Ground state solution of fractional Schr\"odinger equations with a general nonlinearity

TL;DR

This paper establishes the existence of a positive ground state solution for a fractional Schrödinger equation with a general nonlinearity using minimax methods, broadening understanding of such equations in mathematical physics.

Contribution

It introduces a new approach to find ground state solutions for fractional Schrödinger equations with broad nonlinear conditions, nearly optimal in scope.

Findings

Existence of positive ground state solution proven

Applicable to general nonlinearities under broad conditions

Uses minimax arguments for solution construction

Abstract

In this paper, we study the following fractional Schr\"odinger equation: \[ \left\{\begin{gathered} {(- \Delta)^s}u + mu = f(u){\text{in}}{\mathbb{R}^N}, \hfill u \in {H^s}({\mathbb{R}^N}),{\text{}}u > 0{\text{on}}{\mathbb{R}^N}, \hfill \\ \end{gathered} \right. \] where $m>0$, $N>2s$, ${(- \Delta)^s}$, $s \in (0,1)$ is the fractional Laplacian. Using minimax arguments, we obtain a positive ground state solution under general conditions on $f$ which we believe to be almost optimal.

Equations

$$\{\begin{array}{c}(-\Delta {)}^{s}u+mu=f(u)\text{ in }{\mathbb{R}}^N,\text{hfill}\\ u\in H^s({\mathbb{R}}^N),u>0\text{ on }{\mathbb{R}}^N,\text{hfill}\end{array}$$

$$\{\begin{array}{c}-\text{div}(y^{1-2s}\nabla w)=0\text{ in }{\mathbb{R}}_{+}^{N+1},\text{hfill}\\ -k_s\underset{y\to 0^{+}}{\lim }{y}^{1-2s}\frac{\partial w}{\partial y}(x,y)=-mw+f(w)\text{ on }{\mathbb{R}}^N\times \{0\}.\text{hfill}\end{array}$$

$$I_m(w)=\frac{k_s}{2}{\int }_{{\mathbb{R}}_{+}^{N+1}}{y}^{1-2s}|\nabla w{|}^{2}dxdy+\frac{m}{2}{\int }_{{\mathbb{R}}^N}{w}^2(x,0)dx-{\int }_{{\mathbb{R}}^N}F(w(x,0))dx,w\in X^{1,s}({\mathbb{R}}_{+}^{N+1}).$$

$${\|w\|_{{X^{1,s}}(\mathbb{R}_{+}^{N+1})}}={\Bigl{(}{\int_{\mathbb{R}_{+}^{N+1}}{{y^{1-2s}}|\nabla w{|^{2}}}dxdy+\int_{{\mathbb{R}^{N}}}{{w^{2}}(x,0)}dx}\Bigr{)}^{1/2}}.$$

$$I_m\circ \Phi (\theta ,w):=I_m(w(e^{-\theta }x,e^{-\theta }y)),(\theta ,w)\in \mathbb{R}\times X^{1,s}({\mathbb{R}}_{+}^{N+1}),$$

$$\Vert u{\Vert }_{D^s({\mathbb{R}}^N)}={\left({\int }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^2}{|x-y{|}^{N+2s}}dxdy\right)}^{1/2}$$

$$\Vert u{\Vert }_{L^{2_s^{\ast }}({\mathbb{R}}^N)}\le C(N,s)\Vert u{\Vert }_{D^s({\mathbb{R}}^N)}$$

$$H^s({\mathbb{R}}^N)=\left\{u\in L^2({\mathbb{R}}^N):{\int }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^2}{|x-y{|}^{N+2s}}dxdy\right\}$$

$$\Vert u{\Vert }_{H^s({\mathbb{R}}^N)}=\Vert u{\Vert }_{D^s({\mathbb{R}}^N)}+\Vert u{\Vert }_{L^2({\mathbb{R}}^N)}.$$

$$H^s({\mathbb{R}}^N)\text{ is continuously embedded in }{L}^p({\mathbb{R}}^N)\text{ for }p\in [2,2_s^{\ast }].$$

$$\{\begin{array}{c}-\text{div}(y^{1-2s}\nabla w)=0\text{ in }{\mathbb{R}}_{+}^{N+1},\text{hfill}\\ w=u\text{ on }{\mathbb{R}}^N\times \{0\}\text{hfill}\end{array}$$

