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

We study the nonlinear Schrödinger equations:

$$(P_\lambda)\quad\qquad\qquad-\Delta u+(\lambda^2 a(x)+1)u =\vert u\vert^{p-1}u, \quad u\in H^1(\mathbf{R}^N), \qquad\qquad\qquad$$

where $p>1$ is a subcritical exponent, $a(x)$ is a continuous function satisfying $a(x)\geq 0$, $0<\liminf_{\vert x\vert\to\infty} a(x)\leq \limsup_{\vert x\vert\to\infty}a(x)<\infty$ 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:

$$-\Delta v_i+v_i=\vert v_i\vert^{p-1}v_i\quad \hbox{in} \Omega_i$$

$(i=1,2)$ as $\lambda\to \infty$.