}\times W^{2,r}(\Omega)$, where

[TABLE]

Here $t(q)$ and $w(q,t(q))$ are smooth with respect to $q$ and satisfy $t(1)=t_{\ast}$ and $w(1,t_{\ast})=0$.

Proof. Let us first verify assertion (i). Since $(q_{n},u_{n})\rightarrow(1,t\phi_{1})$ in $\mathbb{R}\times W^{2,r}(\Omega)$ for some $t>0$, we have $\Phi_{q}(1,t)=0$ by the implicit function theorem. By direct computations, we get

[TABLE]

Putting $q=1$ and using that $w(1,t)=0$, we find that

[TABLE]

Thus

[TABLE]

as claimed in assertion (i).

Next, we verify assertion (ii). To this end, we use the fact that the map $(q,t)\mapsto N(q,t)=t^{q}$ is analytic around $(q,t)=(1,t_{\ast})$, and apply the implicit function theorem. We consider partial derivatives of $\Phi$, and check that $\frac{\partial^{k}\Phi}{\partial t^{k}}(1,t)=0$ and $\Phi_{qt}(1,t_{\ast})>0$. In fact, the case $k=1$ is straightforward since $\Gamma_{1}$ is a trivial line of solutions of $(P_{a,q})$. Moreover, for $k\geq 2$, we have that

[TABLE]

for some continuous function $\Phi_{k}$ of $(q,t)$ at $(1,t)$, so that $\frac{\partial^{k}\Phi}{\partial t^{k}}(1,t)=0$ for all $k\in\mathbb{N}$ and $t>0$. Since $(q,t)\mapsto t^{q}=\exp[q\log t]$ is analytic at $(q,t)=(1,t)$, for any $t>0$, a regularity result for the implicit function theorem (see e.g. [19]) ensures that so is $w(q,t)$ around $(1,t_{\ast})$, and thus so is $\Phi(q,t)$. Combining this result with the fact that $\frac{\partial^{k}\Phi}{\partial t^{k}}(1,t)=0$ for all $k\in\mathbb{N}$, we deduce that $\Phi(q,t)$ is given around $(1,t_{\ast})$ by

[TABLE]

Therefore, applying the implicit function theorem to $\hat{\Phi}(q,t)$ at $(1,t_{\ast})$, we infer that the set $\Phi(q,t)=0$ around $(1,t_{\ast})$ is given completely by

[TABLE]

provided that $\Phi_{qt}(1,t_{\ast})\not=0$. The desired conclusion follows.

Finally, we check that $\Phi_{qt}(1,t_{\ast})>0$: by a direct computation from (2.3), we observe that

[TABLE]

Letting $q=1$, it follows that

[TABLE]

We differentiate (2.1) with respect to $t$, and we obtain that

[TABLE]

Letting $q=1$ again, we deduce that

[TABLE]

Hence, $w_{t}(1,t)=0$, and thus, it follows from (2.5) that

[TABLE]

When $t=t_{\ast}$, we know that $\int_{\Omega}a\left(x\right)\phi^{2}\log(t_{\ast}\phi_{1})=0$ from (2.4), so that

[TABLE]

as desired. $\blacksquare$

Next, as we did for $q<1$ close to $1$, we show that the Lyapunov-Schmidt reduction is useful for the case $q>0$ close to [math]. Indeed, we exhibit how to construct $a$ such that $(P_{a,q})$ possesses a solution in $\mathcal{P}^{\circ}$ for $q>0$ arbitrarily close to [math]. Consider the problem

[TABLE]

It is easy to check that $(P_{a,0})$ has a solution if and only if $\int_{\Omega}a=0$, in which case all solutions are of the form $u+c$, where $c$ is any constant and $u$ is a particular solution.

Assume now that $a\not\equiv 0$ and $\int_{\Omega}a=0$ (in particular, $a$ changes sign). We set $X:=\{u\in W^{2,r}(\Omega):\frac{\partial u}{\partial\nu}=0\mbox{ on }\partial\Omega\}$, and write $X=\langle 1\rangle+X_{2}$, where $\langle 1\rangle$ is the set of constant functions and $X_{2}:=\{w\in W^{2,r}(\Omega):\int_{\Omega}w=0\}$. Let $w_{0}$ be the unique solution of $(P_{a,0})$ such that $w_{0}\in X_{2}$, and $t_{0}>0$ be such that $u_{0}=t_{0}+w_{0}>0$ on $\overline{\Omega}$. Then $u_{0}\in\mathcal{P}^{\circ}$ solves $(P_{a,0})$.