$$w(x,y)=P_y^s\ast u(x)={\int }_{{\mathbb{R}}^N}{P}_y^s(x-\xi )u(\xi )d\xi ,$$

$$P_y^s(x):=c(N,s)\frac{y^{2s}}{{(|x{|}^2+y^2)}^{(N+2s)/2}}$$

$${\|w\|_{{X^{s}}(\mathbb{R}_{+}^{N+1})}}:={\Bigl{(}{\int_{\mathbb{R}_{+}^{N+1}}{{k_{s}}{y^{1-2s}}|\nabla w{|^{2}}}dxdy}\Bigr{)}^{1/2}}.$$

$$\Vert u{\Vert }_{D^s({\mathbb{R}}^N)}=\Vert w{\Vert }_{X^s({\mathbb{R}}_{+}^{N+1})}.$$

$$\Vert u{\Vert }_{D^s({\mathbb{R}}^N)}\le \Vert w{\Vert }_{X^s({\mathbb{R}}_{+}^{N+1})}.$$

$$S(s,N){\Bigl{(}{\int_{{\mathbb{R}^{N}}}{|u{|^{2_{s}^{*}}}}dx}\Bigr{)}^{2/2_{s}^{*}}}\leq\int_{\mathbb{R}_{+}^{N+1}}{{y^{1-2s}}|\nabla w{|^{2}}}dxdy,$$

$$S(s,N)=\frac{2{\pi }^s\Gamma (1-s)\Gamma ((N+2s)/2)\Gamma {(N/2)}^{2s/N}}{\Gamma (s)\Gamma ((N-2s)/2)\Gamma {(N)}^{2s/N}}$$

$$u_{\delta }(x)={\delta }^{(N-2s)/2}(|x{|}^2+{\delta }^2{)}^{-(N-2s)/2}$$

$$\{\begin{array}{c}-\text{div}(y^{1-2s}\nabla w)=0\text{ in }{\mathbb{R}}_{+}^{N+1},\text{hfill}\\ -k_s\underset{y\to 0^{+}}{\lim }{y}^{1-2s}\frac{\partial w}{\partial y}(x,y)=-mw+f(w)\text{ on }{\mathbb{R}}^N\times \{0\}\text{hfill}\end{array}$$

$$I_m(w)=\frac{k_s}{2}{\int }_{{\mathbb{R}}_{+}^{N+1}}{y}^{1-2s}|\nabla w{|}^{2}dxdy+\frac{m}{2}{\int }_{{\mathbb{R}}^N}{w}^2(x,0)dx-{\int }_{{\mathbb{R}}^N}F(w(x,0))dx,w\in X^{1,s}({\mathbb{R}}_{+}^{N+1}).$$

$$P_m(w)=\frac{k_s(N-2s)}{2}{\int }_{{\mathbb{R}}_{+}^{N+1}}{y}^{1-2s}|\nabla w{|}^{2}dxdy+\frac{mN}{2}{\int }_{{\mathbb{R}}^N}{w}^2(x,0)dx-N{\int }_{{\mathbb{R}}^N}F(w(x,0))dx=0.$$

$$f(w)\le \delta |w|+C_{\delta }|w{|}^{2_s^{\ast }-1}\text{ and }F(w)\le \delta |w{|}^2+C_{\delta }|w{|}^{2_s^{\ast }}.$$

$$I_m(w)\ge \frac{1}{4}\Vert w{\Vert }_{X^{1,s}({\mathbb{R}}_{+}^{N+1})}^2-C\Vert w{\Vert }_{X^{1,s}({\mathbb{R}}_{+}^{N+1})}^{2_s^{\ast }},$$

$$w_{R,T}(x,y)=\{\begin{array}{c}T,\text{ if }(x,y)\in B_R^{+}(0),\text{hfill}\\ T(R+1-\text{(}|x{|}^2+y^2{)}^{1/2}),\text{ if }(x,y)\in B_{R+1}^{+}(0)\backslash B_R^{+}(0),\text{hfill}\\ 0,\text{ if }(x,y)\in {\mathbb{R}}_{+}^{N+1}\backslash B_{R+1}^{+}(0),\text{hfill}\end{array}$$

