The Existence of Solution of a Critical Fractional Equation
Abstract: In this paper, we study the existence of solution of a critical fractional equation; we will use a variational approach to find the solution. Firstly, we will find a suitable functional to our problem; next, by using the classical concept and properties of the genus, we construct a mini-max class of critical points.

1. Introduction

In this paper, we focus our attention on the following problem:

$\left\{\begin{array}{l}{\left(-\Delta \right)}^{s}u=\lambda V\left(x\right){|u|}^{p-1}+\beta K\left(x\right){|u|}^{{2}_{s}^{*}-1}\text{in}\text{\hspace{0.17em}}\Omega \\ u=0\text{}\text{ }\text{in}\text{\hspace{0.17em}}{R}^{n}\\Omega \end{array}$ (1.1)

where $\Omega$ is a bounded domain in ${R}^{n}$ , $\lambda >0$ , $0 and $n>2s$ , $1 , $K\left(x\right)\in C\left({R}^{n}\right)\cap {L}^{\infty }\left({R}^{n}\right)$ , $V\left(x\right)\ge 0$ and $V\left(x\right)\in C\left({R}^{n}\right)\cap {L}^{q}\left({R}^{n}\right)$ with $q=\frac{{2}_{s}^{*}}{{2}_{s}^{*}-p}$ here ${\left(-\Delta \right)}^{s}$ denotes the fractional Laplace operator defined, up to a normalization factor, by

${\left(-\Delta \right)}^{s}u\left(x\right)=\underset{{R}^{n}}{\int }\frac{u\left(x\right)-u\left(y\right)}{{|x-y|}^{n+2s}}\text{d}y,\text{}x\in {R}^{n}$. (1.2)

The aim of this paper is to study the existence of solutions, we will see that if $1 , then by concentration-compactness principle, together with mini-max arguments, we can prove the existence of solutions for (1.1). We now summarize the main result of the paper.

Theorem 1.1. Let $1 , $K\left(x\right)\in C\left({R}^{n}\right)\cap {L}^{\infty }\left({R}^{n}\right)$ and $0\le V\left(x\right)\in C\left({R}^{n}\right)\cap {L}^{q}\left({R}^{n}\right)$ with $q=\frac{{2}_{s}^{*}}{{2}_{s}^{*}-p}$. Moreover, $V\left(x\right)>0$ is bounded on $\Omega$. Then

1) For any $\lambda >0$ , there exists $\stackrel{˜}{\beta }>0$ , then for any $0<\beta <\stackrel{˜}{\beta }$ , (1.1) has a consequence of weak solutions $\left\{{u}_{n}\right\}$.

2) For any $\beta >0$ , there exist $\stackrel{˜}{\lambda }>0$ , then for any $0<\lambda <\stackrel{˜}{\lambda }$ , (1.1) has a consequence of weak solutions $\left\{{u}_{n}\right\}$.

We denote by ${H}^{s}\left({R}^{n}\right)$ the usual fractional Sobolev space endowed with the so-called Gagliardo norm

${‖u‖}_{{H}^{s}\left({R}^{n}\right)}={‖u‖}_{{L}^{2}\left({R}^{n}\right)}+{\left(\underset{{R}^{n}×{R}^{n}}{\int }\frac{{|u\left(x\right)-u\left(y\right)|}^{2}}{{|x-y|}^{n+2s}}\text{d}x\text{d}y\right)}^{\frac{1}{2}},$ (1.3)

Then we defined

${X}_{0}^{s}\left(\Omega \right)=\left\{u\in {H}^{s}\left({R}^{n}\right):u=0\text{}a.e.\text{in}\text{\hspace{0.17em}}{R}^{n}\\Omega \right\}$ (1.4)

endowed with the norm

${‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}={\left(\underset{{R}^{n}×{R}^{n}}{\int }\frac{{|u\left(x\right)-u\left(y\right)|}^{2}}{{|x-y|}^{n+2s}}\text{d}x\text{d}y\right)}^{\frac{1}{2}},$ (1.5)

we refer to [1] for a general definition of ${X}_{0}^{s}\left(\Omega \right)$ and its properties.

Observe that by [ [2] , Proposition 3.6] we have the following identity

${‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}={‖{\left(-\Delta \right)}^{\frac{s}{2}}u‖}_{{L}^{2}\left({R}^{n}\right)}.$ (1.6)

In this work, the Sobolev constant is given by (can be seen in [ [3] , theorem 7.58])

$S\left(n,s\right):=\underset{u\in {H}^{s}\left({R}^{n}\right)\\left\{0\right\}}{\mathrm{inf}}{Q}_{n,s}\left(u\right)>0,$ (1.7)

where

${Q}_{n,s}\left(u\right):=\frac{\underset{{R}^{n}×{R}^{n}}{\int }\frac{|u\left(x\right)-u{\left(y\right)}^{2}|}{{|x-y|}^{n+2s}}\text{d}x\text{d}y}{{\left(\underset{{R}^{n}×{R}^{n}}{\int }{|u\left(x\right)|}^{{2}_{s}^{*}}\text{d}x\right)}^{\frac{2}{{2}_{s}^{*}}}}\text{,}u\in {H}^{s}\left({R}^{n}\right)$ (1.8)

2. Statements of the Result

We will use a variational approach to find a solution of (1.1). Firstly, we will associate a suitable functional to our problem, the Euler-Lagrange functional related to problem (1) is given by $J:{X}_{0}^{s}\left(\Omega \right)\to R$ defined as follow

$J\left({u}_{n}\right)=\frac{1}{2}{‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\frac{\lambda }{p}\underset{\Omega }{\int }V\left(x\right){|{u}_{n}|}^{p}\text{d}x-\frac{\beta }{{2}_{s}^{*}}\underset{\Omega }{\int }K\left(x\right){|{u}_{n}|}^{{2}_{s}^{*}}\text{d}x.$ (2.1)

To proof that J satisfy the Palais Smale condition at level c, we need the following lemma.

