Fractional elliptic equations with Hardy potential and critical nonlinearities

TL;DR

This paper studies fractional elliptic equations with Hardy potential and critical nonlinearities, establishing the existence of nonnegative solutions under certain conditions.

Contribution

It introduces new existence results for solutions to fractional elliptic equations involving Hardy potentials and critical nonlinearities.

Findings

Existence of nonnegative solutions under specific conditions.

Application of variational methods to fractional elliptic equations.

Extension of classical results to fractional and Hardy potential settings.

Abstract

In this paper, we consider the fractional elliptic equation \begin{align*} \left\{\begin{aligned} &(-\Delta)^s u-\mu\frac{u}{|x|^{2s}} = \frac{|u|^{2_s^\ast (\alpha)-2}u}{|x|^{\alpha}} + f(x,u), && \mbox{in} \ \Omega,\\ &u=0, && \mbox{in} \ \mathbb{R}^{n}\backslash \ \Omega, \end{aligned}\right. \end{align*} where $\Omega\subset R^n$ is a smooth bounded domain, $0\in\Omega$, $0<s<1$, $0<\alpha<2s<n$, $2_{s}^{\ast}(\alpha)=\frac{2(n-\alpha)}{n-2s}$. Under some assumptions on $\mu$ and $f$, we obtain the existence of nonnegative solutions.

Equations

$$\{\begin{array}{rlrl}& (-\Delta {)}^{s}u-\mu \frac{u}{|x{|}^{2s}}=\frac{|u{|}^{2_s^{\ast }(\alpha )-2}u}{|x{|}^{\alpha }}+f(x,u),& & \text{mbox}in\Omega ,\\ & u=0,& & \text{mbox}in{\mathbb{R}}^n\backslash \Omega ,\end{array}$$

$$(-\Delta {)}^{s}u(x)=c_{n,s}P.V.{\int }_{{\mathbb{R}}^n}\frac{u(x)-u(y)}{|x-y{|}^{n+2s}}dy,$$

$$(-\Delta {)}^{s}u-\mu \frac{u}{|x{|}^{2s}}=g(u)\text{mbox}in\Omega ,$$

$$\{\begin{array}{rlrl}& (-\Delta {)}^{s}u-\mu \frac{u}{|x{|}^{2s}}-\lambda u=\frac{|u{|}^{2_s^{\ast }(\alpha )-2}u}{|x{|}^{\alpha }}+h(x)u^{q-1},& & \text{mbox}in\Omega ,\\ & u=0,& & \text{mbox}in{\mathbb{R}}^n\backslash \Omega ,\end{array}$$

$$\Vert u{\Vert }_{H_0^s(\Omega )}^2={\int }_{{\mathbb{R}}^n}|(-\Delta {)}^{\frac{s}{2}}u{|}^{2}dx=\frac{c_{n,s}}{2}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\frac{|u(x)-u(y){|}^2}{|x-y{|}^{n+2s}}dxdy,$$

$$\Vert |u\Vert |={\left(\frac{c_{n,s}}{2}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\frac{|u(x)-u(y){|}^2}{|x-y{|}^{n+2s}}dxdy-\mu {\int }_{\Omega }\frac{|u{|}^2}{|x{|}^{2s}}dx\right)}^{\frac{1}{2}}.$$

$${\mu }_H(s){\int }_{{\mathbb{R}}^n}\frac{|u{|}^2}{|x{|}^{2s}}dx\le {\int }_{{\mathbb{R}}^n}|(-\Delta {)}^{\frac{s}{2}}u{|}^{2}dx,u\in H_0^s({\mathbb{R}}^n),$$

$$(-\Delta {)}^{s}u-\mu \frac{u}{|x{|}^{2s}}=\frac{u_{+}^{2_s^{\ast }(\alpha )-1}}{|x{|}^{\alpha }}+f_{+}(x,u),\text{mbox}in\Omega .$$

$$I(u)=\frac{1}{2}\Vert |u|{\Vert }^2-\frac{1}{2_s^{\ast }(\alpha )}{\int }_{\Omega }\frac{u_{+}^{2_s^{\ast }(\alpha )}}{|x{|}^{\alpha }}dx-{\int }_{\Omega }{F}_{+}(x,u)dx,u\in H_0^s(\Omega ),$$

