A supercritical scalar field equation with a forcing term

TL;DR

This paper classifies the solvability of a supercritical scalar field elliptic equation with a forcing term in Euclidean space, depending on the parameters and measure conditions, extending known results up to the Joseph-Lundgren exponent.

Contribution

It provides a complete classification of solutions for the scalar field equation with forcing term for subcritical exponents, including the supercritical range up to the Joseph-Lundgren exponent.

Findings

Solved the elliptic problem for p < p_{JL} under measure conditions.

Established existence and non-existence criteria based on measure integrability.

Extended classification results to supercritical exponents.

Abstract

This paper is concerned with the elliptic problem for a scalar field equation with a forcing term \begin{equation} \tag{P}-\Delta u+u=u^p+ \kappa \mu \quad \mbox{in} \quad{\bf R}^N, \quad u>0 \quad \mbox{in} \quad {\bf R}^N, \quad u(x)\to 0\quad \mbox{as} \quad |x| \to \infty, \end{equation} where $N\ge 2$, $p>1$, $\kappa>0$ and $\mu$ is a Radon measure in ${\bf R}^N$ with a compact support. Under a suitable integrability condition on $\mu$, we give a complete classification of the solvability of problem~(P) with $1<p<p_{JL}$. Here $p_{JL}$ is the Joseph-Lundgren exponent defined by $$ p_{JL} :=\infty\quad\mbox{if}\quad N\le 10, \qquad p_{JL}:=\frac{(N-2)^2-4N+8\sqrt{N-1}}{(N-2)(N-10)}\quad \text{if} \quad N\ge 11. $$

Equations

$$-\Delta u+u=u^p+\kappa \mu \text{mbox}in{\mathbf{R}}^N,u>0\text{mbox}in{\mathbf{R}}^N,u(x)\to 0\text{mbox}as|x|\to \infty ,$$

$$p_{JL}:=\infty \text{mbox}ifN\le 10,p_{JL}:=\frac{(N-2{)}^2-4N+8\sqrt{N-1}}{(N-2)(N-10)}\text{if}N\ge 11.$$

$$\left\{\begin{array}[]{ll}-\Delta u+u=u^{p}+\kappa\mu&\quad\mbox{in}\quad{\bf R}^{N},\vspace{3pt}\\ u>0&\quad\mbox{in}\quad{\bf R}^{N},\vspace{3pt}\\ u(x)\to 0&\quad\mbox{as}\quad|x|\to\infty,\end{array}\right.$$

$$p_S:=\infty \text{mbox}ifN=2,p_S:=\frac{N+2}{N-2}\text{mbox}ifN\ge 3.$$

$$p_{JL}:=\infty \text{mbox}ifN\le 10,p_{JL}:=\frac{(N-2{)}^2-4N+8\sqrt{N-1}}{(N-2)(N-10)}\text{if}N\ge 11.$$

$$\mu \in H^{-1}({\mathbf{R}}^N)\text{mbox}and|x{|}^{N-2}\mu \in L^{\infty }({\mathbf{R}}^N).$$

$$\mu =\sum _{j=1}^n{c}_j{\delta }_{a_j},$$

$$1<p<\infty \text{mbox}ifN=2,1<p<\frac{N}{N-2}\text{mbox}ifN\ge 3.$$

$$\begin{split}&C_{0}({\bf R}^{N}):=\Big{\{}f\in C({\bf R}^{N})\,:\,\lim_{|x|\to\infty}f(x)=0\Big{\}},\\ &L^{r}_{\rm c}({\bf R}^{N}):=\{f\in L^{r}({\bf R}^{N})\,:\,\mbox{$f$ has a compact support in ${\bf R}^{N}$}\},\end{split}$$

$$G(x):=\frac{1}{(2\pi {)}^{N/2}|x{|}^{(N-2)/2}}{K}_{(N-2)/2}(|x|),x\in {\mathbf{R}}^N\setminus \{0\},$$

$$u(x)=[G\ast u^p](x)+\kappa [G\ast \mu ](x)>0\text{for almost all}x\in {\mathbf{R}}^N.$$

$$u(x)\ge [G\ast u^p](x)+\kappa [G\ast \mu ](x)>0\text{for almost all}x\in {\mathbf{R}}^N.$$

$$G*\mu\in L^{q}({\bf R}^{N})\,\,\,\,\mbox{for some $\displaystyle{q>\max\biggr{\{}p,\frac{N(p-1)}{2}\biggr{\}}}$}.$$

$$4\nu (1-\nu )p>1,\frac{p_S+1}{2\nu }>\frac{N}{2}(p-1).$$

$$\begin{split}&G(x)\asymp\left\{\begin{array}[]{ll}|x|^{-(N-2)}&\mbox{if}\quad N\geq 3,\vspace{3pt}\\ -\log|x|&\mbox{if}\quad N=2,\end{array}\right.\quad\mbox{as}\quad|x|\to 0,\\ &G(x)\asymp|x|^{-\frac{N-1}{2}}e^{-|x|}\qquad\qquad\qquad\quad\,\mbox{as}\quad|x|\to\infty.\end{split}$$

$$\Vert G\ast v{\Vert }_{L^r({\mathbf{R}}^N)}\le C_r\Vert v{\Vert }_{L^1({\mathbf{R}}^N)},v\in L^1({\mathbf{R}}^N);$$

$$\Vert G\ast v{\Vert }_{L^{r^{\prime }}({\mathbf{R}}^N)}\le C_r^{\prime }\Vert v{\Vert }_{L^r({\mathbf{R}}^N)},v\in L^r({\mathbf{R}}^N),$$

$$G\ast v\in C_0({\mathbf{R}}^N)\text{mbox}and\Vert G\ast v{\Vert }_{L^{\infty }({\mathbf{R}}^N)}\le C\Vert v{\Vert }_{L^r({\mathbf{R}}^N)}$$

$$G\ast v\in C_0({\mathbf{R}}^N)\cap L^1({\mathbf{R}}^N)\cap H^1({\mathbf{R}}^N).$$

$$g:=G\ast {\chi }_{B(0,1)},$$

$$g\in C^{\infty },g(x)>0\text{mbox}in{\mathbf{R}}^N,g(x)\asymp G(x)\text{mbox}as|x|\to \infty ,\frac{|\nabla g|}{g}\in BC({\mathbf{R}}^N).$$

$$0<[G\ast g^{\sigma }](x)\le Cg(x)\text{mbox}forx\in {\mathbf{R}}^N.$$

$$-\Delta \varphi +\varphi =\lambda a(x)\varphi \text{mbox}in{\mathbf{R}}^N,\varphi \in H^1({\mathbf{R}}^N),$$

$$\lambda_{1}=\inf\left\{\|\psi\|_{H^{1}({\bf R}^{N})}^{2}\,\biggr{/}\,\int_{{\bf R}^{N}}a\psi^{2}\,dx\,:\,\psi\in H^{1}({\bf R}^{N}),\,\,\int_{{\bf R}^{N}}a\psi^{2}\,dx\not=0\right\}.$$

$$-\Delta v+v=a(x)v+f\text{mbox}in{\mathbf{R}}^N,v\in H^1({\mathbf{R}}^N).$$

$$\mu_{0}(x):=[G*\mu](x)>0\quad\mbox{for almost all $x\in{\bf R}^{N}$},\quad\mu_{0}\in C^{\infty}({\bf R}^{N}\setminus B(0,R)).$$

$$\begin{array}[]{ll}U_{0}^{\kappa}:=\kappa\mu_{0},&U^{\kappa}_{j}:=G*(U^{\kappa}_{j-1})^{p}+\kappa\mu_{0},\qquad\,\,\,\,j=1,2,\dots,\vspace{3pt}\\ V_{0}^{\kappa}:=U_{0}^{\kappa},&V^{\kappa}_{j}:=U^{\kappa}_{j}-U^{\kappa}_{j-1},\qquad\qquad\qquad j=1,2,\dots.\end{array}$$

$$\begin{array}{rl}& 0<U_j^{\kappa }(x)\le U_{j+1}^{\kappa }(x),\\ & 0<U_j^{\kappa }(x)\le U_j^{{\kappa }^{\prime }}(x)\text{mbox}if\kappa \le {\kappa }^{\prime },\\ & U_j^{\kappa }(x)\le u(x),\end{array}$$

$$0<V_j^{\kappa }(x)<V_j^{{\kappa }^{\prime }}(x)$$

$$U_0^{\kappa }(x)=\kappa {\int }_{{\mathbf{R}}^N}G(x-y)d\mu (y)\ge \kappa \underset{y\in B(0,R)}{\inf }G(x-y){\int }_{B(0,R)}d\mu \ge C$$