Lemma 2.1 [4] Letting $\varphi$ be a regular function that satisfies that for some $\stackrel{˜}{c}>0$

$|\varphi \left(x\right)|\le \frac{\stackrel{˜}{c}}{1+{|x|}^{n+s}}\text{,}x\in {R}^{n}$ (2.2)

and

$|\nabla \varphi \left(x\right)|\le \frac{\stackrel{˜}{c}}{1+{|x|}^{n+s}}\text{,}x\in {R}^{n}$ (2.3)

Let $B:{X}_{0}^{\frac{s}{2}}\left(\Omega \right)×{X}_{0}^{\frac{s}{2}}\left(\Omega \right)\to R$ be a bilinear form defined by

$B\left(f,g\right)\left(x\right):=2\underset{R}{\int }\frac{\left(f\left(x\right)-f\left(y\right)\right)\left(g\left(x\right)-g\left(y\right)\right)}{{|x-y|}^{n+s}}\text{d}y.$ (2.4)

then, for every $s\in \left(0,1\right)$ , there exist positive constant ${c}_{1}$ and ${c}_{2}$ , such that for $x\in {R}^{n}$ , one has

$|{\left(-\Delta \right)}^{\frac{s}{2}}\varphi \left(x\right)|\le \frac{c}{1+{|x|}^{n+s}}$ and $|B\left(\varphi ,\varphi \right)\left(x\right)|\le \frac{c}{1+{|x|}^{n+s}}$. (2.5)

To establish the next auxiliary result we consider a radial, nonincreasing cut-off function

$\varphi \in {C}_{0}^{\infty }\left({R}^{n}\right)$ and ${\varphi }_{\epsilon }\left(x\right):=\varphi \left(\frac{x}{\epsilon }\right)$ (2.6)

Lemma 2.2. [4] Letting $\left\{{u}_{m}\right\}$ be a uniformly bounded in ${X}_{0}^{s}\left(\Omega \right)$ and ${\varphi }_{\epsilon }\in {C}_{0}^{\infty }\left({R}^{n}\right)$ the function defined in (2.6). Then,

$\underset{\epsilon \to 0}{\mathrm{lim}}\underset{m\to 0}{\mathrm{lim}}|\underset{{R}^{n}}{\int }{u}_{m}\left(x\right){\left(-\Delta \right)}^{\frac{s}{2}}{\varphi }_{\epsilon }\left(x\right){\left(-\Delta \right)}^{\frac{s}{2}}{u}_{m}\left(x\right)\text{d}x|=0.$ (2.7)

Lemma 2.3. [4] With the same assumptions of Lemma 2.8 we have that

$\underset{\epsilon \to 0}{\mathrm{lim}}\underset{m\to 0}{\mathrm{lim}}|\underset{{R}^{n}}{\int }{\left(-\Delta \right)}^{\frac{s}{2}}{u}_{m}\left(x\right)\text{d}xB\left({u}_{m},{\varphi }_{\epsilon }\right)\left(x\right)|=0.$ (2.8)

where B is defined in (2.4).

Lemma 2.4. [5] (Minimax principle) Assume that $E\in C\left(X,ℝ\right)$ , and $\mathcal{A}$ is a family of nonempty subset of X, denote

$c=\underset{A\in \mathcal{A}}{\mathrm{inf}}\underset{x\in A}{\mathrm{sup}}E\left(x\right)$ (2.9)

If the following conditions holds:

1) c is a finite real number;

2) there exists an $\stackrel{¯}{\epsilon }>0$ , such that $\mathcal{A}$ is invariant with respect to the family of mappings;

$\mathcal{T}=\left\{T\in \left(X,X\right)|T\left(x\right)=x,\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}E\left(x\right) , (2.10)

that is, for any $T\in \mathcal{T}$ , there holds

$A\in \mathcal{A}⇒\mathcal{T}\left(A\right)\in \mathcal{A}$

Then, E possesses a ${\left(PS\right)}_{c}$ sequence at level c define as (6.1.1); Furthermore, if E satisfies the ${\left(PS\right)}_{c}$ condition (or the ${\left(PS\right)}_{c}$ condition at level c), then c is a critical value of E.

3. Proof of Theorem 1.1

Firstly, recalling that J is said to satisfy the Palais Smale condition at level c if any sequence $\left\{{u}_{n}\right\}\in {X}_{0}^{s}\left(\Omega \right)$ such that $J\left({u}_{n}\right)\to c$ and ${J}^{\prime }\left(u\right)\to 0$ has a convergent subsequence.

Lemma 3.1. The ${\left(PS\right)}_{c}$ sequence $\left\{{u}_{n}\right\}$ for J is bounded.

Proof. Note that $\left\{{u}_{n}\right\}\subset {X}_{0}^{s}\left(\Omega \right)$ satisfies

$\begin{array}{l}J\left({u}_{n}\right)=\frac{1}{2}{‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\frac{\lambda }{p}\underset{\Omega }{\int }V\left(x\right){|{u}_{n}|}^{p}\text{d}x-\frac{\beta }{{2}_{s}^{*}}\underset{\Omega }{\int }K\left(x\right){|{u}_{n}|}^{{2}_{s}^{*}}\text{d}x\\ \text{}=c+{o}_{n}\left(1\right)\end{array}$ (3.1)

and

$\begin{array}{l}〈{J}^{\prime }\left({u}_{n}\right),\varphi 〉=\underset{\Omega }{\int }{\left(-\Delta \right)}^{s}{u}_{n}\text{d}x-\lambda \underset{\Omega }{\int }V\left(x\right){|{u}_{n}|}^{p-2}u\varphi \text{d}x-\beta \underset{\Omega }{\int }K\left(x\right){|{u}_{n}|}^{{2}_{s}^{*}-2}u\varphi \text{d}x\\ \text{}={o}_{n}\left(1\right){‖\varphi ‖}_{{X}_{0}^{s}\left(\Omega \right)},\text{}\forall \varphi \in {X}_{0}^{s}\left(\Omega \right)\end{array}$ (3.2)

where ${o}_{n}\left(1\right)\to 0$ as $n\to \infty$. Choose $\varphi ={u}_{n}\in {X}_{0}^{s}\left(\Omega \right)$ as test function in (3.2), we get that