Given $\varepsilon,\delta>0$ and $q\in(-\delta,\delta)$, we consider the following perturbation of $(P_{a,0})$:

[TABLE]

Note that if $\varepsilon$ is sufficiently small, then $a-\varepsilon$ changes sign, and $\int_{\Omega}(a(x)-\varepsilon)<0$. Note also that $(P_{a-\varepsilon,q})$ admits $(q,\varepsilon,u)=(0,0,t_{0}+w_{0})$ as a solution. Our aim is to look for positive solutions of $(P_{a-\varepsilon,q})$ in a neighborhood of $(0,0,t_{0}+w_{0})$. Let $Y_{2}:=\{f\in L^{r}(\Omega):\int_{\Omega}f=0\}$ and $Q$ be the usual projection of $L^{r}(\Omega)$ to $Y_{2}$, given by $Q[f]:=f-\frac{1}{|\Omega|}\int_{\Omega}f$. Following the Lyapunov-Schmidt approach already used in Theorem 2.2, we reduce $(P_{a-\varepsilon,q})$ to the following coupled equations

[TABLE]

Associated with (2.6), we define the mapping

[TABLE]

by

[TABLE]

where $B_{\rho_{0}}(w_{0})$ is the ball in $X_{2}$ with center $w_{0}$ and radius $\rho_{0}$. We note that $F(0,0,t_{0},w_{0})=0$, since $Q[a]=a$. The Fréchet derivative $F_{w}(q,\varepsilon,t,w):X_{2}\rightarrow Y_{2}$ is given by

[TABLE]

Taking $(q,\varepsilon,t,w)=(0,0,t_{0},w_{0})$, we see that $F_{w}(0,0,t_{0},w_{0})\varphi=-\Delta\varphi$, so that $F_{w}(0,0,t_{0},w_{0})$ is bijective, and the implicit function theorem applies. Consequently, we have

[TABLE]

Using $w(q,\varepsilon,t)$, we derive from (2.7) the equation

[TABLE]

Note that $\Psi(0,0,t_{0})=0$. We prove now the following result:

Proposition 2.3**.**

*Given $\varepsilon>0$ sufficiently small, there exists $q_{\varepsilon}>0$ such that $u_{\varepsilon}=t_{0}+w(q_{\varepsilon},\varepsilon,t_{0})>0$ on $\overline{\Omega}$, and $\Psi(q_{\varepsilon},\varepsilon,t_{0})=0$. Moreover, $q_{\varepsilon}\rightarrow 0^{+}$ as $\varepsilon\rightarrow 0^{+}$. Consequently, $u_{\varepsilon}\in\mathcal{P}^{\circ}$ is a solution of $(P_{a-\varepsilon,q_{\varepsilon}})$.

Proof. We apply the implicit function theorem for $\Psi$ at $(0,0,t_{0})$. We observe that

[TABLE]

and therefore

[TABLE]

Since $u_{0}$ is a solution in $\mathcal{P}^{\circ}$ of $(P_{a,0})$, we see that

[TABLE]

which implies that $\frac{\partial\Psi}{\partial q}(0,0,t_{0})>0$. Hence, the implicit function theorem ensures that

[TABLE]

Next we show that $\frac{\partial q}{\partial\varepsilon}(0,t_{0})>0$. To this end, we differentiate $\Psi$ with respect to $\varepsilon$, obtaining that

[TABLE]

and so

[TABLE]

It follows that

[TABLE]

Using the mean value theorem, we deduce that for $\varepsilon>0$ small enough,

[TABLE]

for some $0<\theta<1$. Thus

[TABLE]

as desired. $\blacksquare$

Remark 2.4**.**

Let us analyze the asymptotic behavior of nontrivial nonnegative solutions of $(P_{a,q})$ as $q\rightarrow 0^{+}$ under $(H_{0})$ and $(H_{1})$. From Proposition 2.1, we know that bifurcation from zero or from infinity does not occur as $q\rightarrow 0^{+}$. It is thus natural to investigate the limit of a sequence $\{u_{n}\}$ of nontrivial nonnegative solutions of $(P_{a,q_{n}})$ with $q_{n}\rightarrow 0^{+}$. Since $\{u_{n}\}$ is bounded in $L^{\infty}(\Omega)$, it follows, by elliptic regularity, that up to a subsequence, $u_{n}\rightarrow u_{0}$ in $\mathcal{C}^{1}(\overline{\Omega})$ with $u_{0}\not\equiv 0$. We point out that $u_{0}$ must vanish in a nonempty subset of $\Omega$ with positive measure (in other words, $u_{0}$ has a nonempty dead core). Indeed, if $u_{0}>0$ a.e. in $\Omega$, then, passing to the limit, we have that

[TABLE]

i.e. $u_{0}$ is a positive solution of $(P_{a,0})$. Integrating this equation, we deduce that $\int_{\Omega}a=0$, which is a contradiction.

2.1 Stability properties

We conclude this section discussing the stability of the bifurcating positive solutions provided by Theorem 2.2 (ii).

Proposition 2.5**.**

Assume $(H_{0})$. If $\mu_{1}(a)=1$, then the bifurcating positive solution $u(q)=t(q)\phi_{1}+w(q,t(q))$ given by Theorem 2.2 (ii) is asymptotically stable (respect. unstable) for $q<1$ (respect. $q>1$).

