Existence of weak solutions to a class of Schr\"{o}dinger type equations involving the fractional p-Laplacian in $\Bbb R^{N}$

J. Korean Math. Soc. Published online 2019 Apr 10

Jae-Myoung Kim, Yun-Ho Kim, and Jongrak Lee
Yonsei University, Sangmyung University, Ewha Womans University

Abstract : We are concerned with the following elliptic equations:
\begin{equation*}
(-\Delta)_p^{s}u+V(x)|u|^{p-2}u=\lambda g(x,u) \quad \text{in} \quad
\Bbb R^{N},
\end{equation*}
where $(-\Delta)^s_p$ is the fractional $p$-Laplacian operator with $0<s<1<p<+\infty$, $sp<N$, the potential function $V:\Bbb R^{N}\to(0,\infty)$ is a continuous potential function, and $g:\Bbb R^{N}\times\Bbb R \to \Bbb R$ satisfies a Carath\'eodory condition. We show the existence of at least one weak solution for the problem above without the Ambrosetti and Rabinowitz condition. Moreover, we give a positive interval of the parameters $\lambda$ for which the problem admits at least one nontrivial weak solution when the nonlinearity $g$ has the
subcritical growth condition.

Keywords : Fractional p-Laplacian; Variational methods; Critical point theory