Sign-changing multi-bump solutions for nonlinear Schrödinger equations with steep potential wells

Abstract:

We study the nonlinear Schrödinger equations: \begin{equation*}(P_\lambda )\quad \qquad \qquad -\Delta u+(\lambda ^2 a(x)+1)u =|u|^{p-1}u, \quad u\in H^1(\mathbf {R}^N), \qquad \qquad \qquad \end{equation*} where $p>1$ is a subcritical exponent, $a(x)$ is a continuous function satisfying $a(x)\geq 0$, $0<\liminf _{|x|\to \infty } a(x)\leq \limsup _{|x|\to \infty }a(x)<\infty$ and $a^{-1}(0)$ consists of 2 connected bounded smooth components $\Omega _1$ and $\Omega _2$.

We study the existence of solutions $(u_\lambda )$ of $(P_\lambda )$ which converge to $0$ in $\mathbf {R}^N\setminus (\Omega _1\cup \Omega _2)$ and to a prescribed pair $(v_1(x),v_2(x))$ of solutions of the limit problem: \[ -\Delta v_i+v_i=|v_i|^{p-1}v_i\quad \mathrm {in}\; \Omega _i \] $(i=1,2)$ as $\lambda \to \infty$.