$${\Lambda }_{\mu ,s,\alpha }=\underset{u\in H^s({\mathbb{R}}^n)\backslash \{0\}}{\inf }\frac{{\int }_{{\mathbb{R}}^n}(|(-\Delta {)}^{\frac{s}{2}}u{|}^2-\mu \frac{|u{|}^2}{|x{|}^{2s}})dx}{({\int }_{{\mathbb{R}}^n}\frac{|u{|}^{2_s^{\ast }(\alpha )}}{|x{|}^{\alpha }}dx{)}^{\frac{2}{2_s^{\ast }(\alpha )}}}.$$

$$\{\begin{array}{rlrl}& (-\Delta {)}^{s}u-\mu \frac{u}{|x{|}^{2s}}=\frac{|u{|}^{2_s^{\ast }(\alpha )-2}u}{|x{|}^{\alpha }},& & \text{mbox}in{\mathbb{R}}^n,\\ & u\ge 0,u\not\equiv 0& & \text{mbox}in{\mathbb{R}}^n.\end{array}$$

$$\Vert |u_{\epsilon }\Vert {|}^2=({\Lambda }_{\mu ,s,\alpha }{)}^{\frac{n-\alpha }{2s-\alpha }}+O({\epsilon }^{2{\beta }_{+}(\mu )+2s-n}),$$

$${\left({\int }_{\Omega }\frac{|u_{\epsilon }{|}^{2_s^{\ast }(\alpha )}}{|x{|}^{\alpha }}dx\right)}^{\frac{2}{2_s^{\ast }(\alpha )}}=({\Lambda }_{\mu ,s,\alpha }{)}^{\frac{n-\alpha }{2s-\alpha }}+O({\epsilon }^{2_s^{\ast }(\alpha ){\beta }_{+}(\mu )+\alpha -n}),$$

$${\int }_{\Omega }|u_{\epsilon }{|}^{q}dx=\{\begin{array}{rlrl}& O({\epsilon }^{n-\frac{q(n-2s)}{2}}),& & q>\frac{n}{{\beta }_{+}(\mu )},\\ & O({\epsilon }^{n-\frac{q(n-2s)}{2}}|\log \epsilon |),& & q=\frac{n}{{\beta }_{+}(\mu )},\\ & O({\epsilon }^{q({\beta }_{+}(\mu )-\frac{n-2s}{2})}),& & q<\frac{n}{{\beta }_{+}(\mu )},\end{array}$$

$$c+1+o(1)\Vert |u_n\Vert |$$

$$=\left(\frac{1}{2}-\frac{1}{\theta }\right)\Vert |u_n\Vert {|}^2+\left(\frac{1}{\theta }-\frac{1}{2_s^{\ast }(0)}\right){\int }_{\Omega }\frac{(u_n{)}_{+}^{2_s^{\ast }(\alpha )}}{|x{|}^{\alpha }}dx$$

$$-{\int }_{\Omega }\left(F_{+}(x,u_n)-\frac{1}{\theta }{f}_{+}(x,u_n)u_n\right)dx$$

$$\ge \left(\frac{1}{2}-\frac{1}{\theta }\right)\Vert |u_n\Vert {|}^2-\frac{\eta }{\theta }\Vert u_n{\Vert }_2^2$$

$$=\Vert |u_n\Vert {|}^2\left(\frac{\theta -2}{2\theta }-\frac{\eta }{\theta }\Vert v_n{\Vert }_2^2\right),$$

$$0$$

$$\le \underset{n\to \infty }{\lim }\left(\frac{1}{2}-{\int }_{\Omega }\frac{F_{+}(x,u_n)}{u_n^2}{v}_n^{2}dx\right)$$

$$\le \underset{n\to \infty }{\lim }\left(\frac{1}{2}-K{\int }_{\Omega }{v}_n^{2}dx\right)$$

$$=\frac{1}{2}-K{\int }_{\Omega }{v}^{2}dx<0,$$

$$\Vert |u_n\Vert {|}^2-{\int }_{\Omega }\frac{(u_n{)}_{+}^{2_s^{\ast }(\alpha )}}{|x{|}^{\alpha }}dx=o(1).$$