Proof. Consider

[TABLE]

where $\gamma_{1}(q):=\gamma_{1}(q,u(q))$, and $\varphi_{1}(q)$ is a positive eigenfunction associated to $\gamma_{1}(q)$. We see that $\gamma_{1}(1)=0$ and $\varphi_{1}(1)=\phi_{1}$. To analyse $\gamma_{1}(q)$ for $q\neq 1$, we differentiate (2.8) with respect to $q$, to obtain that

[TABLE]

Letting $q=1$ here, it follows that

[TABLE]

and thus, by the divergence theorem,

[TABLE]

Since $\int_{\Omega}a(x)\phi_{1}^{2}\log(t_{\ast}\phi_{1})=0$, we obtain that

[TABLE]

The desired conclusion follows from the fact that $\gamma_{1}(1)=0$. $\blacksquare$

Remark 2.6**.**

1. (i)

The stability result of Proposition 2.5 also follows from [5, Theorem 1]. Even though this result assumes $a$ to be smooth and the nonlinearity to be $C^{2}$ at [math], one may easily see that under our assumptions it also applies to solutions of $(P_{a,q})$ in $\mathcal{P}^{\circ}$. More generally, it shows that any such solution is asymptotically stable for every $0<q<1$. 2. (ii)

When $q>1$, we can deduce (by a well known approach) that every solution $u\in\mathcal{P}^{\circ}$ of $(P_{a,q})$ is unstable. Indeed, linearizing $(P_{a,q})$ at $u$ we obtain $-\Delta\varphi=qa(x)u^{q-1}\varphi+\gamma\varphi$. The divergence theorem yields that

[TABLE]

where $\gamma_{1}=\gamma_{1}(q)$ and $\varphi_{1}=\varphi_{1}(q)$. Hence, we obtain

[TABLE]

3 Proofs of Theorems 1.1, 1.3 and 1.5

Proof of Theorem 1.1: Let us first observe that by Theorem 2.2, there exists $q_{0}=q_{0}(a)<1$ such that $(P_{a,q})$ has a solution $u_{q}\in\mathcal{P}^{\circ}$ for $q_{0}<q<1$. Moreover, the proof of [4, Lemma 3.1] can be adapted to our setting, so that $(P_{a,q})$ has no other positive solution for $q_{0}<q<1$. We consider now the asymptotic behavior of $u_{q}$ as $q\rightarrow 1^{-}$. Assertion (i) is a direct consequence of Theorem 2.2 (ii) and elliptic regularity.

Assume now that $\mu_{1}=\mu_{1}(a)\neq 1$ and set $v:=\mu_{1}^{\frac{1}{1-q}}u$. Note that if $u$ solves $(P_{a,q})$ then $v$ solves $(P_{\tilde{a},q})$, where $\tilde{a}:=\mu_{1}a$. Indeed,

[TABLE]

Moreover, we easily see that $\mu_{1}(\tilde{a})=1$. By item (i), we get a positive solution $v_{q}$ of $(P_{\tilde{a},q})$ such that $v_{q}\rightarrow t_{\ast}(\tilde{a})\phi_{1}(\tilde{a})$, where $\phi_{1}(\tilde{a})$ is a positive eigenfunction of $(E_{1,\tilde{a}})$, which is nothing but $(E_{\mu_{1},a})$, i.e. $\phi_{1}(\tilde{a})=\phi_{1}(a)$ and $t_{\ast}(\tilde{a})=t_{\ast}(a)$ . In this way, we obtain a positive solution $u_{q}=\mu_{1}^{\frac{1}{q-1}}v_{q}$ of $(P_{a,q})$ for $q$ close to $1$. In particular, we see that if $\mu_{1}>1$ then $\mu_{1}^{\frac{1}{q-1}}\rightarrow 0$, so that $u_{q}\rightarrow 0$ in $\mathcal{C}^{1}(\overline{\Omega})$ as $q\rightarrow 1^{-}$. On the other hand, if $\mu_{1}<1$, then $\mu_{1}^{\frac{1}{q-1}}\rightarrow\infty$, so that $\displaystyle\min_{\overline{\Omega}}u_{q}\rightarrow\infty$ when $q\rightarrow 1^{-}$.

Finally, the asymptotic stability of $u_{q}$ is a direct consequence of Proposition 2.5. $\blacksquare$

When proving Theorems 1.3, 1.5 and 1.8, we shall repeatedly use the following remark:

Remark 3.1**.**

1. (i)

Since $(P_{a,q})$ is homogeneous, we see that $(P_{a,q})$ has a nonnegative (respect. positive) solution if and only if, for any $\sigma>0$ fixed, $(P_{\sigma a,q})$ has a nonnegative (respect. positive) solution. 2. (ii)

Lemma 2.4 in [4] (which is proved using Proposition 2.1 therein) gives the existence of arbitrarily large supersolutions of $(P_{a,q})$ provided that $\int_{\Omega}a<0$. Although it is assumed that $a$ is Hölder continuous in [4], one can see that Lemma 2.4 and Proposition 2.1 still hold (with the same proof) if $a\in L^{\infty}(\Omega)$.

Proof of Theorem 1.3: We proceed in several steps. By Remark 3.1, it is enough to provide a positive (in $\Omega$) weak subsolution for $(P_{b,q})$, where $b:=\gamma a$ and $\gamma:=1/\left(1-q\right)$. Observe that $\gamma q=\gamma-1$. We note also that, since $\int_{B_{R}}a<0$, it holds that $R_{0}<R$. Let us first define

[TABLE]

Then, $w\left(R\right)=w^{\prime}\left(R\right)=0$ and $w\left(r\right)>0$ for all $r\in\left[R_{0},R\right)$. Also, a few computations show that

[TABLE]

Let now $z\left(r\right):=w^{\gamma}\left(r\right)$. We claim that

[TABLE]

Indeed, since $z$ is radial, there holds

[TABLE]

and also

[TABLE]

Thus, in order to prove the claim it is enough to verify that

[TABLE]

Now, taking into account (3.1) and that $w=C\phi$, the above inequality is equivalent to

[TABLE]

We observe next that $F\left(R\right)=G\left(R\right)=0$ and $F^{\prime}\left(r\right)\leq 0$ for all $r\in\left[R_{0},R\right]$ (recall that $\phi^{\prime}\leq 0$). So, in order to check (3.3) it suffices to see that $F^{\prime}\left(r\right)\leq G^{\prime}\left(r\right)$ for such $r$. Now,

[TABLE]

where we used the fact that $a$ is differentiable and nonincreasing in $A_{R_{0},R}$. Therefore, $F^{\prime}\left(r\right)\leq G^{\prime}\left(r\right)$ provided that

[TABLE]

i.e.