$$\begin{gathered}{\text{ }}{I_{m}}({w_{R,T}}(x/\theta,y/\theta))\hfill\\ =\frac{{{k_{s}}}}{2}{\theta^{N-2s}}\int_{\mathbb{R}_{+}^{N+1}}{{y^{1-2s}}|\nabla{w_{R,T}}{|^{2}}}dxdy+{\theta^{N}}\Bigl{[}{\frac{m}{2}\int_{{\mathbb{R}^{N}}}{w_{R,T}^{2}(x,0)}dx-\int_{{\mathbb{R}^{N}}}{F({w_{R,T}}(x,0))}dx}\Bigr{]}\hfill\\ \leq\frac{{{k_{s}}}}{2}{\theta^{N-2s}}{T^{2}}\int_{B_{R+1}^{+}(0)\backslash B_{R}^{+}(0)}{{y^{1-2s}}}dxdy+{\theta^{N}}\Bigl{[}{\frac{m}{2}\int_{\Gamma_{R+1}^{0}(0)}{w_{R,T}^{2}(x,0)}dx-\frac{1}{{2_{s}^{*}}}\int_{\Gamma_{R}^{0}(0)}{w_{R,T}^{2_{s}^{*}}(x,0)}dx}\Bigr{]}\hfill\\ \leq C{\theta^{N-2s}}{T^{2}}\int_{R}^{R+1}{{r^{N+1-2s}}}dr+{\theta^{N}}\Bigl{[}{C\Bigl{(}{\frac{m}{2}{T^{2}}-\frac{1}{{2_{s}^{*}}}{T^{2_{s}^{*}}}}\Bigr{)}{R^{N}}+C{T^{2}}({{(R+1)}^{N}}-{R^{N}})}\Bigr{]}\hfill\\ \leq C{T^{2}}{R^{N+1-2s}}{\theta^{N-2s}}+\Bigl{(}{C\Bigl{(}{\frac{m}{2}{T^{2}}-\frac{1}{{2_{s}^{*}}}{T^{2_{s}^{*}}}}\Bigr{)}{R^{N}}+C{T^{2}}{R^{N-1}}}\Bigr{)}{\theta^{N}}.\hfill\\ \end{gathered}$$

$$c_m:=\underset{\gamma \in {\Gamma }_m}{\inf }\underset{t\in [0,1]}{\sup }{I}_m(\gamma (t)),$$

$${\Gamma }_m:=\left\{\gamma \in C([0,1],X^{1,s}({\mathbb{R}}_{+}^{N+1})):\gamma (0)=0\text{ and }{I}_m(\gamma (1))<0\right\}.$$

$$b_m:=\inf \{I_m(w):w\in X^{1,s}({\mathbb{R}}_{+}^{N+1})\backslash \{0\}\text{ be a nontrivial solution of eqref3.1}\}.$$

Full text

[TABLE]

where $m>0$, $N>2s$, ${(-\Delta)^{s}}$, $s\in(0,1)$ is the fractional Laplacian. Using minimax arguments, we obtain a positive ground state solution under general conditions on $f$ which we believe to be almost optimal.

**Key words **: ground state solution; fractional Schrödinger equation; critical growth.

**2010 Mathematics Subject Classification **: Primary 35J20, 35J60, 35J92

Yi He 111Corresponding Author: Yi He. Email addresses: heyi19870113@163.com (Y. He).

1. Introduction and Main Result

We cunsider the following fractional Schrödinger equation:

[TABLE]

where $m>0$, $N>2s$, ${(-\Delta)^{s}}$, $s\in(0,1)$ is the fractional Laplacian. The nonlinearity $f:\mathbb{R}\to\mathbb{R}$ is a continuous function. Since we are looking for positive solutions, we assume that $f(t)=0$ for $t<0$. Furthermore, we need the following conditions:

$(f_{1})$ $\mathop{\lim}\limits_{t\to{0^{+}}}f(t)/t=0$;

$(f_{2})$ $\mathop{\lim}\limits_{t\to+\infty}f(t)/{t^{2_{s}^{*}-1}}=1$ where $2_{s}^{*}=2N/(N-2s)$;

$(f_{3})$ $\exists\lambda>0$ and $2<q<{2_{s}^{*}}$ such that $f(t)\geq\lambda{t^{q-1}}+{t^{2_{s}^{*}-1}}$ for $t\geq 0$.

Note that, for the case $s=1$, $(f_{1})$-$(f_{3})$ were first introduced by J. Zhang, Z. Chen and W. Zou [25]. This hypothesis can be regarded as an extension of the celebrated Berestycki-Lions’ type nonlinearity (see [5, 6]) to the fractional Schrödinger equations with critical growth.