$$\Vert |u_n\Vert {|}^2\ge {\Lambda }_{\mu ,s,\alpha }{\left({\int }_{\Omega }\frac{|u_n{|}^{2_s^{\ast }(\alpha )}}{|x{|}^{\alpha }}dx\right)}^{\frac{2}{2_s^{\ast }(\alpha )}}.$$

$$o(1)\ge \Vert |u_n\Vert {|}^2\left(1-({\Lambda }_{\mu ,s,\alpha }{)}^{-\frac{2_s^{\ast }(\alpha )}{2}}\Vert |u_n\Vert {|}^{2_s^{\ast }(\alpha )-2}\right).$$

$$\Vert |u_n\Vert {|}^2\ge ({\Lambda }_{\mu ,s,\alpha }{)}^{\frac{n-\alpha }{2s-\alpha }}+o(1).$$

$$I(u_n)$$

$$=\frac{2s-\alpha }{2(n-\alpha )}\Vert |u_n\Vert {|}^2+o(1)$$

$$\ge \frac{2s-\alpha }{2(n-\alpha )}({\Lambda }_{\mu ,s,\alpha }{)}^{\frac{n-\alpha }{2s-\alpha }}+o(1),$$

$$\underset{t\ge 0}{\sup }I(t{u}_0)<\frac{2s-\alpha }{2(n-\alpha )}({\Lambda }_{\mu ,s,\alpha }{)}^{\frac{n-\alpha }{2s-\alpha }}.$$

$$g(t)=I(t{u}_{\epsilon })=\frac{t^2}{2}\Vert |u_{\epsilon }\Vert {|}^2-\frac{t^{2_s^{\ast }(\alpha )}}{2_s^{\ast }(\alpha )}{\int }_{\Omega }\frac{u_{\epsilon }^{2_s^{\ast }(\alpha )}}{|x{|}^{\alpha }}dx-{\int }_{\Omega }{F}_{+}(x,t{u}_{\epsilon })dx.$$

$$g(t)=\frac{t^2}{2}\Vert |u_{\epsilon }\Vert {|}^2-\frac{t^{2_s^{\ast }(\alpha )}}{2_s^{\ast }(\alpha )}\left[({\Lambda }_{\mu ,s,\alpha }{)}^{\frac{n-\alpha }{2s-\alpha }}+O({\epsilon }^{2_s^{\ast }(\alpha ){\beta }_{+}(\mu )+\alpha -n})\right]-{\int }_{\Omega }{F}_{+}(x,t{u}_{\epsilon })dx.$$

$$\bar{g}(t)=\frac{t^2}{2}\Vert |u_{\epsilon }\Vert {|}^2-\frac{t^{2_s^{\ast }(\alpha )}}{2_s^{\ast }(\alpha )}({\Lambda }_{\mu ,s,\alpha }{)}^{\frac{n-\alpha }{2s-\alpha }}.$$

Full text

[TABLE]

where $\Omega\subset R^{n}$ is a smooth bounded domain, $0\in\Omega$, $0<s<1$, $0<\alpha<2s<n$, $2_{s}^{\ast}(\alpha)=\frac{2(n-\alpha)}{n-2s}$. Under some assumptions on $\mu$ and $f$, we obtain the existence of nonnegative solutions.

2010 Mathematics Subjects Classification: 35B33, 58E30

1. Introduction

In this paper, we are concerned with the following critical problem

[TABLE]

where $\Omega\subset\mathbb{R}^{n}$ is a smooth bounded domain, $0\in\Omega$, $0<s<1$, $0<\alpha<2s<n$, $0\leq\mu<\mu_{H}(s)=4^{s}\frac{\Gamma^{2}(\frac{n+2s}{4})}{\Gamma^{2}(\frac{n-2s}{4})}$, $2_{s}^{\ast}(\alpha)=\frac{2(n-\alpha)}{n-2s}$ is the fractional critical Hardy-Sobolev exponent. The fractional Laplacian $(-\Delta)^{s}$ is defined by

[TABLE]

where $P.V.$ is the principal value and $c_{n,s}$ is a constant that depends on $n$ and $s$. If $s=1$, $(-\Delta)^{s}$ is defined as $-\Delta$, where $\Delta$ is the Laplacian. In fact, for any $u\in C_{0}^{\infty}(\mathbb{R}^{n})$, $\lim_{s\rightarrow 1-}(-\Delta)^{s}u=-\Delta u$, see Proposition 4.4 in [4]. For $s=1$, there are many results for problem (1.1), we refer to [2, 8, 3, 12].