Full text

[TABLE]

where $N\geq 2$, $p>1$, $\kappa>0$ and $\mu$ is a Radon measure in ${\bf R}^{N}$ with a compact support. Under a suitable integrability condition on $\mu$, we give a complete classification of the solvability of problem (P) with $1<p<p_{JL}$. Here $p_{JL}$ is the Joseph-Lundgren exponent defined by

[TABLE]

Addresses:

K. I.: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan.

E-mail: ishige@ms.u-tokyo.ac.jp

S. O.: Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan.

E-mail: okabes@m.tohoku.ac.jp

T. S.: Miyagi University of Education, Aoba, Sendai 980-0845, Japan.

E-mail: tokusi-s@staff.miyakyo-u.ac.jp

2010 AMS Subject Classifications: Primary 35B09, 35J61; Secondary 35B32

Keywords: scalar field equation, supercritical, the Joseph-Lundgren exponent

1 Introduction

This paper is concerned with the solvability of the elliptic problem for a scalar field equation with a forcing term

[TABLE]

where $N\geq 2$, $p>1$, $\kappa>0$ and $\mu$ is a nontrivial (nonnegative) Radon measure in ${\bf R}^{N}$ with a compact support. In particular, we are interested in problem (1.1) in the supercritical case $p>p_{S}$, where

[TABLE]

In general, the existence of the solutions of elliptic problems with supercritical nonlinearity is widely open since it is difficult to find the Sobolev embedding fitting suitably to a weak formulation of the solutions and many direct tools of calculus of variations are not applicable. See [7] and [8], which include a nice survey and recent progresses for supercritical elliptic problems.

In this paper, under a suitable integrability condition on $\mu$, we prove the existence of the critical constant $\kappa^{*}>0$ in the following sense (see Theorem 1.1):

• (a)

Problem (1.1) possesses a solution if $0<\kappa<\kappa^{*}$;

• (b)

Problem (1.1) possesses no solutions if $\kappa>\kappa^{*}$.

Furthermore, we show the following result (see Theorem 1.2), which is the main result of this paper:

• (c)

Let $1<p<p_{JL}$ and $\kappa=\kappa^{*}$. Then problem (1.1) possesses a unique solution.

Here

[TABLE]

The exponent $p_{JL}$ is called the Joseph-Lundgren exponent (see [19]) and $p_{JL}>p_{S}$ for $N\geq 3$. It is a well-known critical exponent appearing in the study of the bifurcation structure of the radially symmetric solutions and the stability of the solutions of Lane-Emden-Fowler equation $\Delta u+u^{p}=0$ (see e.g., [13], [14], [15], [19], [20], [23] and references therein). Throughout the proof of assertion (c) we give a new characterization of the Joseph-Lundgren exponent $p_{JL}$.

We recall some results closely related to this paper. Deng and Li [9, 10] proved assertions (a) and (b) in $H^{1}({\bf R}^{N})$ under the assumption that

[TABLE]

Furthermore, they proved assertion (c) in the case of $1<p\leq p_{S}$ and the following:

• (d)

Assume either $1<p<p_{S}$ or $p=p_{S}$ with $3\leq N\leq 5$. Then problem (1.1) possesses at least two solutions in $H^{1}({\bf R}^{N})$ if $0<\kappa<\kappa^{*}$;

• (e)

Let $p=p_{S}$ and $N\geq 6$. Under a suitable symmetric condition on $\mu$, problem (1.1) possesses a unique solution in $H^{1}({\bf R}^{N})$ for all sufficiently small $\kappa>0$.

On the other hand, the third author of this paper and Naito [21] considered problem (1.1) with

[TABLE]

where $n\in\{1,2,\dots\}$, $c_{j}>0$, $a_{j}\in{\bf R}^{N}$ and $\delta_{a_{j}}$ is the Dirac measure supported at $a_{j}$. They proved assertions (a), (b), (c) and (d) in the case of

[TABLE]

We remark that problem (1.1) with (1.2) possesses no solutions if $p\geq N/(N-2)$. There are many related results on assertions (a)–(e). See e.g., [1], [3]–[6], [9]–[12], [16]–[18], [21], [22], [24], [26]–[29] and references therein. However, unfortunately, they are not applicable to the proof of assertion (c) in the supercritical case even if $\mu\in C({\bf R}^{N})$ and $\mu$ has a compact support.

In this paper we prove assertions (a)–(c) and give a complete classification of the solvability of problem (1.1) in the case of $1<p<p_{JL}$. As far as we know, there are no available results for complete classifications of the solvability of problem (1.1) in the supercritical case. Multiple existence of solutions concerning assertions (d) and (c) will be discussed in a forthcoming paper.

We introduce some notation and formulate a definition of solutions of (1.1). For $x\in{\bf R}^{N}$ and $r>0$, let $B(x,r):=\{y\in{\bf R}^{N}\,:\,|x-y|<r\}$. Define

[TABLE]

where $1\leq r\leq\infty$. We denote by $G$ the fundamental solution of $-\Delta v+v=0$ in ${\bf R}^{N}$, that is

[TABLE]

where $K_{(N-2)/2}$ is the modified Bessel function of order $(N-2)/2$.

Definition 1.1

Let $\mu$ be a Radon measure in ${\bf R}^{N}$, $\kappa>0$ and $1<p\leq q<\infty$.*

1. (i)

We say that $u$ is a $(C_{0}+L^{q}_{\rm c})$-solution of (1.1) if $u\in C_{0}({\bf R}^{N})+L^{q}_{\rm c}({\bf R}^{N})$ and $u$ satisfies

[TABLE] 2. (ii)

We say that $u$ is a $(C_{0}+L^{q}_{\rm c})$-supersolution of (1.1) if $u\in C_{0}({\bf R}^{N})+L^{q}_{\rm c}({\bf R}^{N})$ and $u$ satisfies

[TABLE] 3. (iii)

We say that $u$ is a minimal $(C_{0}+L^{q}_{\rm c})$-solution of (1.1) if $u(x)\leq v(x)$ for almost all $x\in{\bf R}^{N}$ for any $(C_{0}+L^{q}_{\rm c})$-solution $v$ of (1.1).

Now we are ready to state our results of this paper. Theorem 1.1 is concerned with assertions (a) and (b).