Equation (1.1) has been derived as models of many physical phenomena, such as phase transition, conservation laws, especially in fractional quantum mechanics, etc., [16]. (1.1) was introduced by N. Laskin [19, 20] as an extension of the classical nonlinear Schrödinger equations $s=1$ in which the Brownian motion of the quantum paths is replaced by a Lévy flight. We refer to [15] for more physical backgrounds.

In recent years, the study of fractional Schrödinger equations has attracted much attention from many mathematicians. In [9, 10, 23], L. Caffarelli, L. Silvestre $et~{}al$ investigated free boundary problems of fractional Schrödinger equations and obtained some regularity estimates. In [7, 8], X. Cabré and Y. Sire studied the existence, uniqueness, symmetry, regularity, maximum principle and qualitative properties of solutions to the fractional Schrödinger equations in the whole space. For more results, we refer to [1, 3, 4, 12, 15, 16, 18, 22].

Our main result is as follows:

Theorem 1.1**.**

Assume that the nonlinearity $f$ satisfies $(f_{1})$-$(f_{3})$. If $N\geq 4s$, $2<q<{2_{s}^{*}}$ or $2s<N<4s$, $4s/(N-2s)<q<2_{s}^{*}$, then for every $\lambda>0$, (1.1) possesses a positive ground state solution. Moreover, the same conclusion holds provided that $2s<N<4s$, $2<q\leq 4s/(N-2s)$ and $\lambda>0$ sufficiently large.

We note that, to the best of our knowledge, there is no result on the existence of positive ground state solutions for fractional Schrödinger equation under $(f_{1})$-$(f_{3})$.

The proof of Theorem 1.1 is based on variational method. The main difficulties lie in two aspects: (i) The facts that the nonlinearity $f(u)$ does not satisfy $({\text{AR}})$ condition and the function $f(s)/s$ is not increasing for $s>0$ prevent us from obtaining a bounded Palais-Smale sequence ((PS) sequence in short) and using the Nehari manifold respectively. (ii) The unboundedness of the domain $\mathbb{R}^{N}$ and the nonlinearity $f(u)$ with critical growth lead to the lack of compactness.

To complete this section, we sketch our proof.

To treat the nonlocal problem (1.1), we use the L. Caffarelli and L. Silvestre extension method [11] to study a corresponding extension problem

[TABLE]

with the corresponding functional

[TABLE]

where $F(s):=\int_{0}^{s}{f(t)}dt$ and ${X^{1,s}}(\mathbb{R}_{+}^{N+1})$ is defined as the completion of $C_{0}^{\infty}(\overline{\mathbb{R}_{+}^{N+1}})$ under the norm

[TABLE]

Motivated by J. Hirata, N. Ikoma and K. Tanaka [17], by applying the General Minimax principle (Theorem 2.8 of [24]) to the composite functional

[TABLE]

we construct a bounded ${{\text{(PS)}}_{{c_{m}}}}$ sequence $\{{w_{n}}\}_{n=1}^{\infty}\subset{X^{1,s}}(\mathbb{R}_{+}^{N+1})$ with an extra property ${P_{m}}({w_{n}})\to 0$ as $n\to\infty$ where $c_{m}$ is the mountain pass level of $I_{m}$ and ${P_{m}}(w)=0$ is the Pohozaev’s identity of (1.2) (Proposition 3.2 below). Proceeding by standard arguments, the existence of ground state solutions for (1.2) follows.

This paper is organized as follows, in Section 2, we give some preliminary results. In Section 3, we prove the main result Theorem 1.1.

2. Preliminaries

In this section, we collect some preliminary results. Recall that for $s\in(0,1)$, ${D^{s}}({\mathbb{R}^{N}})$ is defined by the completion of $C_{0}^{\infty}({\mathbb{R}^{N}})$ with respect to the Gagliardo norm

[TABLE]

and the embedding ${D^{s}}({\mathbb{R}^{N}})\hookrightarrow{L^{2_{s}^{*}}}({\mathbb{R}^{N}})$ is continuous, that is

[TABLE]

by Theorem 1 of [21]. The fractional Sobolev space ${H^{s}}({\mathbb{R}^{N}})$ is defined by

[TABLE]

endowed with the norm

[TABLE]

For $N>2s$, we see from Lemma 2.1 of [1] that

[TABLE]