$\begin{array}{c}{o}_{n}\left(1\right){‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}=〈{J}^{\prime }\left({u}_{n}\right),{u}_{n}〉\\ ={‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\lambda \underset{\Omega }{\int }V\left(x\right){|{u}_{n}|}^{p}\text{d}x-\beta \underset{\Omega }{\int }K\left(x\right){|{u}_{n}|}^{{2}_{s}^{*}}\text{d}x\\ =c+{o}_{n}\left(1\right).\end{array}$ (3.3)

therefore, by (3.1) and (3.2), we have

$\begin{array}{l}c+{o}_{n}\left(1\right)-\frac{1}{{2}_{s}^{*}}{o}_{n}\left(1\right){‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}\\ =\frac{1}{2}{‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\frac{\lambda }{p}\underset{\Omega }{\int }V\left(x\right){|{u}_{n}|}^{p}\text{d}x-\frac{\beta }{{2}_{s}^{*}}\underset{\Omega }{\int }K\left(x\right){|{u}_{n}|}^{{2}_{s}^{*}}\text{d}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-\frac{1}{{2}_{s}^{*}}{‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\frac{\lambda }{{2}_{s}^{*}}\underset{\Omega }{\int }V\left(x\right){|{u}_{n}|}^{p}\text{d}x-\frac{\beta }{{2}_{s}^{*}}\underset{\Omega }{\int }K\left(x\right){|{u}_{n}|}^{{2}_{s}^{*}}\text{d}x\\ \ge \frac{s}{n}{‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\left(\frac{\lambda }{p}-\frac{1}{{2}_{s}^{*}}\right){‖V\left(x\right)‖}_{{L}^{q}\left(\Omega \right)}{‖{u}_{n}‖}_{{L}^{{2}_{s}^{*}}}^{p}\\ \ge \frac{s}{n}{‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\left(\frac{\lambda }{p}-\frac{1}{{2}_{s}^{*}}\right)S{\left(n,s\right)}^{-\frac{p}{2}}{‖V\left(x\right)‖}_{{L}^{q}\left(\Omega \right)}{‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}^{p}.\end{array}$ (3.4)

which yields the boundeness of $\left\{{u}_{n}\right\}$ in ${X}_{0}^{s}\left(\Omega \right)$ ,since $1.

If $K\left(x\right)\in {L}^{\infty }\left({ℝ}^{n}\right)$ , then for $2 , similar to the proof of $1 , we get

$c+{o}_{n}\left(1\right)+{o}_{n}\left(1\right){‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}\ge \left(\frac{p-2}{2p}\right){‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\frac{\left(p-{2}_{s}^{*}\right)\beta }{{2}_{s}^{*}}{S}^{-\frac{{2}_{s}^{*}}{2}}{‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}^{{2}_{s}^{*}}$

Which also yields the boundedness of ${\left(PS\right)}_{c}$ sequence $\left\{{u}_{n}\right\}$.

Lemma 3.2. Assume that $c<0$. Then

1) For any $\lambda >0$ , there exists ${\beta }_{0}>0$ , such that for any $0<\beta <{\beta }_{0}$ , then J satisfies ${\left(PS\right)}_{c}$.

2) For any $\beta >0$ there exists ${\lambda }_{0}>0$ such that for any $0<\lambda <{\lambda }_{0}$ , then J satisfies ${\left(PS\right)}_{c}$.

Proof. By Lemma3.1 $\left\{{u}_{n}\right\}$ is bounded in ${X}_{0}^{s}\left(\Omega \right)$ , up to a subsequence, we get that

${u}_{n}\to u$ $x\in {X}_{0}^{s}\left(\Omega \right)$.

${u}_{n}\to u$ $x\in {L}^{r}\left(\Omega \right)$ , $1\le r<{2}_{s}^{*}$. (3.5)

${u}_{n}\to u$ a.e. $x\in \Omega$.

Following [6] it is easy to prove that ${X}_{0}^{s}\left(\Omega \right)$ could also be the ${X}_{0}^{s}\left(\Omega \right)$ -norm. Applying [ [7] , Theorem1.5], we have that the exist an index. Set $I\subseteq N$ a sequence of point ${\left\{{x}_{k}\right\}}_{x\in I\subset \Omega }$ and two sequences of nonnegative real numbers ${\left\{{\mu }_{k}\right\}}_{k\in I},{\left\{{v}_{k}\right\}}_{k\in I}$ , such that

${|{\left(-\Delta \right)}^{\frac{s}{2}}{u}_{n}|}^{2}\to \mu {|{\left(-\Delta \right)}^{\frac{s}{2}}u|}^{2}+\underset{k\in I}{\sum }{\mu }_{k}{\delta }_{{x}_{k}}$. (3.6)

moreover

${|{u}_{n}|}^{{2}_{s}^{*}}\to \mu {|u|}^{{2}_{s}^{*}}+\underset{k\in I}{\sum }{v}_{k}{\delta }_{{x}_{k}}$. (3.7)

in the sense of measures, with

${v}_{k}\le S{\left(s,n\right)}^{-\frac{{2}_{s}^{*}}{2}}{\mu }_{k}^{\frac{{2}_{s}^{*}}{2}}$ for every $k\in I$ (3.8)

here ${\delta }_{{x}_{k}}$ denotes the Dirac Delta at ${x}_{k}$ , while $S\left(n,s\right)$ is the constant given in (1.7), we consider $\varphi \in {C}_{0}^{\infty }\left({R}^{n}\right)$ a nonincreasing cut-off function satisfying