Theorem 1.1

Let $\mu$ be a nontrivial Radon measure in ${\bf R}^{N}$ with $\mbox{supp}\,\mu\subset B(0,R)$ for some $R>0$. Let $p>1$ and assume that*

[TABLE]

Then there exists $\kappa^{*}\in(0,\infty)$ with the following properties$\colon$

• (i)

If $1<\kappa<\kappa^{*}$, then problem (1.1) possesses a minimal $(C_{0}+L^{q}_{\rm c})$-solution $u^{\kappa}$. Furthermore, $u^{\kappa}(x)=O(G(x))$ as $|x|\to\infty$;

• (ii)

If $\kappa>\kappa^{*}$, then problem (1.1) possesses no $(C_{0}+L^{q}_{\rm c})$-solutions.

Remark 1.1

We give some comments on assumption (1.5).

(i) Let $\mu\in H^{-1}({\bf R}^{N})$. Then $G*\mu\in H^{1}({\bf R}^{N})$, which together with the Sobolev embedding implies that $\mu$ satisfies (1.5) in the case of $1<p<p_{S}$.

(ii) Let $\mu$ satisfy (1.2). Then condition (1.5) holds under assumption (1.3).*

Due to Remark 1.1, Theorem 1.1 is somewhat new even in the subcritical case (compare with [1, 5, 9, 10, 12, 21]). The proof of Theorem 1.1 is based on the construction of approximate solutions and the supersolution-subsolution method.

In Theorem 1.2 we show the unique solvability of problem (1.1) with $\kappa=\kappa^{*}$ in the case of $1<p<p_{JL}$.

Theorem 1.2

Let $1<p<p_{JL}$ and assume the same conditions as in Theorem 1.1. Then problem (1.1) with $\kappa=\kappa^{*}$ possesses a unique $(C_{0}+L^{q}_{\rm c})$-solution.*

The main ingredient in the proof of the existence of the solution with $\kappa=\kappa^{*}$ is to prove uniform local estimates of $\{w^{\kappa}\}_{0<\kappa<\kappa^{*}}$, where $w^{\kappa}:=u^{\kappa}-U_{j_{*}}^{\kappa}\geq 0$, $u^{\kappa}$ is the minimal solution given in Theorem 1.1 and $U_{j_{*}}^{\kappa}$ is an approximate solution to (1.1) (see (3.2) and (3.7)). Here $w^{\kappa}\in H^{1}({\bf R}^{N})$ and it is a weak solution of a nonlinear elliptic problem (see (4.3)). Applying elliptic regularity theorems to $\{(w^{\kappa})^{1/2\nu}\}_{0<\kappa<\kappa^{*}}$, where $\nu\in(0,1)$, with the aid of a delicate inequality (see Lemma 5.1), we obtain a uniform local estimate of $\{(w^{\kappa})^{1/2\nu}\}_{0<\kappa<\kappa^{*}}$, instead of $\{w^{\kappa}\}_{0<\kappa<\kappa^{*}}$. This argument gives a new characterization of $p_{JL}$. Indeed, the argument requires to find $\nu\in(0,1)$ satisfying

[TABLE]

The existence of $\nu\in(0,1)$ satisfying (1.6) is equivalent to $1<p<p_{JL}$ (see Lemmas 5.4 and 5.5). Consequently, in the case of $1<p<p_{JL}$, we obtain a uniform local estimate of $\{(w^{\kappa})^{1/2\nu}\}_{0<\kappa<\kappa^{*}}$ for some $\nu\in(0,1)$. Furthermore, we apply elliptic regularity theorems again to prove that $u^{\kappa}$ converges, as $\kappa\to\kappa^{*}$, to a solution $u^{\kappa^{*}}$ of (1.1) with $\kappa=\kappa^{*}$. The proof of the uniqueness of the solution with $\kappa=\kappa^{*}$ is by contradiction and the construction of supersolutions.

The rest of this paper is organized as follows. In Section 2 we prepare some inequalities of the fundamental solution $G$ and recall two lemmas on eigenvalue problems. In Section 3 we construct approximate solutions to (1.1) and obtain some estimates of the approximate solutions. Section 4 is devoted to the proof of Theorem 1.1. In Sections 5 and 6 we prove Theorem 1.2.

2 Preliminaries

In this section we recall some properties on the fundamental solution $G=G(x)$. In what follows, for any nonnegative functions $f$ and $g$ in ${\bf R}^{N}$, we say that $f(x)\asymp g(x)$ as $x\to 0$ if there exists $c>0$ such that $c^{-1}g(x)\leq f(x)\leq cg(x)$ in a neighborhood of [math]. Similarly, we say that $f(x)\asymp g(x)$ as $x\to\infty$ if there exists $c>0$ such that $c^{-1}g(x)\leq f(x)\leq cg(x)$ in a neighborhood of the space infinity.

2.1 Fundamental solution $G$

We collect some properties of the fundamental solution $G$. It follows from (1.4) that

[TABLE]

By the Hölder inequality, the Hardy-Littlewood-Sobolev inequality and the Sobolev inequality we have the following properties (see [21, Appendix]):

• (G1)

For $r\in[1,N/(N-2))$, there exists $C_{r}>0$ such that

[TABLE]

• (G2)

For $r\in(1,N/2)$, there exists $C_{r}^{\prime}>0$ such that

[TABLE]

where $1/r^{\prime}=1/r-2/N$;

• (G3)

Let $r>N/2$. Then

[TABLE]

for $v\in L^{r}({\bf R}^{N})$;

• (G4)

For $v\in L^{1}({\bf R}^{N})\cap L^{r}({\bf R}^{N})$ with some $r>N/2$, then

[TABLE]

Let

[TABLE]

where $\chi_{B(0,1)}$ denotes the characteristic function of the ball $B(0,1)$. Then

[TABLE]

Furthermore, for any $\sigma>1$, there exists a constant $C>0$ such that

[TABLE]

2.2 Eigenvalue problem

We recall two lemmas of the eigenvalue problem

[TABLE]

where $a\in L^{N/2}({\bf R}^{N})\cap L^{r}({\bf R}^{N})$ for some $r>N/2$ and $a(x)>0$ for almost all $x\in{\bf R}^{N}$. See [21, Lemmas B2 and B3].

Lemma 2.1

Then problem (2.4) has the first eigenvalue $\lambda_{1}>0$ and the corresponding eigenfunction $\phi_{1}$ with $\phi_{1}>0$ in ${\bf R}^{N}$. Furthermore,*

[TABLE]

Lemma 2.2

Let $\lambda_{1}$ be the first eigenvalue to problem (2.4) and assume that $\lambda_{1}>1$. Then, for any $f\in H^{-1}({\bf R}^{N})$, there exists a unique solution $v$ of*

[TABLE]

3 Approximate solutions

Let $\mu$ be a nontrivial Radon measure in ${\bf R}^{N}$ with $\mbox{supp}\,\mu\subset B(0,R)$ for some $R>0$. Assume (1.5). Let $u$ be a $(C_{0}+L^{q}_{\rm c})$-solution of (1.1). Then

[TABLE]

For any $\kappa>0$, we define $\{U_{j}^{\kappa}\}_{j=0}^{\infty}$ and $\{V_{j}^{\kappa}\}_{j=0}^{\infty}$ by

[TABLE]

By induction we easily obtain

[TABLE]

for $j\in\{0,1,2,\dots\}$ and almost all $x\in{\bf R}^{N}$. In what follows, the letter $C$ denotes generic positive constants and it may have different values also within the same line.

Lemma 3.1

Assume the same conditions as in Theorem 1.1.*

• (i)

For any $\kappa>0$, there exists $c>0$ such that $U_{j}^{\kappa}(x)\geq cg(x)$ for $j\in\{1,2,\dots\}$ and almost all $x\in{\bf R}^{N}$.

• (ii)

For any $0<\kappa<\kappa^{\prime}$ and $j\in\{0,1,2,\dots\}$,

[TABLE]

for almost all $x\in{\bf R}^{N}$.