Recently, the existence of nontrivial solutions for nonlinear fractional elliptic equations with Hardy potential

[TABLE]

have been studied by several authors. Barrios, Medina and Peral [1] studied (1.2) with $g(u)=u^{p}+\lambda u^{q}$ $(0<q<1,1<p<p(\mu,s))$ and discussed the existence and multiplicity of solutions depending on the value of $p$. Fall [5] studied (1.2) with $g(u)=u^{p}$ $(p>1)$ and obtained the existence and nonexistence of nonnegative distributional solutions. Shakerian [10] studied the following problem

[TABLE]

where $\lambda\in\mathbb{R}$, $h\in C^{0}(\bar{\Omega})$, $h\geq 0$, $q\in(2,2_{s}^{*}(0))$. A sufficient condition is established for the existence of a positive solution for (1.3). Zhang and Hsu [11] investigated a system of fractional elliptic equation involving critical Sobolev-Hardy exponents and concave-convex nonlinearities, the existence and multiplicity of positive solutions was proved by variational methods. Motivated by the above papers, we will study the problem (1.1), here is the main result of this paper.

Theorem 1**.**

*Suppose that $0<s<1$, $0<\mu<\mu_{H}(s)$, $0<\alpha<2s<n$, $\frac{n-2s}{2}<\beta_{+}(\mu)<\frac{n}{2}$, where $\beta_{+}(\mu)$ is the unique solution of $\Psi_{n,s}(\beta)=4^{s}\frac{\Gamma(\frac{n-\beta}{2})\Gamma(\frac{2s+\beta}{2})}{\Gamma(\frac{n-2s-\beta}{2})\Gamma(\frac{\beta}{2})}-\mu=0$ in $(\frac{n-2s}{2},n-2s]$,

$(f_{1})$ $f\in C(\Omega\times\mathbb{R},\mathbb{R})$ and there exist constants $c_{1},c_{2}>0$ and $p\in(2,2_{s}^{*}(0))$ such that $|f(x,t)|\leq c_{1}|t|+c_{2}|t|^{p-1}$ for all $x\in\Omega$, $t\in\mathbb{R}$,

$(f_{2})$ There exists a constant $K>0$ big enough such that $F(x,t)=\int_{0}^{t}f(x,s)ds\geq Kt^{2}$ for all $x\in\Omega$, $t\in\mathbb{R}$,

$(f_{3})$ There exist constants $\rho>2$, $\eta>0$ such that $\rho F(x,t)\leq f(x,t)t+\eta t^{2}$ for all $x\in\Omega$, $t\in\mathbb{R}$,

then problem (1.1) has at least a nonnegative solution.*

2. Preliminaries

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{n}$ with 0 in its interior. For $p\in[1,\infty)$, we denote by $L^{p}(\Omega)$ the usual Lebesgue space with the norm $\|u\|_{p}=\left(\int_{\Omega}|u|^{p}dx\right)^{\frac{1}{p}}$. For $s\in(0,1)$, the fractional Sobolev space $H^{s}_{0}(\Omega)$ is defined as the closure of $C_{0}^{\infty}(\Omega)$ with respect to the norm

[TABLE]

where $c_{n,s}=\frac{4^{s}\Gamma(\frac{n+2s}{2})}{\pi^{\frac{n}{2}}|\Gamma(-s)|}$.

For $0<\mu<\mu_{H}(s)$, define another norm

[TABLE]

By the fractional Hardy inequality (see [6])

[TABLE]

we see that $\||\cdot\||$ is well defined and for $0<\mu<\mu_{H}(s)$, $\||\cdot\||$ is equalivalent to the norm $\|\cdot\|_{H^{s}_{0}(\Omega)}$.