$\varphi =1\text{in}\text{\hspace{0.17em}}{B}_{1}\left({x}_{{k}_{0}}\right)$ and $\varphi =0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{B}_{2}{\left({x}_{{k}_{0}}\right)}^{c}$ (3.9)

Set ${\varphi }_{\epsilon }\left(x\right)=\varphi \left(\frac{x}{\epsilon }\right),\text{\hspace{0.17em}}x\in {R}^{n}$ taking the derivative of (1.6), for any $u,\varphi \in {X}_{0}^{s}\left(\Omega \right)$. We obtain that

$\underset{{R}^{n}×{R}^{n}}{\int }\frac{\left(u\left(x\right)-u\left(y\right)\right)\left(\varphi \left(x\right)-\varphi \left(y\right)\right)}{{|x-y|}^{n+2s}}\text{d}x\text{d}y=\underset{{R}^{n}}{\int }\varphi \left(x\right){\left(-\Delta \right)}^{s}u\left(x\right)\text{d}x$ (3.10)

Then, taking ${\varphi }_{\epsilon }{u}_{n}$ as a test function in ${J}^{\prime }\left({u}_{n}\right)\to 0$

$\underset{n\to 0}{\mathrm{lim}}\underset{{ℝ}^{n}}{\int }{\varphi }_{\epsilon }{u}_{n}\left(-\Delta \right){u}_{n}\text{d}x-\left(\lambda \underset{{B}_{2\epsilon }\left({x}_{{k}_{0}}\right)}{\int }V\left(x\right){u}_{n}^{p}{\varphi }_{\epsilon }\text{d}x+\beta \underset{{B}_{2\epsilon }\left({x}_{{k}_{0}}\right)}{\int }K\left(x\right){u}_{n}^{{2}_{s}^{*}}{\varphi }_{\epsilon }\text{d}x\right)=0$ (3.11)

by (3.10), we have

$\begin{array}{l}\underset{n\to \infty }{\mathrm{lim}}\underset{{ℝ}^{n}}{\int }{u}_{n}\left(x\right){\left(-\Delta \right)}^{\frac{s}{2}}{u}_{n}\left(x\right){\left(-\Delta \right)}^{\frac{s}{2}}{\varphi }_{\epsilon }\left(x\right)\text{d}x\\ \text{ }-2\underset{{ℝ}^{n}}{\int }{\left(-\Delta \right)}^{\frac{s}{2}}{u}_{n}\left(x\right)\underset{{ℝ}^{n}}{\int }\frac{\left({\varphi }_{\epsilon }\left(x\right)-{\varphi }_{\epsilon }\left(y\right)\right)\left({u}_{n}\left(x\right)-{u}_{n}\left(y\right)\right)}{{|x-y|}^{n+s}}\text{d}x\text{d}y\\ =\underset{n\to \infty }{\mathrm{lim}}\lambda \underset{{B}_{2\epsilon }\left({x}_{{k}_{0}}\right)}{\int }V\left(x\right){|{u}_{n}|}^{p}\left(x\right){\varphi }_{\epsilon }\left(x\right)\text{d}x+\beta {\underset{{B}_{2\epsilon }\left({x}_{{k}_{0}}\right)}{\int }K\left(x\right)|{u}_{n}|}^{{2}_{s}^{*}}\left(x\right){\varphi }_{\epsilon }\left(x\right)\text{d}x\\ \text{}-\underset{{B}_{2\epsilon }\left({x}_{{k}_{0}}\right)}{\int }{\left({\left(-\Delta \right)}^{\frac{s}{2}}{u}_{n}\right)}^{2}{\varphi }_{\epsilon }\left(x\right)\text{d}x.\end{array}$ (3.12)

therefore, by (3.5) (3.6) and (3.7) we get

$\begin{array}{l}\underset{\epsilon \to 0}{\mathrm{lim}}\underset{n\to \infty }{\mathrm{lim}}\underset{{ℝ}^{n}}{\int }{u}_{n}\left(x\right){\left(-\Delta \right)}^{\frac{s}{2}}{u}_{n}\left(x\right){\left(-\Delta \right)}^{\frac{s}{2}}{\varphi }_{\epsilon }\left(x\right)\text{d}x\\ \text{ }-2\underset{{ℝ}^{n}}{\int }{\left(-\Delta \right)}^{\frac{s}{2}}{u}_{n}\left(x\right)\underset{{ℝ}^{n}}{\int }\frac{\left({\varphi }_{\epsilon }\left(x\right)-{\varphi }_{\epsilon }\left(y\right)\right)\left({u}_{n}\left(x\right)-{u}_{n}\left(y\right)\right)}{{|x-y|}^{n+s}}\text{d}x\text{d}y\\ =\underset{\epsilon \to 0}{\mathrm{lim}}\lambda \underset{{B}_{2\epsilon }\left({x}_{{k}_{0}}\right)}{\int }V\left(x\right){|{u}_{n}|}^{p}\left(x\right){\varphi }_{\epsilon }\left(x\right)\text{d}x+\underset{{B}_{2\epsilon }\left({x}_{{k}_{0}}\right)}{\int }{\varphi }_{\epsilon }\left(x\right)\text{d}v\\ \text{}-\beta \underset{{B}_{2\epsilon }\left({x}_{{k}_{0}}\right)}{\int }K\left(x\right){\varphi }_{\epsilon }\left(x\right)\text{d}\mu .\end{array}$ (3.13)

Since $\varphi$ is regular function with compact support, it is easy to see that it satisfies the hypothesis of Lemma 2.1, by Lemma 2.2 and Lemma 2.3 applied to the sequence $\left\{{u}_{n}\right\}$ , it follows that the left hand side of (3.13) goes to zero. We obtain that