Proof. We prove assertion (i). Let $R>0$ be as in Theorem 1.1. Let $z\in{\bf R}^{N}\setminus B(0,R+2)$. It follows that

[TABLE]

for almost all $x\in B(z,1)$. Since $g=G*\chi_{B(0,1)}$, by (2.2), (3.2) and (3.3) we have

[TABLE]

for $j\in\{1,2,\ldots\}$ and almost all $x\in{\bf R}^{N}$. Thus assertion (i) follows.

We prove assertion (ii). Let $0<\kappa<\kappa^{\prime}$. It follows from (3.1) and (3.2) that

[TABLE]

for almost all $x\in{\bf R}^{N}$. Then (3.4) holds for $j=0,1$.

Assume that (3.4) holds for some $j=j_{0}\in\{1,2,\dots\}$. It follows from (3.1) and (3.2) that

[TABLE]

On the other hand, the function

[TABLE]

is monotone increasing for any fixed $t\geq 0$. Since (3.4) holds for $j=j_{0}$, we deduce from (3.3) and (3.5) that

[TABLE]

Thus (3.4) holds for $j=j_{0}+1$. By induction we obtain (3.4) for $j\in\{0,1,2,\dots\}$ and Lemma 3.1 follows. $\Box$

Since $q>\max\{p,N(p-1)/2\}$, we can find $r_{*}\in(1,\infty)$ such that

[TABLE]

Define a sequence $\{q_{j}\}_{j=0}^{\infty}$ by

[TABLE]

By (3.6) there exists $j_{*}\in\{1,2,\dots\}$ such that

[TABLE]

Lemma 3.2

Assume the same condition as in Theorem 1.1. Let $K>0$. Then there exists $c_{1}>0$ such that*

[TABLE]

for $j=0,\dots,j_{*}-1$ and $0<\kappa<K$. Furthermore, for any $j\in\{j_{*},j_{*}+1,\dots\}$, there exists $c_{2}>0$ such that

[TABLE]

for $0<\kappa<K$.

Proof. Since $\mbox{supp}\,\mu$ is closed, we can find $R^{\prime}\in(0,R)$ such that $\mbox{supp}\,\mu\subset B(0,R^{\prime})$. Let $\{R_{j}\}_{j=0}^{j_{*}}$ be such that $R^{\prime}<R_{0}<R_{1}<\cdots<R_{j_{*}}<R$. Then it follows from (1.5) and (2.2) that

[TABLE]

for $0<\kappa<K$. Since $V_{0}^{\kappa}=U_{0}^{\kappa}$, by (3.11) we have (3.9) for $j=0$.

Assume $j_{*}\geq 2$ and that

[TABLE]

hold for some $j=j_{0}\in\{0,\dots,j_{*}-2\}$ and all $\kappa\in(0,K)$. It follows from (3.2) and (3.3) that

[TABLE]

for almost all $x\in{\bf R}^{N}$. Since $r_{*}<q/(p-1)$, we observe from (3.12) that

[TABLE]

Then, by (3.12) we have

[TABLE]

for $0<\kappa<K$, where $1/r_{j_{0}}=1/r_{*}+1/q_{j_{0}}$. Since

[TABLE]

by (3.13) and (3.15) we apply (G2) to obtain

[TABLE]

for $0<\kappa<K$. This together with (2.2) implies that

[TABLE]

for $x\in{\bf R}^{N}\setminus B(0,R_{j_{0}+1})$. Furthermore, by (2.3) and (3.12) we see that

[TABLE]

By (3.13), (3.18) and (3.19) we have

[TABLE]

for $0<\kappa<K$. In addition, by (3.2), (3.12), (3.17) and (3.20) we obtain

[TABLE]

for $0<\kappa<K$. This together with (3.17) and (3.20) implies (3.12) for $j=j_{0}+1$. Since (3.9) holds for $j=0$, by induction we obtain (3.12) for $j\in\{0,\dots,j_{*}-1\}$. This yields (3.9) for $j\in\{0,\dots,j_{*}-1\}$.

It remains to prove (3.10). Similarly to (3.15), by (3.12) with $j=j_{*}-1$, we have

[TABLE]

for $0<\kappa<K$, where $1/r_{j_{*}-1}=1/r_{*}+1/q_{j_{*}-1}<1$. On the other hand, it follows from (3.7) and (3.8) that

[TABLE]

Then, similarly to (3.13), we see that

[TABLE]

Combining (3.2) with (3.21), we observe from (G3) that

[TABLE]

Furthermore, similarly to (3.20), we obtain

[TABLE]

This together with (3.9) implies (3.10) for $j=j_{*}$. Repeating this argument, we obtain (3.10). Thus Lemma 3.2 follows. $\Box$

We obtain estimates on $U^{\kappa+\epsilon}_{j}-U^{\kappa}_{j}$ and $V^{\kappa+\epsilon}_{j}-V^{\kappa}_{j}$ in the following two lemmas.

Lemma 3.3

Assume the same condition as in Theorem 1.1. Then, for any $j\in\{0,1,2,\dots\}$ and $\kappa>0$, there exists $c>0$ such that*

[TABLE]

Proof. It follows from (3.2) that $0\leq U_{0}^{\kappa+\epsilon}-U_{0}^{\kappa}=\epsilon\mu_{0}=\kappa^{-1}\epsilon U_{0}^{\kappa}$ in ${\bf R}^{N}$. So (3.22) holds for $j=0$ and $0<\epsilon\leq 1$.

Assume that (3.22) holds for some $j=j_{0}\in\{0,1,2,\dots\}$ and all $0<\epsilon\leq 1$. Similarly to (3.13), by (3.2) we have

[TABLE]

for $0<\epsilon\leq 1$. This implies that (3.22) holds for $j=j_{0}+1$ and $0<\epsilon\leq 1$. Therefore, by induction (3.22) holds for $j\in\{0,1,2,\dots\}$ and $0<\epsilon\leq 1$. Thus Lemma 3.3 follows. $\Box$

Lemma 3.4

Assume the same conditions as in Theorem 1.1. Let $\kappa>0$. Then there exists $c_{1}>0$ such that*

[TABLE]

for $j=0,\dots,j_{*}-1$ and $0<\epsilon\leq 1$. Furthermore, for any $j\in\{j_{*},j_{*}+1,\dots\}$, there exists $c_{2}>0$ such that

[TABLE]

for $0<\epsilon\leq 1$.

Proof. Let $\{R_{j}\}_{j=0}^{j_{*}}$ and $\{r_{j}\}$ be as in the proof of Lemma 3.2. In the case of $j=0$, since $V_{j}^{\kappa+\epsilon}-V_{j}^{\kappa}=\kappa^{-1}\epsilon U^{\kappa}_{0}$, by Lemma 3.2 we have (3.23) for $0\leq\epsilon\leq 1$.

Let $j_{*}\geq 1$ and $j=1$. By (3.2) we see that

[TABLE]

Then Lemma 3.2 implies (3.23) with $j=1$.

Let $j_{*}\geq 2$ and assume that

[TABLE]

for some $j=j_{0}\in\{0,\dots,j_{*}-2\}$ and all $0<\epsilon\leq 1$. It follows from (3.2) that

[TABLE]

where

[TABLE]

On the other hand,

[TABLE]

It follows from Lemma 3.3, (3.3) and (3.14) that

[TABLE]

Then, similarly to (3.15), by (3.25) we obtain

[TABLE]

for $0<\epsilon\leq 1$. On the other hand, by Lemma 3.3 and (3.3) we see that

[TABLE]

for $0<\epsilon\leq 1$. In the case of $1<p\leq 2$, we have

[TABLE]

On the other hand, in the case of $p>2$,

[TABLE]

Similarly to (3.14), by (3.9), (3.30) and (3.31) we see that

[TABLE]

for $0<\epsilon\leq 1$. Therefore, by (3.27), (3.29) and (3.32) we have

[TABLE]

for $0<\epsilon\leq 1$. Then, similarly to (3.17), we deduce from (G2), (3.16) and (3.26) that

[TABLE]

for $0<\epsilon\leq 1$.

