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May 2018, 17(3): 1201-1217. doi: 10.3934/cpaa.2018058
Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential
Song Peng and Aliang Xia ,
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
Received March 2017 Revised June 2017 Published January 2018
Fund Project: This work was supported by the Foundation of Jiangxi Provincial Education Department (No: GJJ160335), the NSFC (No. 11701239) and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University
In this article, we prove the existence, multiplicity and concentration of non-trivial solutions for the following indefinite fractional elliptic equation with concave-convex nonlinearities:
$\left\{\begin{array}{*{20}{l}}(-Δ)^α u+V_λ(x)u = a(x)|u|^{q-2}u+b(x)|u|^{p-2}u &{\rm in}\,\,\mathbb{R}^N,\\u≥0\,\,&{\rm in}\,\,\mathbb{R}^N, \end{array} \right.$
where $0<α<1$, $N>2α$, $1<q<2<p<2^*_α$ with $ 2^*_α = 2N/(N-2α)$, the potential $V_λ(x) = λ V^+(x)-V^-(x)$ with $V^± = \max\{± V, 0\}$ and the parameter $λ>0$. Our multiplicity results are based on studying the decomposition of the Nehari manifold.
Keywords: Fractional Laplacian, steep potential, Nehari manifold, concave-convex term.
Mathematics Subject Classification: Primary: 35J60, 47G20.
Citation: Song Peng, Aliang Xia. Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1201-1217. doi: 10.3934/cpaa.2018058
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Song Peng Aliang Xia