$\begin{array}{l}\underset{\epsilon \to 0}{\mathrm{lim}}\left(\lambda \underset{{B}_{2\epsilon }\left({x}_{{k}_{0}}\right)}{\int }V\left(x\right){|{u}_{n}|}^{p}\left(x\right){\varphi }_{\epsilon }\left(x\right)\text{d}x+\underset{{B}_{2\epsilon }\left({x}_{{k}_{0}}\right)}{\int }{\varphi }_{\epsilon }\left(x\right)\text{d}v-\beta \underset{{B}_{2\epsilon }\left({x}_{{k}_{0}}\right)}{\int }K\left(x\right){\varphi }_{\epsilon }\left(x\right)\right)\text{d}\mu \\ =\beta K\left({x}_{{k}_{0}}\right){v}_{{k}_{0}}-{\mu }_{{k}_{0}}=0.\end{array}$ (3.14)

Clearly, if $K\left(x\right)\le 0$ , we get ${\mu }_{{k}_{0}}={v}_{{k}_{0}}=0$ ; if $K\left({x}_{{k}_{0}}\right)>0$ , by (3.8), we get ${v}_{{k}_{0}}=0$ or ${v}_{{k}_{0}}\ge {\left[\frac{S\left(n,s\right)}{\beta K\left({x}_{{k}_{0}}\right)}\right]}^{\frac{n}{2s}}$.

suppose that ${v}_{{k}_{0}}\ne 0$ , we know that

$0>c=\underset{n\to \infty }{\mathrm{lim}}\left[J\left({u}_{n}\right)-\frac{1}{{2}_{s}^{*}}〈{J}^{\prime }\left({u}_{n}\right),{u}_{n}〉\right]$ (3.15)

according to the embedded theorem, we have

$\begin{array}{l}0>c\ge \left(\frac{1}{2}-\frac{1}{{2}_{s}^{*}}\right){‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\left(\frac{\lambda }{p}-\frac{\lambda }{{2}_{s}^{*}}\right)\underset{\Omega }{\int }V\left(x\right){|{u}_{n}|}^{p}\text{d}x\\ \text{}=\frac{s}{n}{‖{u}_{n}‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\left(\frac{\lambda }{p}-\frac{\lambda }{{2}_{s}^{*}}\right)\underset{\Omega }{\int }V\left(x\right){|{u}_{n}|}^{p}\text{d}x\\ \text{}\ge \frac{s}{n}{S}^{-1}\left(n,s\right){‖{u}_{n}‖}_{{L}^{{2}_{s}^{*}}\left(\Omega \right)}^{2}-\left(\frac{\lambda }{p}-\frac{\lambda }{{2}_{s}^{*}}\right){S}^{-\frac{p}{2}}\left(n,s\right){‖V\left(x\right)‖}_{{L}^{q}\left(\Omega \right)}{‖{u}_{n}‖}_{{L}^{{2}_{s}^{*}}}^{p}.\end{array}$ (3.16)

This yields that

${‖u‖}_{{L}^{{2}_{s}^{*}}\left(\Omega \right)}\le C{\lambda }^{\frac{1}{2-p}}$. (3.17)

Thus, if ${v}_{{k}_{0}}\ge {\left[\frac{S\left(n,s\right)}{\beta K\left({x}_{{k}_{0}}\right)}\right]}^{\frac{n}{2s}}$ , we get that

$\begin{array}{l}0>c=\underset{n\to \infty }{\mathrm{lim}}\left[J\left({u}_{n}\right)-\frac{1}{{2}_{s}^{*}}〈{J}^{\prime }\left({u}_{n}\right),{u}_{n}〉\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\ge \left(\frac{1}{2}-\frac{1}{{2}_{s}^{*}}\right){‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}+\frac{s}{n}{\mu }_{{k}_{0}}-\left(\frac{\lambda }{p}-\frac{\lambda }{{2}_{s}^{*}}\right)\underset{\Omega }{\int }V\left(x\right){|u|}^{p}\text{d}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\ge \frac{s}{n}{S}^{-1}\left(n,s\right){‖u‖}_{{L}^{{2}_{s}^{*}}\left(\Omega \right)}^{2}+\frac{s}{n}{\mu }_{{k}_{0}}-\left(\frac{\lambda }{p}-\frac{\lambda }{{2}_{s}^{*}}\right){S}^{-\frac{p}{2}}\left(n,s\right){‖V\left(x\right)‖}_{{L}^{q}\left(\Omega \right)}{‖u‖}_{{L}^{{2}_{s}^{*}}}^{p}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\ge \frac{s}{n}{S}^{-1}\left(n,s\right){‖u‖}_{{L}^{{2}_{s}^{*}}\left(\Omega \right)}^{2}+\frac{s}{n}{\mu }_{{k}_{0}}-\left(\frac{\lambda }{p}-\frac{\lambda }{{2}_{s}^{*}}\right){S}^{-\frac{p}{2}}\left(n,s\right){‖V\left(x\right)‖}_{{L}^{q}\left(\Omega \right)}{‖u‖}_{{L}^{{2}_{s}^{*}}}^{p}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\ge \frac{s}{n}S\left(n,s\right){v}_{{k}_{0}}^{-\frac{{2}_{s}^{*}}{2}}-\left(\frac{\lambda }{p}-\frac{\lambda }{{2}_{s}^{*}}\right){S}^{-\frac{p}{2}}\left(n,s\right){‖V\left(x\right)‖}_{{L}^{p}\left(\Omega \right)}{‖u‖}_{{L}^{{2}_{s}^{*}}}^{p}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\ge \frac{s}{n}{S}^{\frac{n}{2s}}\left(n,s\right){\left[\beta K\left({x}_{{k}_{0}}\right)\right]}^{\frac{2s-n}{2s}}-C{\lambda }^{\frac{2}{2-p}}.\end{array}$ (3.18)

However, if $\beta >0$ is given, we can choose ${\lambda }_{0}>0$ so small for every $0<\lambda <{\lambda }_{0}$ that last term on the right-hand side above is greater than 0 which is contradiction when $2