On the other hand, it follows that

[TABLE]

By (3.33) we have

[TABLE]

for $x\in{\bf R}^{N}\setminus B(0,R_{j_{0}+1})$. Since (3.12) holds for $j=j_{0}$, from (3.25), (3.28), (3.30) and (3.31) we deduce that

[TABLE]

for $x\in{\bf R}^{N}\setminus B(0,R_{j_{0}})$. This together with (3.27) implies that

[TABLE]

By (3.35) and (3.36) we obtain

[TABLE]

for $0<\epsilon\leq 1$. This together with (3.34) implies that (3.25) holds for $j=j_{0}+1$. Therefore, by induction we see that (3.25) holds for $j\in\{0,1,\dots,j_{*}-1\}$ and $0<\epsilon\leq 1$.

We prove (3.24). By (3.25) with $j=j_{*}-1$, similarly to (3.33), we have

[TABLE]

for $0<\epsilon\leq 1$. It follows from (3.21) that $r_{j_{*}-1}>N/2$. Then we deduce from (G2) and (3.26) that

[TABLE]

for $0<\epsilon\leq 1$. Furthermore, similarly to (3.37), we have

[TABLE]

for $0<\epsilon\leq 1$. By (3.38) and (3.39) we have (3.24). Thus Lemma 3.4 follows. $\Box$

4 Proof of Theorem 1.1

For any $\kappa>0$, define

[TABLE]

According to Definition 1.1, (3.2) and (3.3), the following lemma holds.

Lemma 4.1

Assume the same condition as in Theorem 1.1. Let $\kappa>0$. Then the following conditions are equivalent*:**

• (i)

$u=w+U^{\kappa}_{j_{*}}$* is a $(C_{0}+L^{q}_{\rm c})$-solution of (1.1)*;**

• (ii)

$w\in C_{0}({\bf R}^{N})$* is positive in ${\bf R}^{N}$ and $w$ satisfies*

[TABLE]

Furthermore, we have:

Lemma 4.2

Assume the same condition as in Theorem 1.1. Let $\kappa>0$. If $w\in C_{0}({\bf R}^{N})$ is a positive solution of (4.1), then there exists $c>0$ such that*

[TABLE]

Furthermore, $w\in H^{1}({\bf R}^{N})$ and $w$ is a weak solution of

[TABLE]

that is

[TABLE]

for $\psi\in H^{1}({\bf R}^{N})$. Here $s_{+}:=\max\{s,0\}$ for $s\in{\bf R}$.

Proof. Let $R>0$ be as in Theorem 1.1. Let $\zeta\in C^{\infty}({\bf R}^{N})$ be such that

[TABLE]

Let $W:=(w+U_{j_{*}}^{\kappa})^{p}-(U_{j_{*}-1}^{\kappa})^{p}$. Set

[TABLE]

Since

[TABLE]

it follows from (3.9) and (3.10) that $\zeta W\in L^{1}({\bf R}^{N})\cap L^{r}({\bf R}^{N})$ for some $r>N/2$. Thus we deduce from (G4) that

[TABLE]

Furthermore,

[TABLE]

for $x\in{\bf R}^{N}\setminus B(0,R+3)$.

On the other hand, by (2.1), (3.1), (3.2) and Lemma 3.2, we deduce from a bootstrap argument that $w_{2}\in C^{2,\theta}({\bf R}^{N})$, where $0<\theta<1$, and $w_{2}$ is a classical solution of

[TABLE]

Let $x_{0}\in{\bf R}^{N}\setminus B(0,R+3)$. Since $w\in C_{0}({\bf R}^{N})$, by Lemma 3.1 (i), (3.1), (3.3) and (3.10) we can find a positive constant $C>0$ such that

[TABLE]

Thus we have

[TABLE]

Since $w=w_{1}+w_{2}$, it follows from (3.10) and (4.6) that

[TABLE]

Let $\epsilon\in(0,1)$. Since $w\in C_{0}({\bf R})$, by (3.10) and (4.7) we can find $L>R+3$ such that

[TABLE]

Let $\gamma>0$ be such that

[TABLE]

Set

[TABLE]

Since

[TABLE]

we deduce from the maximum principle that $z\leq 0$ in ${\bf R}^{N}\setminus B(0,L)$. Then we have

[TABLE]

Since $\epsilon$ is arbitrary and $p>1$, we can find $\sigma>1$ such that

[TABLE]

Then it follows from (2.3) that

[TABLE]

for $x\in{\bf R}^{N}\setminus B(0,L+1)$. Since $w$ is continuous in ${\bf R}^{N}$, we obtain (4.2). Moreover, we observe that $W\in L^{1}({\bf R}^{N})\cap L^{r}({\bf R}^{N})$ for some $r>N/2$. Then (G4) implies that $w\in H^{1}({\bf R}^{N})$. Furthermore,

[TABLE]

Therefore $w$ is a weak solution of (4.3) in ${\bf R}^{N}$. Thus Lemma 4.2 follows. $\Box$

We show that the existence of supersolutions of (1.1) ensures the existence of the solution of (1.1).

Lemma 4.3

Assume the same condition as in Theorem 1.1. Let $v$ be a $(C_{0}+L^{q}_{\rm c})$-supersolution of (1.1). Then*

[TABLE]

for almost all $x\in{\bf R}^{N}$ and all $j\in\{0,1,2,\dots\}$. Furthermore,

[TABLE]

exists for almost all $x\in{\bf R}^{N}$ and it is a minimal $(C_{0}+L^{q}_{\rm c})$-solution of (1.1).

Proof. Let $R$ be as in Theorem 1.1. Since $v$ is a $(C_{0}+L^{q}_{\rm c})$-supersolution of (1.1), we can find $L>R$ such that

[TABLE]

Similarly to (3.3), by induction we obtain (4.8). Furthermore, by (3.2), (3.3) and (4.8) we see that $U^{\kappa}_{\infty}$ exists and satisfies

[TABLE]

for almost all $x\in{\bf R}^{N}$. Setting $w_{\infty}=U^{\kappa}_{\infty}-U^{\kappa}_{j_{*}}$, we have

[TABLE]

Let $\zeta\in C^{\infty}({\bf R}^{N})$ be such that

[TABLE]

Let $W_{\infty}=(w_{\infty}+U^{\kappa}_{j_{*}})^{p}-(U^{\kappa}_{j_{*}-1})^{p}$ and set

[TABLE]

By (4.9) and (4.10) we see that $U^{\kappa}_{\infty}\in L^{q}(B(0,L+2))$. Similarly to (4.5), it follows from (G3) and $q>N/2$ that $I\in C_{0}({\bf R}^{N})$. Since $v\in C_{0}({\bf R}^{N})+L^{q}_{{\rm c}}({\bf R}^{N})$, by Lemma 3.2 and (4.10) we have $(1-\zeta)W_{\infty}\in L^{\infty}({\bf R}^{N})$. Then we deduce from (1.4) that $J\in C({\bf R}^{N})$. Therefore we obtain $U^{\kappa}_{\infty}\in C({\bf R}^{N}\setminus B(0,L+2))$. Combining $v\in C_{0}({\bf R}^{N})+L^{q}_{{\rm c}}({\bf R}^{N})$ with (4.10), we observe that $U^{\kappa}_{\infty}(x)\to 0$ as $x\to\infty$. Thus $U_{\infty}^{\kappa}$ is a minimal $(C_{0}+L^{q}_{\rm c})$-solution of (1.1), and the proof is complete. $\Box$

By the use of Lemma 4.3 we prove the following lemma.