[TABLE]

But (3.4) holds by our election of $C$. Indeed, since $\phi^{\prime}\leq 0$, one only has to observe that $2\left(\gamma-1\right)=\frac{1}{C}-1.$

On the other side, let $v$ be a solution of

[TABLE]

Such $v$ can be easily constructed by the sub and supersolutions method, since $a\geq 0$ in $B_{R_{0}}$. Moreover, $v$ is radial. Indeed, this follows from either the fact that the sub and supersolutions can be chosen radial, or because the solution of (3.5) is unique (cf. [7]) and $v\left(Sx\right)$ is also a solution if $S$ is an isometry of $\mathbb{R}^{N}$. Furthermore, it is also easy to check that $r\rightarrow v(r)$ is nonincreasing in $(0,R_{0})$ because $a\geq 0$ in $B_{R_{0}}$. Hence, by the divergence theorem (as stated e.g. in [6], p. 742),

[TABLE]

On the other hand, recalling that $\gamma-1=\gamma q$, we obtain that

[TABLE]

and so

[TABLE]

Next we observe that $v^{\prime}\left(R_{0}\right)\leq z^{\prime}\left(R_{0}\right)$. Indeed, taking into account (3.6), (3.7) and the definition of $C$, we see that this is true by (1.2).

To conclude the existence assertion, we define $u:=z$ in $\overline{A}_{R_{0},R}$ and $u:=v$ in $B_{R_{0}}$. Then $u\in H_{0}^{1}\left(\Omega\right)\cap L^{\infty}\left(\Omega\right)$, $u>0$ in $\Omega$ and $\frac{\partial u}{\partial\nu}=0$ on $\partial\Omega$. Moreover, recalling (3.2), (3.5) and that $v^{\prime}\left(R_{0}\right)\leq z^{\prime}\left(R_{0}\right)$, the divergence theorem yields that $u$ is a weak subsolution of $(P_{b,q})$.

Finally, the uniqueness assertion is a consequence of [4, Theorem 3.1] (see Remark 1.7). $\blacksquare$

Proof of Theorem 1.5: Given $\varepsilon\in\left[0,q\right)$, we define

[TABLE]

We note that both $\varepsilon\mapsto\gamma_{\varepsilon}$ and $\varepsilon\mapsto C_{\varepsilon}$ are continuous, that

[TABLE]

and that

[TABLE]

Given $r\in\left[0,R_{0}\right]$ and $\delta\geq 0$, we set $u_{\delta,\varepsilon}\left(r\right):=C_{\varepsilon}r^{2}+\delta$. We now observe that we can fix $\varepsilon>0$ small enough such that

[TABLE]

Indeed, $u_{0,\varepsilon}^{\prime}\left(R_{0}\right)=2C_{\varepsilon}R_{0}$ and $u_{0,\varepsilon}^{\varepsilon}\left(R_{0}\right)=\left(C_{\varepsilon}R_{0}^{2}\right)^{\varepsilon}$, and so (3.10) holds if and only if

[TABLE]

Now, by (1.4) and (3.9),

[TABLE]

Thus, (3.11) (and consequently (3.10)) holds for $\varepsilon>0$ sufficiently small.

Next, we note that, by definition,

[TABLE]

Therefore, for all $r\in\left[0,R_{0}\right]$,

[TABLE]

In view of this inequality, we may fix $\delta>0$ such that

[TABLE]

Indeed, we pick first any $\delta_{0}>0$ small enough such that, for all $\delta\in\left(0,\delta_{0}\right]$,

[TABLE]

Then there exists $r_{0}=r_{0}\left(\delta_{0}\right)>0$ such that

[TABLE]

and thus (3.13) clearly holds for all $r\in\left[0,r_{0}\right]$. Suppose now that $r\in\left[r_{0},R_{0}\right]$. Then, from (3.12) we derive that

[TABLE]

Now, since by Dini’s theorem the left-hand side of (3.13) converges to the left-hand side of (3.14) uniformly in $r\in\left[r_{0},R_{0}\right]$ as $\delta\rightarrow 0^{+}$, then, decreasing $\delta_{0}$ if necessary, we also see that (3.13) holds for all $r\in\left[r_{0},R_{0}\right]$.

Finally, since $u_{0,\varepsilon}^{\prime}=u_{\delta,\varepsilon}^{\prime}$ and $u_{0,\varepsilon}<u_{\delta,\varepsilon}$, recalling (3.10), we get that

[TABLE]

We fix for the rest of the proof $\varepsilon,\delta>0$ such that (3.13) and (3.15) hold.

