Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells

Abstract

In this paper, we are concerned with the existence of least energy solutions of nonlinear Schrodinger equations involving the half Laplacian \begin{eqnarray} (-\Delta)^{1/2}u(x)+\lambda V(x)u(x)=u(x)^{p-1}, u(x)\geq 0, \quad x\in R^N, \end{eqnarray} for sufficiently large $\lambda$, $2 < p < \frac{2N}{N-1}$ for $N \geq 2$. $V(x)$ is a real continuous function on $R^N$. Using variational methods we prove the existence of least energy solution $u(x)$ which localize near the potential well int$(V^{-1}(0))$ for $\lambda$ large. Moreover, if the zero sets int$(V^{-1}(0))$ of $V(x)$ include more than one isolated components, then $u_\lambda(x)$ will be trapped around all the isolated components. However, in Laplacian case, when the parameter $\lambda$ large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrary small in other components of int$(V^{-1}(0))$. This is the essential difference with the Laplacian problems since the operator $(-\Delta)^{1/2}$ is nonlocal.