A fractional Laplacian problem with critical nonlinearity

Xiuhong Long; Jixiu Wang; Xiuhong Long; Jixiu Wang

doi:10.3934/math.2021488

AIMS Mathematics

2021, Volume 6, Issue 8: 8415-8425. doi: 10.3934/math.2021488

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Research article

A fractional Laplacian problem with critical nonlinearity

Xiuhong Long ¹,
Jixiu Wang ^{2
,
,}

1.
School of Information and Mathematics, Yangtze University, Jingzhou 434023, China
2.
School of Mathematics and Statistics, Hubei University of Arts and Science, Xiangyang 441053, China

Received: 22 March 2021 Accepted: 18 May 2021 Published: 01 June 2021
MSC : 35B09, 35J20

In this paper, using the mountain pass theorem we obtain a positive solution to the fractional Laplacian problem

$ \begin{equation*} \label{eq: main problem} \begin{cases} (-\Delta)^su = g(x)(u-k)_{+}^{q-1}+u^{\frac{n+2s}{n-2s}}& \rm{in}\ \,\Omega,\\ u>0& \rm{in}\ \,\Omega,\\ u = 0& \rm{on}\ \,\partial\Omega, \end{cases} \end{equation*} $

where $ \Omega\subset {\mathbb R}^{n} $ is a bounded smooth domain, $ 0 < s < 1, 2\leq q < {2n}/(n-2s) $ and $ k\in(0, \infty) $ is an arbitrary number. The function $ g:{\Omega}\to {\mathbb R} $ is a nonnegative continuous function satisfying some integrability condition.
- fractional Laplacian,
- critical Sobolev growth,
- positive solutions,
- variational methods
Citation: Xiuhong Long, Jixiu Wang. A fractional Laplacian problem with critical nonlinearity[J]. AIMS Mathematics, 2021, 6(8): 8415-8425. doi: 10.3934/math.2021488

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Abstract

In this paper, using the mountain pass theorem we obtain a positive solution to the fractional Laplacian problem

$ \begin{equation*} \label{eq: main problem} \begin{cases} (-\Delta)^su = g(x)(u-k)_{+}^{q-1}+u^{\frac{n+2s}{n-2s}}& \rm{in}\ \,\Omega,\\ u>0& \rm{in}\ \,\Omega,\\ u = 0& \rm{on}\ \,\partial\Omega, \end{cases} \end{equation*} $

where $ \Omega\subset {\mathbb R}^{n} $ is a bounded smooth domain, $ 0 < s < 1, 2\leq q < {2n}/(n-2s) $ and $ k\in(0, \infty) $ is an arbitrary number. The function $ g:{\Omega}\to {\mathbb R} $ is a nonnegative continuous function satisfying some integrability condition.

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