$\begin{array}{c}0>c=\underset{n\to \infty }{\mathrm{lim}}\left[J\left({u}_{n}\right)-\frac{1}{p}〈{J}^{\prime }\left({u}_{n}\right),{u}_{n}〉\right]\\ =\left(\frac{1}{2}-\frac{1}{p}\right){‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\left(\frac{\beta }{{2}_{s}^{*}}-\frac{\beta }{p}\right)\underset{\Omega }{\int }K\left(x\right){|u|}^{{2}_{s}^{*}}\text{d}x\\ \ge \left(\frac{1}{2}-\frac{1}{p}\right){‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\left(\frac{\beta }{{2}_{s}^{*}}-\frac{\beta }{p}\right)\underset{\Omega \cap \left\{K\left(x\right)<0\right\}}{\int }K\left(x\right){|u|}^{{2}_{s}^{*}}\text{d}x\\ \ge \left(\frac{1}{2}-\frac{1}{p}\right){‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\left(\frac{\beta }{{2}_{s}^{*}}-\frac{\beta }{p}\right){‖K\left(x\right)‖}_{{L}^{\infty }}{‖u‖}_{{L}^{{2}_{s}^{*}}}^{{2}_{s}^{*}}\end{array}$

$\beta$ is the same as $\lambda$ greater than 0. We see that ${v}_{{k}_{0}}\ge {\left[\frac{S\left(n,s\right)}{\beta K\left({x}_{{k}_{0}}\right)}\right]}^{\frac{n}{2s}}$ cannot occur if ${\lambda }_{0}$ or ${\beta }_{0}$ are choose properly. Thus ${\mu }_{k}={v}_{k}=0$. As consequence, we obtain that ${\left({u}_{n}\right)}_{+}-u\to 0$ in ${L}^{{2}_{s}^{*}}\left(\Omega \right)$ , that is $\underset{n\to \infty }{\mathrm{lim}}\underset{{R}^{n}}{\int }{|{\left({u}_{n}\right)}_{+}|}^{{2}_{s}^{*}}\text{d}x=\underset{{R}^{n}}{\int }{|u|}^{{2}_{s}^{*}}\text{d}x$. This implies convergence of $\lambda V\left(x\right){|{u}_{n}|}^{p-1}+\beta K\left(x\right){|{u}_{n}|}^{{2}_{s}^{*}-1}$ in ${L}^{{2}_{s}^{*}}\left(\Omega \right)$. Finally using the continuity of the inverse operator ${\left(-\Delta \right)}^{s}$. We obtain strong convergence of ${u}_{n}$ in ${X}_{0}^{s}\left(\Omega \right)$. #

Next, by using the classical concept and properties of the genus, we construct a min-max class of the critical point.

For a Banach space X, We define the set

$\mathcal{A}=\left\{A\subset X\\left\{0\right\}:A\text{is closed in}X\text{and symmetric with respect to the orign}\right\}$

For $\mathcal{A}\in A$ , define

$\gamma \left(A\right):=\mathrm{inf}\left\{m\in N,\exists \varphi \in C\left(A,{R}^{m}\\left\{0\right\}\right),\varphi \left(x\right)=-\varphi \left(-x\right)\right\}$ (3.19)

If there is no mapping $\varphi$ as above for any $m\in N$ , there $\gamma \left(A\right)=+\infty$. we refer to [8] for the properties of the genus.

Proposition 3.3. [8] Let $A,B\subset Α$ ,

1) If there exists an odd map $f\in C\left(A,B\right)$ , then $\gamma \left(A\right)\le \gamma \left(B\right)$ ;

2) If $A\subset B$ , then $\gamma \left(A\right)\le \gamma \left(B\right)$ ;

3) $\gamma \left(A\cup B\right)\le \gamma \left(A\right)+\gamma \left(B\right)$ ;

4) If S is a sphere centered at the origin in ${R}^{m}$ , then $\gamma \left(s\right)=m$ ;

5) If A is compact, there exists a symmetric Neighborhood N of A, such that $\gamma \left(\stackrel{¯}{N}\right)=\gamma \left(A\right)$.

According Holder inequality, we get that

$\begin{array}{l}J\left(u\right)=\frac{1}{2}{‖u‖}_{{X}_{0}^{s}}^{2}-\frac{\lambda }{p}{\underset{\Omega }{\int }V\left(x\right)|u|}^{p}\text{d}x-\frac{\beta }{{2}_{s}^{*}}\underset{\Omega }{\int }K\left(x\right){|u|}^{{2}_{s}^{*}}\text{d}x\\ \text{}\ge \frac{1}{2}{‖u‖}_{{X}_{0}^{s}}^{2}-{C}_{1}\lambda {‖u‖}_{{X}_{0}^{s}}^{p}-{C}_{2}\beta {‖u‖}_{{X}_{0}^{s}}^{{2}_{s}^{*}}\end{array}$ (3.20)

We define the function

$Q\left(t\right):=\frac{1}{2}{t}^{2}-{C}_{1}\lambda {t}^{p}-{C}_{2}\beta {t}^{{2}_{s}^{*}}$ (3.21)

Then it is easy to see that given $\beta >0$ , there exists ${\lambda }_{1}>0$ so small that for every $0<\lambda <{\lambda }_{1}$ , there exists $0<{T}_{0}<{T}_{1}$ such that $Q\left(t\right)<0$ for $0\le t\le {T}_{0}$ , $Q\left(t\right)>0$ for ${T}_{0}. and $Q\left(t\right)<0$ $t>{T}_{1}$. Analogously, for given $\lambda >0$ , we can choose ${\beta }_{1}>0$ with the property that ${T}_{0},{T}_{1}$ as above for each $0<\beta <{\beta }_{1}$. Clearly, $Q\left({T}_{0}\right)=Q\left({T}_{1}\right)=0$.