An important feature of the operator ${(-\Delta)^{s}}(0<s<1)$ is its nonlocal character. A common approach to deal with this problem was proposed by L. Caffarelli and L. Silvestre [11], allowing to transform (1.1) into a local problem via the Dirichlet-Neumann map in the domain $\mathbb{R}_{+}^{N+1}:=\{(x,t)\in{\mathbb{R}^{N+1}}:t>0\}$. For $u\in{D^{s}}({\mathbb{R}^{N}})$, the solution $w\in{X^{s}}(\mathbb{R}_{+}^{N+1})$ of

[TABLE]

is called $s$-harmonic extension of $u$, denoted by $w={E_{s}}(u)$. The $s$-harmonic extension and the fractional Laplacian have explicit expressions in terms of the Poisson and the Riesz kernels, respectively

[TABLE]

where

[TABLE]

with a constant $c(N,s)$ such that $\int_{{\mathbb{R}^{N}}}{P_{1}^{s}(x)}dx=1$ (see [18]).

Here, the space ${X^{s}}(\mathbb{R}_{+}^{N+1})$ is defined as the completion of $C_{0}^{\infty}(\overline{\mathbb{R}_{+}^{N+1}})$ under the norm

[TABLE]

From [4], the map ${E_{s}}(\cdot)$ is an isometry between ${D^{s}}({\mathbb{R}^{N}})$ and ${X^{s}}(\mathbb{R}_{+}^{N+1})$, i.e. for $w={E_{s}}(u)$,

[TABLE]

On the other hand, for a function $w\in{X^{s}}(\mathbb{R}_{+}^{N+1})$, we shall denote its trace on ${\mathbb{R}^{N}}\times\{0\}$ as $u(x):={\text{Tr}}(w)=w(x,0)$. This trace operator is also well defined and it satisfies

[TABLE]

Lemma 2.1**.**

(Theorem 2.1 of [4]) For every $w\in{X^{s}}(\mathbb{R}_{+}^{N+1})$, it holds that

[TABLE]

where $u={\text{Tr}}(w)$. The best constant takes the exact value

[TABLE]

and it is achieved when $u_{\delta}$ takes the form

[TABLE]

for some $\delta>0$ and $w_{\delta}={E_{s}}(u_{\delta})$.

3. Proof of the main results

In view of [11], (1.1) can be transformed into

[TABLE]

with the corresponding functional

[TABLE]

In view of [12, 22], if $w\in{X^{1,s}}(\mathbb{R}_{+}^{N+1})$ is a weak solution to (3.1), the following Pohozaev’s identity holds:

[TABLE]

Lemma 3.1**.**

${I_{m}}$* possesses the Mountain-Pass geometry (see [2]), i.e.

$(i)$ There exist ${\rho_{0}},{\alpha_{0}}>0$ such that ${I_{m}}(w)\geq{\alpha_{0}}$ for all $w\in{X^{1,s}}(\mathbb{R}_{+}^{N+1})$ with ${\|w\|_{{X^{1,s}}(\mathbb{R}_{+}^{N+1})}}={\rho_{0}}$.

$(ii)$ $\exists{w_{0}}\in{X^{1,s}}(\mathbb{R}_{+}^{N+1})$ such that ${I_{m}}({w_{0}})<0$.*

Proof.

$(i)$ By $(f_{1})$ and $(f_{2})$, $\forall\delta>0$, $\exists{C_{\delta}}>0$ such that

[TABLE]

Choosing $\delta=m/4$ in (3.3), we see from Lemma 2.1 that

[TABLE]

then taking ${\rho_{0}},{\alpha_{0}}>0$ small, $(i)$ holds.

$(ii)$ For $R>0$, $T>0$, we define

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then ${w_{R}}\in{X^{1,s}}(\mathbb{R}_{+}^{N+1})$. By $(f_{3})$ and the polar coordinate transformation, we have

[TABLE]

Choosing a large ${T_{0}}>0$ such that $\frac{m}{2}T_{0}^{2}-\frac{1}{{2_{s}^{*}}}T_{0}^{2_{s}^{*}}<0$, then we can choose a large ${R_{0}}>0$ such that C\Bigl{(}{\frac{m}{2}T_{0}^{2}-\frac{1}{{2_{s}^{*}}}T_{0}^{2_{s}^{*}}}\Bigr{)}R_{0}^{N}+CT_{0}^{2}R_{0}^{N-1}<0, at last, we select a large $\bar{\theta}>0$ to ensure that ${I_{m}}({w_{{R_{0}},{T_{0}}}}(x/\bar{\theta},y/\bar{\theta}))<0$, ${w_{{R_{0}},{T_{0}}}}$ is the desired $w_{0}$. ∎