Let $z_{\delta,\varepsilon}\left(r\right):=u_{\delta,\varepsilon}^{\gamma_{\varepsilon}}\left(r\right)$. Let us show that

[TABLE]

Note that (3.16) implies that $-\Delta z_{\delta,\varepsilon}\leq\gamma_{\varepsilon}a\left(x\right)z_{\delta,\varepsilon}^{q}$ $a.e.$ in $B_{R_{0}}$. We compute

[TABLE]

Thus, in order to prove (3.16) it is enough to see that

[TABLE]

Furthermore, since $\varepsilon=\gamma_{\varepsilon}q-\left(\gamma_{\varepsilon}-1\right)$ (recall (3.8)), this is equivalent to

[TABLE]

But taking into account the definition of $u_{\delta,\varepsilon}$, we see that the above inequality holds thanks to (3.13).

On the other side, let us define

[TABLE]

Note that $v\left(R_{0}\right)=z_{\delta,\varepsilon}\left(R_{0}\right)$, and observe also that

[TABLE]

and hence $w^{\prime}\left(R\right)=0$ (and $v^{\prime}\left(R\right)=0$). We also infer that $w\left(r\right)>0$ for $r\in\left[R_{0},R\right]$ since $w\left(R_{0}\right)>0$ and $w$ is increasing. Moreover,

[TABLE]

We prove now that

[TABLE]

Indeed, since $v$ is radial, (3.18) is equivalent to

[TABLE]

and the above inequality clearly holds by (3.17).

We next verify that

[TABLE]

We have that

[TABLE]

and so it suffices to check that

[TABLE]

We observe now that, by definition, $w^{\gamma_{0}}\left(R_{0}\right)=u_{\delta,\varepsilon}^{\gamma_{\varepsilon}}\left(R_{0}\right)$, and hence

[TABLE]

Therefore, (3.20) can be written as

[TABLE]

Now, $\gamma_{0}-1=q/\left(1-q\right)$ and so, recalling the first equality in (3.8), we see that

[TABLE]

Thus, we have to verify that

[TABLE]

But

[TABLE]

and so (3.21) follows immediately from (3.15).

Taking into account (3.16), (3.18) and (3.19), the proof can now be ended as the proof of Theorem 1.3. $\blacksquare$

Remark 3.2**.**

Note that if we take $\delta=0$ in the proof of Theorem 1.5, then the subsolution vanishes at the origin. This is why we have to choose $\varepsilon>0$ and we cannot pick $\varepsilon=0$.

Remark 3.3**.**

Although Theorems 1.3 and 1.5 hold in particular for $N=1$, in this case one can obtain similar results without assuming that $a$ is even. More precisely, if $\Omega:=\left(\alpha,\beta\right)$ and $\mu\in\Omega$, a quick look at the proofs of the aforementioned theorems shows that one can replace $B_{R_{0}}$ and $A_{R_{0},R}$ by $\left(\alpha,\mu\right)$ and $\left(\mu,\beta\right)$ respectively, in order to reach a similar conclusion.

4 Proof of Theorem 1.8 and some corollaries

Proof of Theorem 1.8 (i): First we note, as a consequence of Theorem 1.1, that $\mathcal{I}\neq\emptyset$ since $(H_{0})$ holds.

1. (i1)

Let $q_{0}\in\mathcal{I}$ and $u_{0}\in\mathcal{P}^{\circ}$ be a corresponding solution of $(P_{a,q_{0}})$. Given $q\in(q_{0},1)$, define

[TABLE]

Then, a brief computation yields that

[TABLE]

and $\frac{\partial w}{\partial\nu}=0$ on $\partial\Omega$. In other words, $w$ is a subsolution of $(P_{a,q})$ belonging to $\mathcal{P}^{\circ}$. So, recalling Remark 3.1 (ii), we obtain a solution $u\in\mathcal{P}^{\circ}$ of $(P_{a,q})$, and thus $q\in\mathcal{I}$. Therefore, defining $q_{i}:=\inf$ $\mathcal{I}$, the first assertion of (i) follows. 2. (i2)

Since $(H_{0})$ holds, one can see by a variational approach that $(P_{a,q})$ has a nontrivial nonnegative solution for any $0<q<1$ (see e.g. the proof of [16, Corollary 1.8]), and thus $\mathcal{A}\subseteq\mathcal{I}$. Assume now $(H_{1}^{\prime})$ and $(H_{+})$. Let $q\in\left(0,1\right)$, and suppose by contradiction that there exist $u$ and $v$ nontrivial nonnegative solutions of $(P_{a,q})$ with $u\not\equiv v$. We claim that $u\not\equiv 0$ in $\Omega_{+}$. Indeed, if not, then $\Delta u\geq 0$ in $\Omega$ and $\frac{\partial u}{\partial\nu}=0$ on $\partial\Omega$, and therefore the maximum principle says that $u\equiv 0$ in $\Omega$, which is not possible. Now, taking into account that $\Omega_{+}$ is connected (by $(H_{1}^{\prime})$), arguing as in Lemma 2.2 in [4] we infer that $u>0$ in $\Omega_{+}$. But, since the same reasoning applies to $v$, this contradicts the uniqueness result of [4, Theorem 3.1] (see Remark 1.7). Finally, recalling that $\mathcal{A}\subseteq\mathcal{I}$, we deduce that $\mathcal{A}=\mathcal{I}$. $\blacksquare$

Proof of Theorem 1.8 (ii): After a dilation and a translation, we can assume that $\Omega:=\left(-2,2\right)$. For any $q\in(0,1)$, we shall construct $a\in\mathcal{C}(\overline{\Omega})$ such that $(P_{a,q})$ has one solution in $\mathcal{P}^{\circ}$ and two nontrivial nonnegative solutions having nonempty dead cores. This result will be proved in two parts, in accordance with the value of $q$.