As in [9] , Let $\tau :{ℝ}^{+}\to \left[0,1\right]$ be a nonincreasing ${C}^{\infty }$ function such that $\tau \left(t\right)=1$ if $0\le \tau \le {T}_{0}$ and $\tau \left(t\right)=0$. if $\tau \ge {T}_{0}$. Set $\Psi \left(u\right)=\tau \left({‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}\right)$ , we make the following truncation of the function J:

$\stackrel{˜}{J}\left(u\right)=\frac{1}{2}{‖u‖}_{{X}_{0}^{s}}^{2}-\frac{\lambda }{p}{\underset{\Omega }{\int }V\left(x\right)|u|}^{p}\text{d}x-\frac{\beta }{{2}_{s}^{*}}\psi \left(u\right)\underset{\Omega }{\int }K\left(x\right){|u|}^{{2}_{s}^{*}}\text{d}x$ (3.22)

then

$\stackrel{˜}{J}\left(u\right)\ge \stackrel{˜}{Q}{‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}.$ (3.23)

where $\stackrel{˜}{Q}\left(t\right):=\frac{1}{2}{t}^{2}-{C}_{1}\lambda {t}^{p}-{C}_{2}\beta {t}^{{2}_{s}^{*}}\psi \left(t\right)$.

It is clear that $\stackrel{˜}{J}\left(u\right)\in {C}^{1}$ and is bounded from below.

Lemma 3.4. [10] 1) For any $\lambda >0$ and $0<\beta <{\beta }_{1}$ or any $\beta >0$ and $0<\lambda <{\lambda }_{1}$ , if $\stackrel{˜}{J}\left(u\right)<0$ , then ${‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}<{T}_{0}$ and $\stackrel{˜}{J}\left(u\right)=J\left(u\right)$.

2) For any $\lambda >0$ , there exists such that if $0<\beta <\stackrel{¯}{\beta }$ and $c<0$ then $\stackrel{˜}{J}$ satisfies ${\left(PS\right)}_{c}$.

3) For any $\beta >0$ ,there exists $\stackrel{˜}{\lambda }>0\left(\stackrel{˜}{\lambda }\le {\lambda }_{1}\right)$ such that if $0<\lambda <\stackrel{˜}{\lambda }$ and $c<0$ then $\stackrel{˜}{J}$ satisfies ${\left(PS\right)}_{c}$.

Lemma 3.5. Denote ${\stackrel{˜}{J}}^{\alpha }:=\left\{u\in {X}_{0}^{s}\left(\Omega \right),\stackrel{˜}{J}\left(u\right)\le \alpha \right\}$. Then for any $m\in N$ , there is ${\epsilon }_{m}<0$ such that $\gamma \left({\stackrel{˜}{J}}^{{\epsilon }_{m}}\right)\ge m$.

Proof. Denote by ${X}_{0}^{s}\left(\Omega \right)$ the closure of ${C}_{0}^{\infty }\left(\Omega \right)$ with the respect to norm ${‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}={\left(\underset{\Omega }{\int }\frac{{|u\left(x\right)-u\left(y\right)|}^{2}}{{|x-y|}^{n+2s}}\text{d}x\text{d}y\right)}^{\frac{1}{2}}$ , $V\left(x\right)>0$ in $\Omega$. Extending functions in

${X}_{0}^{s}\left(\Omega \right)$ by 0 outside $\Omega$. Let ${X}_{m}$ be a m-dimensional subspace of ${X}_{0}^{s}\left(\Omega \right)$. For any $u\in {X}_{m},u\ne 0$. We write $u={r}_{m}w$ with $w\in {X}_{m}$ and ${‖w‖}_{{X}_{0}^{s}\left(\Omega \right)}=1$. From the assumptions of $V\left(x\right)$ , it is easy to see for every $w\in {X}_{m}$ with ${‖w‖}_{{X}_{0}^{s}\left(\Omega \right)}=1$ that there exists ${d}_{m}>0$ such that

$\underset{\Omega }{\int }V\left(x\right){|w|}^{p}\text{d}x\ge {d}_{m}$ (3.24)

For $0<{r}_{m}<{T}_{0}$. Since all the norms are equivalent, we get

$\begin{array}{l}\stackrel{˜}{J}\left(u\right)=J\left(u\right)=\frac{1}{2}{‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\frac{\lambda }{p}\underset{\Omega }{\int }V\left(x\right){|u|}^{p}\text{d}x-\frac{\beta }{{2}_{s}^{*}}\underset{\Omega }{\int }K\left(x\right){|u|}^{{2}_{s}^{*}}\text{d}x\\ \text{}\text{ }\text{ }\text{ }\le \frac{1}{2}{‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}^{2}-\frac{\lambda }{p}\underset{\Omega }{\int }V\left(x\right){|u|}^{p}\text{d}x+\frac{\beta }{{2}_{s}^{*}}|\underset{\Omega }{\int }K\left(x\right){|u|}^{{2}_{s}^{*}}\text{d}x|\\ \text{}\text{ }\text{}\le \frac{1}{2}{r}_{m}^{2}-\lambda c{d}_{m}+c\beta {r}_{m}^{{2}_{s}^{*}}:={\epsilon }_{m}.\end{array}$

Therefore for given $\lambda$ and $\beta$. we can choose ${r}_{m}\in \left(0,{T}_{0}\right)$ sufficiently small so that $\stackrel{˜}{J}\left(u\right)\le {\epsilon }_{m}<0$.#

Let ${S}_{{r}_{m}}=\left\{u\in {X}_{0}^{s}\left(\Omega \right):{‖u‖}_{{X}_{0}^{s}\left(\Omega \right)}={r}_{m}\right\}$. Then ${S}_{{r}_{m}}\cap {X}_{m}\subset {\stackrel{˜}{J}}^{{\epsilon }_{m}}$ , Hence by proposition 3.3 (2) and (4) $r\left({\stackrel{˜}{J}}^{{\epsilon }_{m}}\right)\ge r\left({S}_{{r}_{m}}\cap {X}_{m}\right)\ge m$.

