Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in RN

Mathematics

卷:8

Yun-Ho Kim¹ 

[1] Department of Mathematics Education, Sangmyung University, Seoul 03016, Korea;

关键词: fractional p-Laplacian;    variational methods;    critical point theory;   

DOI  :  10.3390/math8101792

来源: DOAJ

【 摘 要 】

We are concerned with the following elliptic equations: ${(-\Delta )}_p^{s}v+{V\left(x\right)\left|v\right|}^{p-2}v=\lambda a\left(x\right){\left|v\right|}^{r-2}v+g(x,v)in{\mathbb{R}}^N,$ where ${(-\Delta )}_p^s$ is the fractional p-Laplacian operator with $0<s<1<r<p<+\infty$, $sp<N$, the potential function $V:{\mathbb{R}}^N\to (0,\infty )$ is a continuous potential function, and $g:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem.

【 授权许可】

Unknown