In order to study positive solution for (1.1), we consider the existence of nontrivial solutions to the problem

[TABLE]

The energy functional $I:H_{0}^{s}(\Omega)\rightarrow\mathbb{R}$ whose critical points are weak solutions to problem (2.2) is given by

[TABLE]

where $u_{+}=\max\{u,0\}$, $F_{+}(x,t)=\int_{0}^{t}f_{+}(x,s)ds$, $f_{+}(x,t)=\left\{\begin{aligned} &f(x,t),&&t\geq 0,\\ &0,&&t<0.\end{aligned}\right.$

The best constant in fractional Hardy-Sobolev inequality in $\mathbb{R}^{n}$ is

[TABLE]

Ghoussoub and Shakerian [7] proved the existence of extremals for $\Lambda_{\mu,s,\alpha}(\mathbb{R}^{n})$, when $0<\alpha<2s<n$ and $-\infty<\mu<\mu_{H}(s)$. Note that any minimizer for (2.3) is a variational solution of the following borderline problem

[TABLE]

by [7, 10], for $0<\mu<\mu_{H}(s)$, the problem (2.4) has positive radial symmetric solution $U_{\varepsilon}(x)=\varepsilon^{-(n-2s)/2}u_{\mu}(x/\varepsilon)$, where $u_{\mu}$ is a solution of (2.4), $u_{\mu}\in C^{1}(\mathbb{R}^{n}\backslash\{0\})$, $u_{\mu}>0$. Let $\varphi\in C_{0}^{\infty}(\Omega)$, $0\leq\varphi(x)\leq 1$, choose $\rho>0$ small enough such that $\varphi(x)=1$ for $|x|<\rho/2$, $\varphi(x)=0$ for $|x|\geq\rho$. Set $u_{\varepsilon}(x)=\varphi(x)U_{\varepsilon}(x)$.

Lemma 1**.**

([11]).* Assume that $0<s<1$, $n>2s$, $0\leq\mu<\mu_{H}(s)$, $0\leq\alpha<2s$, $1\leq q<2_{s}^{\ast}(\alpha)$. Then, as $\varepsilon\rightarrow 0$, we have the following estimates:*

[TABLE]

[TABLE]

[TABLE]

where $\beta_{+}(\mu)$ is the unique solution of $\Psi_{n,s}(\beta)=4^{s}\frac{\Gamma(\frac{n-\beta}{2})\Gamma(\frac{2s+\beta}{2})}{\Gamma(\frac{n-2s-\beta}{2})\Gamma(\frac{\beta}{2})}-\mu=0$ in $(\frac{n-2s}{2},n-2s]$.

3. Proof of Theorem 1.1

Lemma 2**.**

Suppose that $(f1)$, $(f2)$, $(f3)$ hold. Let $\{u_{n}\}\subset H_{0}^{s}(\Omega)$ be a sequence such that $I(u_{n})\rightarrow c$ and $I^{\prime}(u_{n})\rightarrow 0$ in $(H_{0}^{s}(\Omega))^{-1}$, where $c\in\left(0,\frac{2s-\alpha}{2(n-\alpha)}(\Lambda_{\mu,s,\alpha})^{\frac{n-\alpha}{2s-\alpha}}\right)$. Then there exists $u\in H_{0}^{s}(\Omega)$ such that $u_{n}\rightharpoonup u$, up to a subsequence, $I^{\prime}(u)=0$ and $u$ is a nontrival solution of problem (1.1).

Proof.

We use the method of contradiction to show that $\{u_{n}\}$ is bounded. Suppose that $\||u_{n}\||\rightarrow\infty$ as $n\rightarrow\infty$. Let $v_{n}=u_{n}/\||u_{n}\||$, then $\||v_{n}\||=1$. By $(f_{3})$, we get

[TABLE]

where $\theta=\min\{2_{s}^{*}(\alpha),\rho\}$. Up to a subsequence, we assume that $v_{n}\rightharpoonup v$ in $H_{0}^{s}(\Omega)$. By Corollary 7.2 in [4], $H_{0}^{s}(\Omega)\hookrightarrow L^{q}(\Omega)$ is compact for $1\leq q<2_{s}^{\ast}(0)$. Then $v_{n}\rightarrow v$ in $L^{q}(\Omega)$, $1\leq q<2_{s}^{\ast}(0)$, and $v_{n}\rightarrow v$ a.e. in $\Omega$. Thus, by (3), we get $v\neq 0$. On the other hand,

[TABLE]

since $K$ is big enough, this is a contradiction. Therefore $\{u_{n}\}$ is bounded in $H_{0}^{s}(\Omega)$ and there exists $u\in H_{0}^{s}(\Omega)$ such that $u_{n}\rightharpoonup u$ up to a subsequence. By the weak continuity of $I^{\prime}$, we have $I^{\prime}(u)=0$. Assume that $u=0$ in $\Omega$, since $f(x,u)$ is subcritical, from $\langle I^{\prime}(u_{n}),u_{n}\rangle=o(1)$,