Lemma 4.4

Assume the same conditions as in Theorem 1.1. Then*

• (i)

$S_{\kappa}\not=\emptyset$* for sufficiently small $\kappa>0$;*

• (ii)

If $S_{\kappa}\not=\emptyset$, then problem (1.1) possesses a minimal $(C_{0}+L^{q}_{\rm c})$-solution $u^{\kappa}$.

Proof. We prove assertion (i). Let

[TABLE]

Then $gBC({\bf R}^{N})$ is a Banach space with the norm $|||f|||:=\sup_{x\in{\bf R}^{N}}|g(x)^{-1}f(x)|$. Let $0<\kappa<1$ and define

[TABLE]

For any $v\in gBC({\bf R}^{N})$, we have

[TABLE]

It follows from Lemma 3.2 that

[TABLE]

for $0<\kappa<1$. Thus, by (4.11) and (4.12) we have $(v+U_{j_{*}}^{\kappa})_{+}^{p}-(U_{j_{*}-1}^{\kappa})^{p}\in L^{q/(p-1)}({\bf R}^{N})$. Then it follows from (G3) and $q/(p-1)>N/2$ that $F_{\kappa}[v]\in C_{0}({\bf R}^{N})$. Set

[TABLE]

By (4.11) and (4.12) we have

[TABLE]

for $x\in B(0,R+2)$. Similarly, it follows that

[TABLE]

for $x\in{\bf R}^{N}\setminus B(0,R+2)$. Similarly, we observe from (2.3) and Lemma 3.2 that

[TABLE]

for $x\in{\bf R}^{N}$. From $I_{1}+I_{2}=F_{\kappa}[v]\in C_{0}({\bf R}^{N})$, it follows that

[TABLE]

Since

[TABLE]

similarly to (4.13), we obtain

[TABLE]

for $0<\kappa<1$. Then we can find positive constants $\delta$ and $\kappa^{0}$ such that

[TABLE]

for $0<\kappa<\kappa^{0}$, where $\overline{B_{gBC}(0,\delta)}:=\{f\in gBC({\bf R}^{N})\,:\,|||f|||\leq\delta\}$.

Define a sequence $\{w_{k}\}\subset gBC({\bf R}^{N})$ by

[TABLE]

Due to Lemma 3.1, by induction we see that

[TABLE]

Applying the fixed point theorem, we can find a positive function $w_{\infty}\in gBC({\bf R}^{N})$ such that

[TABLE]

Then, by Lemma 4.1 we see that $u=U_{j_{*}}^{\kappa}+w_{\infty}$ is a $(C_{0}+L^{q}_{\rm c})$-solution. Therefore assertion (i) follows. Assertion (ii) is proved by Lemma 4.3. The proof is complete. $\Box$

Let $u^{\kappa}$ be a minimal $(C_{0}+L^{q}_{\rm c})$-solution of (1.1). Consider the linearized eigenvalue problem of (1.1) at $u^{\kappa}$

[TABLE]

Since $q>N(p-1)/2$ and $u^{\kappa}$ is a $(C_{0}+L^{q}_{\rm c})$-solution of (1.1), applying Lemma 2.1, we see that problem $(\mbox{E}_{\kappa})$ has the first eigenvalue $\lambda_{1}^{\kappa}>0$ and

[TABLE]

Then the following two lemmas hold.

Lemma 4.5

Let $\phi^{\kappa}_{1}$ be the first eigenfunction of problem $(\mbox{E}_{\kappa})$ such that $\phi_{1}^{\kappa}>0$ in ${\bf R}^{N}$. Then $\phi_{1}^{\kappa}\in C_{0}({\bf R}^{N})$ and*

[TABLE]

Furthermore, there exists $c>0$ such that

[TABLE]

Proof. Since $(u^{\kappa})^{p-1}\in L^{q/(p-1)}({\bf R}^{N})$ with $q/(p-1)>N/2$ and $\phi_{1}^{\kappa}\in H^{1}({\bf R}^{N})$, we apply regularity theorems for elliptic equations to see that $\phi_{1}^{\kappa}\in C_{0}({\bf R}^{N})$. Indeed, since $\phi_{1}^{\kappa}\in H^{1}({\bf R}^{N})$, it follows that

[TABLE]

Then, applying [25, Theorems 1 and 8] to $(\mbox{E}_{\kappa})$, we see that $\phi_{1}^{\kappa}\in C_{0}({\bf R}^{N})$.

Set

[TABLE]

Then, by a similar argument as in the proof of Lemma 4.2 we see that

[TABLE]

Then we deduce that $z:=\phi_{1}^{\kappa}-\tilde{\phi}\in H^{1}({\bf R}^{N})$ and $z$ is a weak solution of $-\Delta z+z=0$ in ${\bf R}^{N}$. Then it easily follows that $z=0$ in ${\bf R}^{N}$, i.e., $\phi_{1}^{\kappa}=\tilde{\phi}$ in ${\bf R}^{N}$. Thus we have (4.15).

Since $\phi_{1}^{\kappa}\in C_{0}({\bf R}^{N})$ and $\phi_{1}^{\kappa}>0$ in ${\bf R}^{N}$, Lemma 3.1 (i) together with (4.15) implies that

[TABLE]

This together with (4.17) implies (4.16). Thus Lemma 4.5 follows. $\Box$

Lemma 4.6

Define*

[TABLE]

Let $0<\kappa<\kappa^{*}$ and let $\lambda^{\kappa}_{1}$ be the first eigenvalue to problem ($\mbox{E}_{\kappa}$). Then $\lambda^{\kappa}_{1}>1$ and

[TABLE]

for $\psi\in H^{1}({\bf R}^{N})\setminus\{0\}$. Furthermore,

[TABLE]

Proof. Let $0<\kappa<\kappa^{\prime}<\kappa^{*}$. Set

[TABLE]

By Lemma 3.1 we see that

[TABLE]

Since the function

[TABLE]

is strictly monotone increasing for any fixed $t>0$, we observe from Lemma 3.1 (i) that

[TABLE]

Since $\phi_{1}^{\kappa}$ is the eigenfunction of ($\mbox{E}_{\kappa}$) and positive, by (4.20) we have

[TABLE]

Furthermore, by (4.4) and (4.21) we obtain

[TABLE]

Here we used Lemma 4.2. Combining (4.22) with (4.23), we see that $\lambda^{\kappa}_{1}>1$. This together with (4.14) implies (4.18) and (4.19). Thus Lemma 4.6 follows. $\Box$

Now we are ready to complete the proof of Theorem 1.1.