Hence we define the Mountain-Pass level of ${I_{m}}$:

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where the set of paths is defined as

[TABLE]

By Lemma 3.1(i), we see that ${c_{m}}>0$. Moreover, we denote

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Next, we will construct a (PS) sequence $\{{w_{n}}\}_{n=1}^{\infty}$ for $I_{m}$ at the level $c_{m}$ that satisfies ${P_{m}}({w_{n}})\to 0$ as $n\to\infty$, i.e.

Proposition 3.2**.**

There exists a sequence $\{{w_{n}}\}_{n=1}^{\infty}$ in ${X^{1,s}}(\mathbb{R}_{+}^{N+1})$ such that, as $n\to\infty$,

[TABLE]

Proof.

Define the map $\Phi:\mathbb{R}\times{X^{1,s}}(\mathbb{R}_{+}^{N+1})\to{X^{1,s}}(\mathbb{R}_{+}^{N+1})$ for $\theta\in\mathbb{R}$, $w\in{X^{1,s}}(\mathbb{R}_{+}^{N+1})$ and $(x,y)\in\mathbb{R}_{+}^{N+1}$ by $\Phi(\theta,w)=w({e^{-\theta}}x,{e^{-\theta}}y)$. For every $\theta\in\mathbb{R}$, $w\in{X^{1,s}}(\mathbb{R}_{+}^{N+1})$, the functional ${I_{m}}\circ\Phi$ is computed as

[TABLE]

By Lemma 3.1, $({I_{m}}\circ\Phi)(\theta,w)>0$ for all $(\theta,w)$ with $|\theta|$, ${\|w\|_{{X^{1,s}}(\mathbb{R}_{+}^{N+1})}}$ small and $({I_{m}}\circ\Phi)(0,{w_{0}})<0$, i.e. ${I_{m}}\circ\Phi$ possesses the Mountain-Pass geometry in $\mathbb{R}\times{X^{1,s}}(\mathbb{R}_{+}^{N+1})$. The Mountain-Pass level of ${I_{m}}\circ\Phi$ is defined by

[TABLE]

where the set of paths is

[TABLE]

As ${\Gamma_{m}}=\{{\Phi\circ\tilde{\gamma}:\tilde{\gamma}\in{{\tilde{\Gamma}}_{m}}}\}$, the Mountain-Pass levels of ${I_{m}}$ and ${I_{m}}\circ\Phi$ coincide, i.e. ${c_{m}}={{\tilde{c}}_{m}}$.

By the General Minimax principle (Theorem 2.8 of [24]), there exists a sequence $\{({\theta_{n}},{v_{n}})\}_{n=1}^{\infty}$ in $\mathbb{R}\times{X^{1,s}}(\mathbb{R}_{+}^{N+1})$ such that as $n\to\infty$,

[TABLE]

[TABLE]

[TABLE]

Indeed, set $\varepsilon={\varepsilon_{n}}:=1/{n^{2}}$, $\delta={\delta_{n}}:=1/n$ in Theorem 2.8 of [24], (3.9), (3.10) are direct conclusions from $(a)$, $(c)$ in Theorem 2.8 of [24]. By (3.4) and (3.5), for $\varepsilon={\varepsilon_{n}}:=1/{n^{2}}$, $\exists{\gamma_{n}}\in{\Gamma_{m}}$, such that $\mathop{\sup}\limits_{t\in[0,1]}{I_{m}}({\gamma_{n}}(t))\leq{c_{m}}+1/{n^{2}}$. Set ${{\tilde{\gamma}}_{n}}(t)=(0,{\gamma_{n}}(t))$, then

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From $(b)$ in Theorem 2.8 of [24], $\exists({\theta_{n}},{v_{n}})\in\mathbb{R}\times{X^{1,s}}(\mathbb{R}_{+}^{N+1})$ such that ${\text{dist}}(({\theta_{n}},{v_{n}}),(0,{\gamma_{n}}(t)))\leq 2/n$, then (3.11) holds.

