Abstract:

Consider the following fractional Schrödinger equation

$\begin{equation}\renewcommand{\theequation}{$P_{\lambda}$} (-\Delta)^{s}u+\lambda V(x)u+V_{0}(x)u=P(x)|u|^{p-2}u+Q(x)|u|^{q-2}u, ~~~~~x\in {\Bbb R}^{N}, \end{equation}$

where $\lambda>0$, $s\in(0, 1)$, $N>2s$, $2<q<p<2_{s}^{\ast}$ ($2_{s}^{\ast}=\frac{2N}{N-2s}$), $P\in L^{\infty}$ is positive, $Q\in L^{\infty}$ may be positive, sign-changing or negative, $V$ is steep well potential, and $V_{0}\in L^{\infty}$. When $\lambda$ is large, the existence of nontrivial solutions is obtained via variational methods. Furthermore, if $V(x)\geq0$, concentration results are also obtained. In particular, the potential $V$ is allowed to be sign-changing for the existence.

Key words: Fractional Schrödinger equations, Steep well potential, Sign-changing potential, Concentration