Proof of Theorem 1.1. We show that $\kappa^{*}<\infty$. Consider the eigenvalue problem

[TABLE]

Let $\lambda_{B}$ and $\psi_{B}$ be the first eigenvalue and the first eigenfunction of problem (4.24), respectively, such that $\psi_{B}>0$ in $B(0,1)$. Set $\psi_{B}=0$ outside $B(0,1)$. Then $\psi_{B}\in H^{1}({\bf R}^{N})$ and

[TABLE]

On the other hand, it follows from Lemmas 3.1 and 4.6 that

[TABLE]

for $0<\kappa<\kappa^{*}$. By (3.1), (4.25) and (4.26) we see that

[TABLE]

for $0<\kappa<\kappa^{*}$. Then we deduce that $\kappa^{*}<\infty$. This together with Lemma 4.4 implies assertions (i) and (ii) of Theorem 1.1. Thus the proof of Theorem 1.1 is complete. $\Box$

5 Uniform estimates of $w^{\kappa}$

For $0<\kappa<\kappa^{*}$, let $u^{\kappa}$ be the minimal $(C_{0}+L^{q}_{\rm c})$-solution of (1.1) and set $w^{\kappa}:=u^{\kappa}-U_{j_{*}}^{\kappa}$. In this section we obtain uniform estimates of $\{w^{\kappa}\}_{0<\kappa<\kappa^{*}}$ in $H^{1}({\bf R}^{N})$ and $L^{\infty}({\bf R}^{N})$ and prove the following proposition:

Proposition 5.1

Let $1<p<p_{JL}$ and assume the same conditions as in Theorem 1.1. Then problem (1.1) with $\kappa=\kappa^{*}$ possesses a $(C_{0}+L^{q}_{\rm c})$-solution.*

We prepare an inequality in order to estimate the nonlinear term of (4.3).

Lemma 5.1

Let $p>1$, $\epsilon>0$ and $\delta\geq 0$. Then there exists a constant $c>0$ such that*

[TABLE]

Proof. Let $\delta>0$ and $\epsilon>0$. For sufficiently small $\beta>0$, we have

[TABLE]

for $t>s\geq 0$. This implies that

[TABLE]

for $t\geq s\geq 0$. Thus Lemma 5.1 follows. $\Box$

We obtain a uniform estimate of $\{w^{\kappa}\}_{0<\kappa<\kappa^{*}}$ in $H^{1}({\bf R}^{N})$.

Lemma 5.2

Let $0<\kappa<\kappa^{*}$. Let $u^{\kappa}$ be the minimal $(C_{0}+L^{q}_{\rm c})$-solution of (1.1) and set $w^{\kappa}:=u^{\kappa}-U_{j_{*}}^{\kappa}$. Then*

[TABLE]

Proof. Let $\epsilon$ and $\delta$ be sufficiently small positive constants. By (3.10) we can find $\tilde{V}\in H^{1}({\bf R}^{N})$ such that $V_{j_{*}}^{\kappa^{*}}\leq\tilde{V}$ in ${\bf R}^{N}$. Then it follows from (4.4) and (5.1) that

[TABLE]

where $M:=\|V_{j_{*}}^{\kappa^{*}}\|_{L^{\infty}({\bf R}^{N})}$. This together with Lemma 4.6 implies that

[TABLE]

On the other hand, it follows from $q>\max\{N(p-1)/2,p\}>1$ that

[TABLE]

By (3.10), applying the interpolation inequality in Lebesgue spaces and the Sobolev inequality, we obtain

[TABLE]

Taking a sufficiently small $\epsilon>0$ if necessary, by (5.3) and (5.5) we obtain

[TABLE]

This implies (5.2). Thus Lemma 5.2 follows. $\Box$

Applying Lemma 5.2, we have:

Lemma 5.3

Assume the same conditions as in Theorem 1.1. Let $0<\kappa<\kappa^{*}$ and $z\in{\bf R}^{N}$. Let $\nu\in(0,1)$ be such that*

[TABLE]

Then

[TABLE]

where $g_{z}(x):=g(x-z)$ for $x,z\in{\bf R}^{N}$.

Proof. Let $0<\kappa<\kappa^{*}$, $z\in{\bf R}^{N}$ and $0<\delta<1$. Similarly to the proof of Lemma 5.2, we set $M:=\|V_{j_{*}}^{\kappa^{*}}\|_{L^{\infty}({\bf R}^{N})}$. Then it follows from Lemma 3.1 (ii) that

[TABLE]

Set

[TABLE]

For any $\epsilon>0$, it follows that

[TABLE]

and

[TABLE]

Then, taking $\epsilon>0$ small enough, we observe from (2.2) that

[TABLE]

On the other hand, it follows from (3.3), (4.4) and (5.1) that

[TABLE]

Furthermore, Lemma 4.6 implies that

[TABLE]

By the Hölder inequality and Lemma 3.2 we have

[TABLE]

Therefore, by (5.8), (5.9), (5.10) and (5.11) we obtain

[TABLE]

By (5.6), taking a sufficiently small $\delta>0$ if necessary, we see that

[TABLE]

Then we deduce from the Sobolev inequality that

[TABLE]

This implies that

[TABLE]

Consider the case where $1/\nu\geq 2$. By (5.4), applying the interpolation inequality in Lebesgue spaces, for any $\epsilon^{\prime}>0$, we can find a constant $C>0$ such that

[TABLE]

This together with (5.12) implies that

[TABLE]

In the case where $1/\nu<2$, similarly to (5.13), we have

[TABLE]

Then, by (5.12) we have

[TABLE]

Therefore, by Lemma 5.2, (5.14) and (5.15), we obtain (5.7). Thus Lemma 5.3 follows. $\Box$

We apply regularity theorem for elliptic equations to obtain the following lemma.

Lemma 5.4

Let $0<\kappa<\kappa^{*}$ and $z\in{\bf R}^{N}$. Assume that there exists $\nu\in(0,1)$ such that*

[TABLE]

Then $\{w^{\kappa}\}_{0<\kappa<\kappa^{*}}$ are uniformly bounded in ${\bf R}^{N}$ and equi-continuous for any compact set in ${\bf R}^{N}$.

Proof. Let $0<\kappa<\kappa^{*}$. It follows that

[TABLE]

where $M:=\|V^{\kappa^{*}}_{j_{*}}\|_{L^{\infty}({\bf R}^{N})}$. By Lemma 5.3 and (5.16) we see that

[TABLE]

Furthermore, it follows from Lemma 3.2 that

[TABLE]

Then, applying [25, Theorem 1] to problem (4.3), we obtain

[TABLE]

for $z\in{\bf R}^{N}$ and $0<\kappa<\kappa^{*}$. This together with Lemma 5.2 implies that $\{w^{\kappa}\}_{0<\kappa<\kappa^{*}}$ are uniformly bounded in ${\bf R}^{N}$. Furthermore, applying [25, Theorem 7] to problem (4.3), we see that $\{w^{\kappa}\}_{0<\kappa<\kappa^{*}}$ are equi-continuous for any compact set in ${\bf R}^{N}$. Thus Lemma 5.4 follows. $\Box$

We show that (5.16) holds for some $\nu\in(0,1)$ if $1<p<p_{JL}$.

Lemma 5.5

Let $1<p<p_{JL}$. Then there exists $\nu\in(0,1)$ satisfying (5.16).*

Proof. Let $1<p<p_{JL}$. Let $\nu^{\pm}$ be the roots of $4p(\nu-\nu^{2})=1$ such that $\nu^{-}\leq\nu^{+}$, that is

[TABLE]

Then $4p(\nu-\nu^{2})>1$ if and only if $\nu^{-}<\nu<\nu^{+}$. Since $\nu^{-}<(0,1/2)$, if

[TABLE]

then we can find $\nu\in(0,1)$ satisfying (5.16).

We prove (5.17). If $N=2$, then $p_{S}=\infty$ and (5.17) holds. So it suffices to consider the case of $N\geq 3$. Since

[TABLE]

the inequality (5.17) is equivalent to

[TABLE]

This is reduced to

[TABLE]

If $p<p_{S}$, then $D<0$ and we have (5.18). Thus (5.17) holds in the case of $p<p_{S}$.

Consider the case of $p\geq p_{S}$. Then $D\geq 0$ and (5.18) is equivalent to

[TABLE]

that is

[TABLE]

Let $N=10$. Then (5.19) is equivalent to $p>4/3$. Since $p\geq p_{S}=(N+2)/(N-2)=3/2$, (5.19) holds for $p\geq p_{S}$ if $N=10$.