For every $(h,w)\in\mathbb{R}\times{X^{1,s}}(\mathbb{R}_{+}^{N+1})$,

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Taking $h=1$, $w=0$ in (3.12), we have

[TABLE]

For every $v\in{X^{1,s}}(\mathbb{R}_{+}^{N+1})$, set $w(x,y)=v({e^{{\theta_{n}}}}x,{e^{{\theta_{n}}}}y)$, $h=0$ in (3.12), by (3.11), we get

[TABLE]

Denote ${w_{n}}:=\Phi({\theta_{n}},{v_{n}})$ in (3.9), (3.13) and (3.14), we get (3.6). ∎

Lemma 3.3**.**

Every sequence $\{{w_{n}}\}_{n=1}^{\infty}$ satisfying (3.6) is bounded in ${X^{1,s}}(\mathbb{R}_{+}^{N+1})$.

Proof.

By (3.6),

[TABLE]

we get the upper bound of ${\|{{w_{n}}}\|_{{X^{s}}(\mathbb{R}_{+}^{N+1})}}$, then by Lemma 2.1, we see that $\{{w_{n}}(x,0)\}$ is bounded in ${L^{2_{s}^{*}}}({\mathbb{R}^{N}})$. From (3.6) and (3.3), we see that

[TABLE]

hence $\{{w_{n}}\}$ is bounded in ${{X^{1,s}}(\mathbb{R}_{+}^{N+1})}$. ∎

For the Mountain-Pass level $c_{m}$, we have the following estimate:

Lemma 3.4**.**

If $N\geq 4s$, $2<q<2_{s}^{*}$ or $2s<N<4s$, $4s/(N-2s)<q<2_{s}^{*}$, then for all $\lambda>0$, ${c_{m}}<\frac{s}{N}{({k_{s}}S(s,N))^{N/(2s)}}$. Moreover, if $2s<N<4s$ and $2<q\leq 4s/(N-2s)$, then for $\lambda>0$ sufficiently large, the same conclusion holds.

Proof.

Let ${\phi_{0}}\in{C^{\infty}}({\mathbb{R}_{+}})$ satisfying

[TABLE]

and denote ${w_{\delta}}$ the $s$-harmonic extension of ${u_{\delta}}$ in Lemma 2.1. Denote

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by [3, 14], we see that

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and for any $p\in[2,2_{s}^{*})$,

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By $(f_{3})$,

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In view of (3.17) and (3.18), for $\delta>0$ small, ${g_{\delta}}(t)$ has a unique critical point ${t_{\delta}}>0$ which corresponds to its maximum. Therefore, we check from ${g^{\prime}_{\delta}}({t_{\delta}})=0$ that

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then we see from (3.17) and (3.18) that

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By (3.17) and (3.18), we get

[TABLE]

where ${t^{\prime}_{\delta}}=\|{{v_{\delta}}}\|_{{X^{s}}(\mathbb{R}_{+}^{N+1})}^{1/s}$ is the maximum point of ${h_{\delta}}(t)(t>0)$.

By (3.19) and (3.20), we have

[TABLE]

Next, we distinguish the following cases:

(i) If $N>4s$, then $q>2>N/(N-2s)$, by (3.18) and (3.21), we get

[TABLE]

In view of $(2N-(N-2s)q)/2<2s<(N-2s)$, we get the conclusion for $\delta>0$ small.

(ii) If $N=4s$, then $q>2=N/(N-2s)$, by (3.18) and (3.21), we have

[TABLE]

Since $4s-sq<2s$, we get the conclusion for $\delta>0$ small.

(iii) If $2s<N<4s$ and $N/(N-2s)<q<2_{s}^{*}$, we see from (3.18) and (3.21) that

[TABLE]

If $4s/(N-2s)<q<2_{s}^{*}$, then $(N-2s)>(2N-(N-2s)q)/2$, we get the conclusion for $\delta>0$ small. If $N/(N-2s)<q\leq 4s/(N-2s)$, then $(N-2s)\leq(2N-(N-2s)q)/2$, we choose $\lambda={\delta^{-\theta}}$ with $\theta>(2N-(q+2)(N-2s))/2>0$, we still get the conclusion for $\delta>0$ small.

(iv) If $2s<N<4s$ and $q=N/(N-2s)$, (3.18) and (3.21) yield

[TABLE]

Since $(N-2s)<N/2$, we choose $\lambda={\delta^{-\theta}}$ with $\theta>2s-(N/2)$, we get the conclusion for $\delta>0$ small.