1. (i)

First we consider $q\in\left[\frac{1}{3},1\right)$. We define

[TABLE]

Note that $rq=r-2$. Let $p$ be the polynomial given by

[TABLE]

where

[TABLE]

One can verify that

[TABLE]

Moreover, it also holds that $p$ is increasing in $\left(1,2\right)$, and in particular it follows that $p>0$ in $\left[1,2\right]$. Set

[TABLE]

and observe that $a\in\mathcal{C}(\overline{\Omega})$ since (recall that $rq=r-2$)

[TABLE]

Also, since $p>0$ in $\left[1,2\right]$, it follows from the definition that $a$ changes sign in $\left(1,2\right)$. Furthermore,

[TABLE]

and hence

[TABLE]

since $r\geq 3$. Therefore, $\int_{\Omega}a<0$. Define now

[TABLE]

and $u_{2}\left(x\right):=u_{1}\left(-x\right)$. Taking into account (4.1), we see that $u_{1},u_{2}\in\mathcal{C}^{2}(\overline{\Omega})$. Moreover, one can see that $u_{1}$ and $u_{2}$ are two distinct nonnegative nontrivial solutions of the problem

[TABLE]

Now, a simple integration by parts shows that

[TABLE]

is a strictly positive weak subsolution of (4.2). Thus, since $\int_{\Omega}a<0$, by Remark 3.1 (ii) there exist arbitrary large supersolutions of (4.2) and we then obtain a solution $u\in\mathcal{P}^{\circ}$ of (4.2). It follows that $q\in\mathcal{I}$, but $q\not\in\mathcal{A}$, since $u_{1}$ and $u_{2}$ are nontrivial nonnegative solutions having nonempty dead cores. 2. (ii)

Now we consider $q\in\left(0,\frac{1}{3}\right)$. We proceed as above, the only difference being the definition of $p$. For $K>0$, let

[TABLE]

where

[TABLE]

One can check that

[TABLE]

We observe that $p>0$ in $\left[1,2\right]$. Indeed, since

[TABLE]

if $p\leq 0$ somewhere, then $p^{\prime\prime}$ would vanish at least at three points in $\left(1,2\right)$, which is impossible since $p^{\prime\prime}$ has degree $2$. It follows that

[TABLE]

We claim now that for all $K>0$ large enough, $a_{k}$ changes sign in $\left(1,2\right)$ and $\int_{1}^{2}a_{K}<0$. Indeed, $a_{K}(1)<0$, and for $K$ sufficiently large we have $p_{K}^{\prime\prime}(2)<0$, so that $a_{K}(2)>0$. Hence, the first assertion follows. To show the second one, we first note that

[TABLE]

[TABLE]

and

[TABLE]

Define

[TABLE]

Then, $H^{\prime}=h$. Also, since $2-3q>0$ we see that $\displaystyle\lim_{x\rightarrow 1^{+}}H(x)g(x)^{-q}=0$. Therefore, an integration by parts yields that

[TABLE]

because $g^{\prime}>0$ in $(1,2)$. It follows from (4.4) and (4.5) that

[TABLE]

and therefore the claim is proved. We can then fix some $K>0$ such that $a_{K}$ changes sign in $\Omega$ and $\int_{\Omega}a_{K}<0$, and thus the proof can be completed as in the previous case. $\blacksquare$

Remark 4.1**.**

Let us point out that, by the uniqueness results in [4], for every $q\in\left(0,1\right)$, the problem $(P_{a,q})$ with the weight $a_{q}$ constructed in the above proof has exactly three (nontrivial) nonnegative solutions. Indeed, one can verify that $\Omega_{+}(a_{q})$ has exactly two connected components (taking $K$ large if $q<1/3$), say $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. Now, by [4, Theorem 3.1] there exists at most one nonnegative solution which is positive in $\mathcal{O}_{1}$ and zero in $\mathcal{O}_{2}$, and vice-versa. Also, by the aforementioned theorem, there exists at most one nonnegative solution which is positive in both $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. Since the nontrivial nonnegative solutions $u$ satisfy that either $u>0$ in $\mathcal{O}_{i}$ or $u\equiv 0$ in $\mathcal{O}_{i}$ (see [4, Lemma 2.2]), our assertion follows because from the maximum principle we deduce that there is no nontrivial nonnegative solution vanishing in both $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$.

Let us also remark that the solution in $\mathcal{P}^{\circ}$ is even: indeed, if not, we would have four nontrivial nonnegative solutions. Summing up, for this family of even weights $a_{q}$, there exist two (nontrivial) noneven nonnegative solutions with nonempty dead cores, and one even solution in $\mathcal{P}^{\circ}$.