Let $N\not=10$. Let $p^{\pm}$ be the roof of

[TABLE]

such that $p^{-}\leq p^{+}$, that is

[TABLE]

In the case of $N\geq 11$, since $p_{S}\leq p<p_{JL}=p^{+}$, (5.19) holds if $p^{-}<p_{S}$. Furthermore, $p^{-}<p_{S}$ if and only if

[TABLE]

which holds if $N\geq 11$. Thus (5.19) holds in the case of $N\geq 11$. In the case of $3\leq N\leq 9$, $(N-2)(N-10)<0$ and $p^{+}<p_{S}$. Then we have (5.19) for $p\geq p_{S}$. Therefore (5.19) holds for $p\geq p_{S}$ in the case of $N\not=10$. Thus Lemma 5.5 follows. $\Box$

We are ready to prove Proposition 5.1.

Proof of Proposition 5.1. Let $1<p<p_{JL}$. Thanks to Lemma 5.5, we can find $\nu\in(0,1)$ satisfying (5.16). Therefore, applying Lemmas 5.2 and 5.4, we obtain

[TABLE]

Furthermore, $\{w^{\kappa}\}_{0<\kappa<\kappa^{*}}$ are equi-continuous for any compact set in ${\bf R}^{N}$. Then we can find a solution $w_{*}\in H^{1}({\bf R}^{N})\,\cap\,C({\bf R}^{N})$ of (4.3) with $\kappa=\kappa^{*}$. Furthermore, similarly to the proof of Lemma 4.5, applying [25, Theorem 1] to (4.3), we see that $w_{*}\in C_{0}({\bf R}^{N})$. Therefore, by Lemma 4.1 we can find a $(C_{0}+L^{q}_{\rm c})$-solution of (1.1) with $\kappa=\kappa^{*}$. Thus Proposition 5.1 follows. $\Box$

6 Proof of Theorem 1.2

Let $u^{\kappa^{*}}$ be the minimal solution of (1.1) with $\kappa=\kappa^{*}$ obtained by Proposition 5.1. For the proof of Theorem 1.2, we prepare two lemmas.

Lemma 6.1

Let $1<p<p_{JL}$ and assume the same conditions as in Theorem 1.1. Then $\lambda_{1}^{\kappa^{*}}\geq 1$.*

Proof. It follows from (4.18) that

[TABLE]

for $\psi\in H^{1}({\bf R}^{N})\setminus\{0\}$, $\kappa\in(0,\kappa^{*})$ and $j\in\{0,1,2,\dots\}$. Then we have

[TABLE]

for $\psi\in H^{1}({\bf R}^{N})$ and $j\in\{0,1,2,\dots\}$. This together with Lemma 3.1 (ii) implies that

[TABLE]

for $\psi\in H^{1}({\bf R}^{N})$. Then we see that $\lambda_{1}^{\kappa^{*}}\geq 1$, and the proof is complete. $\Box$

Lemma 6.2

Let $1<p<p_{JL}$ and assume the same conditions as in Theorem 1.1. Let $\kappa\in(0,\kappa^{*}]$ be such that $\lambda_{1}^{\kappa}>1$. Then $\kappa<\kappa^{*}$.*

Proof. Let $\kappa\in(0,\kappa^{*}]$ be such that $\lambda_{1}^{\kappa}>1$. For $\epsilon\in(0,1]$ and $a\in(0,1]$, set

[TABLE]

Then

[TABLE]

where

[TABLE]

It follows that

[TABLE]

for $x\in{\bf R}^{N}$ and $t\in(0,1)$.

It follows that

[TABLE]

Combing Lemmas 3.1, 3.2 and 3.3 with (3.3) and (6.3), we have

[TABLE]

These together with Lemma 3.3 imply that

[TABLE]

for $x\in{\bf R}^{N}$ and $t\in(0,1)$. Furthermore, by Lemma 3.4 we have

[TABLE]

for $x\in{\bf R}^{N}$ and $t\in(0,1)$. Similarly to (6.4), we have

[TABLE]

These imply that

[TABLE]

for $x\in{\bf R}^{N}$ and $t\in(0,1)$.

Let $\delta\in(0,1)$. By (6.2), (6.5), (6.6) and (6.7), taking sufficiently small $a\in(0,1]$ and $\epsilon\in(0,1]$, we obtain

[TABLE]

for $x\in{\bf R}^{N}$ and $t\in(0,1)$. This together with (6.1) implies that

[TABLE]

On the other hand, it follows from $(\mbox{E}_{\kappa})$ that

[TABLE]

Combing (6.8) with (6.9), we see that

[TABLE]

Since $\lambda_{1}^{\kappa}>1$ and $\phi_{1}^{\kappa}>0$ in ${\bf R}^{N}$, taking a sufficiently small $\delta\in(0,1)$ if necessary, we obtain

[TABLE]

This means that $\overline{u}$ is a $(C_{0}+L^{q}_{\rm c})$-supersolution of (1.1) with $\kappa$ replaced by $\kappa+\epsilon$. Then, by Lemma 4.3 there exists a $(C_{0}+L^{q}_{\rm c})$-solution of (1.1) with $\kappa$ replaced by $\kappa+\epsilon$. Therefore we see that $\kappa<\kappa^{*}$ and the proof is complete. $\Box$

Now we are ready to prove Theorem 1.2.

Proof of Theorem 1.2. By Proposition 5.1 it suffices to prove the uniqueness of $(C_{0}+L^{q}_{\rm c})$-solution of (1.1) with $\kappa=\kappa^{*}$.

Let $u^{\kappa^{*}}$ be a minimal $(C_{0}+L^{q}_{\rm c})$-solution of (1.1) with $\kappa=\kappa^{*}$ and set $w^{\kappa^{*}}=u^{\kappa^{*}}-U_{j_{*}}^{\kappa^{*}}$. Let $\tilde{u}$ be a $(C_{0}+L^{p}_{\rm c})$ solution of (1.1) with $\kappa=\kappa^{*}$ and set $\tilde{w}=\tilde{u}-U_{j_{*}}^{\kappa^{*}}$. Then $w^{\kappa^{*}}$ and $\tilde{w}$ are solutions of (4.3) with $\kappa=\kappa^{*}$ and the function $z:=\tilde{w}-w^{\kappa^{*}}$ satisfies

[TABLE]

Let $\phi_{1}^{\kappa^{*}}$ be the eigenfunction of problem $(\mbox{E}_{\kappa^{*}})$ corresponding to $\lambda_{1}^{\kappa^{*}}$ such that $\phi_{1}^{\kappa^{*}}>0$ in ${\bf R}^{N}$. By Lemmas 6.1 and 6.2 we see that $\lambda_{1}^{\kappa^{*}}=1$. Then

[TABLE]

Multiplying (6.11) by $z$ and integrating it on ${\bf R}^{N}$, we obtain

[TABLE]

On the other hand, by (6.10) we see that

[TABLE]

These imply that

[TABLE]

Since $t^{p}>s^{p}+ps^{p-1}(t-s)$ for $t>s\geq 0$, we obtain $\tilde{w}=w^{\kappa^{*}}$ in ${\bf R}^{N}$. Therefore we deduce that $u^{\kappa^{*}}$ is the unique $(C_{0}+L^{q}_{\rm c})$-solution of (1.1) with $\kappa=\kappa^{*}$. Thus Theorem 1.2 follows. $\Box$

Acknowledgements. The first author was partially supported by the Grant-in-Aid for Scientific Research (A)(No. 15H02058) from Japan Society for the Promotion of Science.