(v) If $2s<N<4s$ and $2<q<N/(N-2s)$, (3.18) and (3.21) show that

[TABLE]

We choose $\lambda={\delta^{-\theta}}$ with $\theta>(q-2)(N-2s)/2$, we get the conclusion for $\delta>0$ small. ∎

Lemma 3.5**.**

There is a sequence $\{{x_{n}}\}_{n=1}^{\infty}\subset{\mathbb{R}^{N}}$ and $R>0$, $\beta>0$ such that

[TABLE]

where $\{{w_{n}}\}_{n=1}^{\infty}$ is the sequence given in (3.6).

Proof.

Assuming on the contrary that the lemma does not hold, then by Lemma 2.2 of [16], it follows that

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Since $\langle{I^{\prime}_{m}({w_{n}}),{w_{n}}}\rangle=o(1)$ and ${I_{m}}({w_{n}})\to{c_{m}}$, by $(f_{1})$ and $(f_{2})$, we get

[TABLE]

[TABLE]

Let $l\geq 0$ be such that

[TABLE]

It is trivial that $l>0$, otherwise ${\|{{w_{n}}}\|_{{X^{1,s}}(\mathbb{R}_{+}^{N+1})}}\to 0$ as $n\to\infty$ which contradicts ${c_{m}}>0$. By (3.22), we get

[TABLE]

By Lemma 2.1, we see that

[TABLE]

Letting $n\to\infty$ in (3.25), we get $l\geq{({k_{s}}S(s,N))^{N/2s}}$, then by (3.24), ${c_{m}}\geq\frac{s}{N}{({k_{s}}S(s,N))^{N/(2s)}}$, which contradicts Lemma 3.4. ∎

Proof of Theorem 1.1.

Let $\{{w_{n}}\}_{n=1}^{\infty}$ be the sequence given in (3.6) and denote ${\tilde{w}_{n}}(x,y)={w_{n}}(x+{x_{n}},y)$, where $\{{x_{n}}\}_{n=1}^{\infty}$ is the sequence given in Lemma 3.5. By Lemma 3.3 and Lemma 3.5, we see that, up to a subsequence, $\exists\tilde{w}(x,y)\in{X^{1,s}}(\mathbb{R}_{+}^{N+1})\backslash\{0\}$ such that ${{\tilde{w}}_{n}}(x,y)\rightharpoonup\tilde{w}(x,y)$ in ${X^{1,s}}(\mathbb{R}_{+}^{N+1})$, ${{\tilde{w}}_{n}}(x,0)\to\tilde{w}(x,0)$ in $L_{{\text{loc}}}^{p}({\mathbb{R}^{N}})(1\leq p<2_{s}^{*})$, ${{\tilde{w}}_{n}}(x,0)\to\tilde{w}(x,0)$ a.e. in ${\mathbb{R}^{N}}$ and ${\tilde{w}}$ satisfies (3.1). Hence

[TABLE]

For any $\bar{w}\in{X^{1,s}}(\mathbb{R}_{+}^{N+1})\backslash\{0\}$ a solution of (3.1), we set the path

[TABLE]

Since

[TABLE]

there exists a ${T_{0}}>0$ large such that ${I_{m}}(\bar{\gamma}({T_{0}}))<0$ and ${I_{m}}(\bar{\gamma}(t))$ achieve the strict global maximum at $t=1$. By the definition of $c_{m}$, we see that ${I_{m}}(\bar{w})\geq{c_{m}}$. Since $\bar{w}$ is arbitrary, we see that ${b_{m}}\geq{c_{m}}$. Hence, we conclude from (3.26) that ${I_{m}}(\tilde{w})={c_{m}}={b_{m}}$ and ${I^{\prime}_{m}}(\tilde{w})=0$. Arguing as Proposition 4.1.1 of [13], we see that $\tilde{w}\in{L^{\infty}}({\mathbb{R}^{N}})$. Since $\tilde{w}$ is nonnegative and nontrivial and $f$ is continuous, we can apply the Harnack’s inequality in Lemma 4.9 of [7] to conclude that $\tilde{w}$ is positive, that is, $\tilde{w}$ is in fact a positive ground state solution of (3.1), hence, $u(x):=\tilde{w}(x,0)$ is a positive ground state solution of (1.1).

∎