We denote ${\Gamma }_{m}=\left\{A\in Α:\gamma \left(A\right)\ge m\right\}$ and let

${C}_{m}:=\underset{A\in {\Gamma }_{m}}{\mathrm{inf}}\underset{u\in A}{\mathrm{sup}}J\left(u\right)$ (3.25)

then

$-\infty <{C}_{m}\le {\epsilon }_{m}<0\text{,}m\in N$ (3.26)

because ${\stackrel{˜}{J}}^{{\epsilon }_{m}}\in {\Gamma }_{m}$ and $\stackrel{˜}{J}$ is bounded from below.

Proposition 3.6. Let $\lambda ,\beta$ be as in Lemma 3.5 (2) and (3). Then all ${c}_{m}$ given by (3.25) are critical values of $\stackrel{˜}{J}$ and ${c}_{m}\to 0$ as $m\to 0$.

Proof. Denote ${K}_{\epsilon }=\left\{u\in {X}_{0}^{s}\left(\Omega \right):\stackrel{˜}{J}\left(u\right)=c,{\stackrel{˜}{J}}^{\prime }\left(u\right)=0\right\}$. Then by Lemma 3.4 (2) and (3), if $c<0$ , ${K}_{c}$ is compact. It is clear that ${C}_{m}\le {C}_{m+1}$. By (3.26) ${C}_{m}<0$. Hence ${C}_{m}\to \stackrel{¯}{C}\le 0$. Moreover, since ${\left(PS\right)}_{c}$ satisfied, it follows from a standard argument (see [11] ) that all ${C}_{m}$ are critical values of $\stackrel{˜}{J}$. Now, we claim that $\stackrel{¯}{c}=0$. If $\stackrel{¯}{c}<0$ because ${K}_{\stackrel{¯}{c}}$ is compact and ${K}_{\stackrel{¯}{c}}\in A$ , it follows from Proposition 3.3 (5) that $\gamma \left({K}_{\stackrel{¯}{c}}\right)={m}_{0}<+\infty$ and there exists $\delta >0$ such that $\gamma \left({K}_{\stackrel{¯}{c}}\right)=\gamma \left({N}_{\delta }\left({K}_{\stackrel{¯}{c}}\right)\right)={m}_{0}$. By the deformation Lemma [9] , there exists $\epsilon >0\left(\stackrel{¯}{c}+\epsilon <0\right)$ and an odd homeomorphism $\varsigma \left(\cdot \right):{X}_{0}^{s}\left(\Omega \right)\to {X}_{0}^{s}\left(\Omega \right)$ such that

$\varsigma \left({\stackrel{˜}{J}}^{\stackrel{¯}{c}+\epsilon }\{N}_{\delta }\left({K}_{\stackrel{¯}{c}}\right)\right)\subset {\stackrel{˜}{J}}^{\stackrel{¯}{c}-\epsilon }$ (3.27)

Since ${c}_{m}$ is increasing anad converges to $\stackrel{¯}{c}$. there exists $m\in N$ such that

${c}_{m}>\stackrel{¯}{c}-\epsilon$. (3.28)

And exists a $A\in {\Gamma }_{m+{m}_{0}}$ such that

$\underset{u\in A}{\mathrm{sup}}\stackrel{˜}{J}\left(u\right)<\stackrel{¯}{c}+\epsilon$ (3.29)

By Proposition 3.3 (3), we obtain

$\gamma \left(\stackrel{¯}{A\{N}_{\delta }\left({K}_{\stackrel{¯}{c}}\right)}\right)\ge \gamma \left(A\right)-\gamma \left({N}_{\delta }\left({K}_{\stackrel{¯}{c}}\right)\right)\ge m$ (3.30)

By Proposition 3.3 (1), we obtain

$\gamma \left(\stackrel{¯}{\varsigma \left(A\{N}_{\delta }\left({K}_{\stackrel{¯}{c}}\right)\right)}\right)\ge m$ (3.31)

therefore

$\varsigma \left(A\{N}_{\delta }\left({K}_{\stackrel{¯}{c}}\right)\right)\in {\Gamma }_{m}$

consequently, from (3.28), we get

$\underset{u\in \varsigma \left(A\{N}_{\delta }\left({K}_{\stackrel{¯}{c}}\right)\right)}{\mathrm{sup}}\stackrel{˜}{J}\left(u\right)\ge {c}_{m}>\stackrel{¯}{c}-\epsilon$ (3.32)

on the other hand, by (3.27) and (3.29)

$\varsigma \left(A\{N}_{\delta }\left({K}_{\stackrel{¯}{c}}\right)\right)\subset \varsigma \left({\stackrel{˜}{J}}^{\stackrel{¯}{c}+\epsilon }\{N}_{\delta }\left({K}_{\stackrel{¯}{c}}\right)\right)\subset {\stackrel{˜}{J}}^{\stackrel{¯}{c}-\epsilon }$ (3.33)

which implies that

$\underset{u\in \varsigma \left(A\{N}_{\delta }\left({K}_{\stackrel{¯}{c}}\right)\right)}{\mathrm{sup}}\stackrel{˜}{J}\left(u\right)\le \stackrel{¯}{c}-\epsilon$ (3.34)

this contradicts to (3.32).Hence ${c}_{m}\to 0$. #

By (1) of Lemma 3.4 $\stackrel{˜}{J}\left(u\right)=J\left(u\right)$ if $\stackrel{˜}{J}\left(u\right)<0$. This and Proposition 3.6 give Theorem1.1.

Cite this paper: Chen, H. (2019) The Existence of Solution of a Critical Fractional Equation. Journal of Applied Mathematics and Physics, 7, 243-253. doi: 10.4236/jamp.2019.71020.
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