Remark 4.2**.**

Let $a_{q}$ be as in the first case of the proof of Theorem 1.8 (ii), but now with $q\in\left[0,1\right)$. A quick look at the aforementioned proof shows that $\int_{\Omega}a_{q}>0$ for $q>0$ close enough to [math]. Indeed, this follows easily from the fact that $\int_{\Omega}a_{0}=1$. Furthermore, for such $q$’s, reasoning as therein we obtain two (nontrivial) nonnegative solutions of $(P_{a_{q},q})$. In other words, this result shows that, unlike for the existence of positive solutions, the condition $\int_{\Omega}a<0$ is not necessary in order to have existence of (nontrivial) nonnegative solutions of $(P_{a,q})$. Let us add that this matter has already been noted in [4, Section 2.3].

As an immediate consequence of Theorems 1.5 and 1.8 (i), we have the following result:

Corollary 4.3**.**

Let $\Omega:=B_{R}$ and $a\in L^{\infty}\left(\Omega\right)$ be a radial function such that $\int_{\Omega}a<0$ and $a\geq 0$ in $A_{R_{0},R}$ for some $R_{0}\in\left(0,R\right)$. Then,

[TABLE]

Moreover,

[TABLE]

if $a\leq 0$ in $B_{R_{0}}$.

Proof. Since $\int_{\Omega}a<0$, a direct computation gives that $KN<1$. Let $q\in\left(\frac{1-KN}{1-KN+2K},1\right)$. Then one can check that (1.4) is satisfied and thus there exists $u\in\mathcal{P}^{\circ}$ solution of $(P_{a,q})$, so that $q\in\mathcal{I}$. The last assertion of the corollary is now immediate from Theorem 1.8 (i2). $\blacksquare$

Remark 4.4**.**

Let us point out that $\mathcal{I}(a)$ may approach the whole interval $(0,1)$ as the coefficient $a$ varies. To show this, we may use either the sub and supersolutions method (Corollary 4.3), or a bifurcation analysis (Proposition 2.3). Let us also add that, however, we believe that there is no $a$ such that $\mathcal{I}\left(a\right)=\left(0,1\right)$, but we are not able to prove it.

1. (i)

Given any fixed $q_{0}\in\left(0,1\right)$, Corollary 4.3 provides some cases in which $\left(q_{0},1\right)\subseteq\mathcal{I}=\mathcal{A}$. Indeed, in order to see this it suffices to find $a$ such that $K(a)$ satisfies $1>K(a)N\approx 1$. One may take for instance $\Omega:=B_{1}$ and

[TABLE]

where $\sigma>0$. Since $K(a)N=\sigma\left(2^{N}-1\right)$, it is easy to choose $\sigma$ adequately. 2. (ii)

Let $a\in L^{\infty}(\Omega)$ be given by $a(x):=\sigma\chi_{\Omega_{1}}(x)-\chi_{\Omega_{2}}(x)a_{2}(x)$, where $\sigma>0$, $a_{2}\geq 0$, and $\Omega_{1},\Omega_{2}$ are disjoint subsets of $\Omega$ such that $\int_{\Omega}a=0$. Then, for any $\varepsilon>0$ small, we see that $a-\varepsilon$ changes sign, $\int_{\Omega}(a-\varepsilon)<0$, and $\Omega_{+}(a-\varepsilon)=\Omega_{+}(a)$. By combining Theorem 1.8 (i) and Proposition 2.3, we see that $\mathcal{I}_{a-\varepsilon}$ approaches $(0,1)$ as $\varepsilon\rightarrow 0^{+}$. Additionally, if $\Omega_{+}(a)$ satisfies $(H_{1}^{\prime})$ and $(H_{+})$, then $\mathcal{A}_{a-\varepsilon}=\mathcal{I}_{a-\varepsilon}$ approaches $(0,1)$ as $\varepsilon\rightarrow 0^{+}$.

5 Proof of Theorem 1.10

Proof of Theorem 1.10 (i): Let $\Omega_{0}$ be a tubular neighborhood of $\partial\Omega$ such that $a>0$ $a.e.$ in $\Omega_{0}$, with smooth boundary $\partial\Omega_{0}=\partial\Omega\cup\Gamma_{0}$, where $\Gamma_{0}=\partial\Omega_{0}\cap\Omega$. We consider the following concave mixed problem

[TABLE]

Proceeding as in [2, Lemma 3.3], we see that the comparison principle holds for (5.1), i.e., $\underline{v}\leq\overline{v}$ on $\overline{\Omega_{0}}$ for any nonnegative supersolution $\overline{v}$ and subsolution $\underline{v}$ of (5.1) such that $\overline{v},\underline{v}>0$ in $\Omega_{0}$.

Let $u$ be a nontrivial nonnegative solution of $(P_{a,q})$. Then $u$ is a supersolution of (5.1). In addition, $u>0$ in $\Omega_{0}$. Indeed, recalling $(H_{1}^{\prime})$, we observe that

[TABLE]

It follows that $u\not\equiv 0$ in $\Omega_{+}$. Also, since $\Omega_{+}$ is connected (by $(H_{1}^{\prime})$), the strong maximum principle yields that $u>0$ in $\Omega_{+}$, and consequently $u>0$ in $\Omega_{0}$ as claimed.