[TABLE]

By the definition of $\Lambda_{\mu,s,\alpha}$,

[TABLE]

By (3.2) and (3.3),

[TABLE]

If $\||u_{n}\||\rightarrow 0$, this contradict $c>0$. Then by (3.4),

[TABLE]

It follows from (3.2) and (3.5) that

[TABLE]

this contradicts $c<\frac{2s-\alpha}{2(n-\alpha)}(\Lambda_{\mu,s,\alpha})^{\frac{n-\alpha}{2s-\alpha}}$. Therefore $u\neq 0$ and $u$ is a nontrival solution of problem (1.1). ∎

Lemma 3**.**

Assume that $0<\alpha<2s<n$, $0\leq\mu<\mu_{H}(s)$, $\frac{n-2s}{2}<\beta_{+}(\mu)<\frac{n}{2}$ and $(f_{1})$, $(f_{2})$ hold. Then there exists $u_{0}\in H_{0}^{s}(\Omega)$, $u_{0}\neq 0$, such that

[TABLE]

Proof.

Since $U_{\varepsilon}(x)>0$, then $u_{\varepsilon}(x)\geq 0$ and $(u_{\varepsilon})_{+}=\max\{u_{\varepsilon},0\}=u_{\varepsilon}$. Set

[TABLE]

By (2.6),

[TABLE]

Let

[TABLE]

Note that $\lim_{t\rightarrow+\infty}g(t)=-\infty$, $g(0)=0$, $g(t)>0$ for $t>0$ small enough, thus $\sup_{t\geq 0}g(t)$ is attained for some $t_{\varepsilon}>0$. Since

[TABLE]

we have

[TABLE]

then

[TABLE]

By $(f_{1})$, we have

[TABLE]

Choosing $\varepsilon$ small enough, by (3), (2.5) and (2.7), we have

[TABLE]

On the other hand, $\bar{g}(t)$ attains its maximum at $t_{\varepsilon}^{0}=\left[\frac{\||u_{\varepsilon}\||^{2}}{(\Lambda_{\mu,s,\alpha})^{\frac{n-\alpha}{2s-\alpha}}}\right]^{\frac{1}{2_{s}^{*}(\alpha)-2}}$ and is increasing in $[0,t_{0}]$. Since $\frac{n-2s}{2}<\beta_{+}(\mu)<\frac{n}{2}$, by (2.5), (2.7), (3.7) and $(f_{2})$,

[TABLE]

Since $K$ is big enough, we have

[TABLE]

∎

Proof of Theorem 1.1. From fractional Hardy-Sobolev inequality and fractional Sobolev inequality, it follows that

[TABLE]

By (3.8), $(f_{1})$,

[TABLE]

Thus, there exists $\delta>0$ such that $I(u)\geq\delta$ for all $u\in\partial B_{r}(0)=\{u\in H_{0}^{s}(\Omega),\||u|\|=r\}$, where $r>0$ small enough. Since $F_{+}(x,t)\geq 0$, for $u_{0}\neq 0$, we have

[TABLE]

then $\lim_{t\rightarrow+\infty}I(tu_{0})\rightarrow-\infty$. We can choose $t_{0}$ such that $\|t_{0}u_{0}\|>r$ and $I(t_{0}u_{0})<0$. By the Mountain Pass Lemma [9], there is a sequence $\{u_{n}\}\subset H_{0}^{s}(\Omega)$ satisfying

[TABLE]

where

[TABLE]

By Lemma 2 and Lemma 3, we get a $(PS)_{c}$ sequence $\{u_{n}\}\subset H_{0}^{s}(\Omega)$ and $u\in H_{0}^{s}(\Omega)$ such that $I^{\prime}(u)=0$. Then $u$ is a solution of (2.2) and $\langle I^{\prime}(u),u^{-}\rangle=0$, where $u^{-}=\min\{u,0\}$. Therefore $u\geq 0$, we get a nonnegative solution of (1.1).

4. Acknowledgements

This work is partially supported by NSFC under the grant 11571269, China Postdoctoral Science Foundation Funded Project under grants 2015M572539, 2016T90899.