On the other hand, in order to construct a subsolution, we consider the following mixed eigenvalue problem:

[TABLE]

By $\sigma_{1}>0$, we denote the smallest eigenvalue of (5.2), and by $\psi_{1}$ an eigenfunction associated to $\sigma_{1}$ satisfying $\psi_{1}>0$ on $\overline{\Omega_{0}}\setminus\Gamma_{0}$. Then, we see that $\varepsilon_{q}\psi_{1}$ is a subsolution of (5.2) for some $\varepsilon_{q}>0$ small. By the comparison principle, we deduce that $\varepsilon_{q}\psi_{1}\leq u$ on $\overline{\Omega_{0}}$, from which the desired conclusion follows. $\blacksquare$

The following result will be used in the proof of Theorem 1.10 (ii):

Lemma 5.1**.**

*Under the conditions of Theorem 1.10 (ii), let $\overline{q}\in(0,1)$ and $\delta_{2}>0$ such that $\int_{\Omega}a_{\delta_{2}}<0$. Then, there exists $C>1$ such that $\|u\|_{L^{\infty}(\Omega)}<C$ for all nonnegative solutions $u$ of $(P_{a_{\delta},q})$ with $q\in(0,\overline{q}]$ and $\delta\geq\delta_{2}$.

**Proof. **Let $u$ be a nonnegative solution of $(P_{a_{\delta},q})$ with $q\in(0,\overline{q}]$ and $\delta\geq\delta_{2}$. Then $u$ is a subsolution of $(P_{a_{\delta},q})$ for $\delta=\delta_{2}$. In view of Remark 3.1 (ii), we can construct a supersolution $w$ of $(P_{a_{\delta_{2}},q})$ such that $u\leq w$. Hence, the sub and supersolutions method ensures the existence of a nonnegative solution $v$ of $(P_{a_{\delta_{2}},q})$ such that $u\leq v\leq w$. By Proposition 2.1 (ii), $(P_{a_{\delta_{2}},q})$ has an a priori bound for nonnegative solutions in $L^{\infty}(\Omega)$, which is uniform in $q\in(0,\overline{q}]$. The lemma now follows. $\blacksquare$

**Proof of Theorem 1.10 (ii): ** We proceed as in the proofs of [9, Theorem 1(iv)] and [8, Theorem 3.1]. Let $\overline{q}\in(0,1)$ and $\delta_{2}>0$ be the constant given by Lemma 5.1, and $u$ be a nontrivial nonnegative solution of $(P_{a_{\delta},q})$ with $q\in(0,\overline{q}]$ and $\delta\geq\delta_{2}$. Given $\sigma>0$, we pick $a_{0}>0$ such that

[TABLE]

where we have used the continuity of $b_{2}$. Let us fix $x\in G_{\sigma/2}$, and consider $d(x):=\text{dist}(x,\partial G_{\sigma/2})$, where $d(x)>0$ since $G_{\sigma/2}$ is open. We then define

[TABLE]

Let $\alpha:=2/(1-q)$, so that $2-\alpha+\alpha q=0$. If $\left|y\right|<1$ then, using (5.3), a brief computation yields

[TABLE]

Here, we have used the fact that $x+d(x)y\in G_{\sigma/2}$. If $\left|y\right|=1$, then we have

[TABLE]

where $C>1$ is provided by Lemma 5.1.

Given $\varepsilon>0$, we now consider the problem

[TABLE]

We observe from (5.5) that $v_{1}$ is a subsolution of $(Q_{\delta,\varepsilon})$ if

[TABLE]

Next, we construct a supersolution of $(Q_{\delta,\varepsilon})$. For $r=\left|y\right|$, we define

[TABLE]

where $A$ is a positive constant to be determined. Since $\alpha>2$, we have $z_{1}\in\mathcal{C}^{2}(\overline{B_{1}})$, and in addition,

[TABLE]

if $A\alpha(\alpha-1)+(N-1)A\alpha\leq\delta a_{0}A^{q}$, i.e.,

[TABLE]

Moreover, we note that

[TABLE]

provided that

[TABLE]

Hence, from (5.6), (5.8) and (5.9), it follows that if

[TABLE]

i.e.,

[TABLE]

then $v_{1}$ is a subsolution of $(Q_{\delta,\varepsilon})$, and in addition, $z_{1}$ is a supersolution of $(Q_{\delta,\varepsilon})$. Since $2<\alpha\leq 2/(1-\overline{q})=:\overline{\alpha}$, this occurs for some $\varepsilon=\varepsilon_{\delta,x}$ if

[TABLE]

Now, using the comparison principle for $(Q_{\delta,\varepsilon})$ (which is deduced from the weak maximum principle) we derive that $v_{1}\leq z_{1}$, so that

[TABLE]

and consequently, $u(x)=0$. Therefore, we have proved that if $x\in G_{\sigma/2}$ satisfies (5.10), then $u(x)=0$ for any nontrivial nonnegative solution $u$ of $(P_{a_{\delta},q})$ with $q\in(0,\overline{q}]$. Since $d_{\delta}$ in (5.10) does not depend on $u$ or $q$, and converges to [math] as $\delta\rightarrow\infty$, we have the desired conclusion. $\blacksquare$

Acknowledgements

U. Kaufmann was partially supported by Secyt-UNC. H. Ramos Quoirin was supported by Fondecyt 1161635. K. Umezu was supported by JSPS KAKENHI Grant Number 15K04945.