Multiple solutions to some singular nonlinear Schrödinger equations (Q2703364)

The paper deals with existence and multiplicity results for the semilinear elliptic problem NEWLINE\[NEWLINE -h^2 \Delta u+V_\varepsilon (x)u=|u|^{p-2}u\quad \text{ in } {\mathbb R}^N,\qquad \lim_{|x|\to\infty}u(x)=0 NEWLINE\]NEWLINE provided \(h\) small enough. Here \(V_\varepsilon (x)=V(x)-\varepsilon (h)W(x),\) \(0<\inf_{{\mathbb R}^N}V<\liminf_{|x|\to\infty}V(x),\) \(\varepsilon (h)=O(h^2)\) as \(h\to 0\) and \(W: {\mathbb R}^N\to [0,\infty)\) is a measurable function such that NEWLINE\[NEWLINE \int_{{\mathbb R}^N} W(x)|u|^2\leq \alpha_1 \|\nabla u\|^2_2 +\alpha_2 \|u\|^2_2 NEWLINE\]NEWLINE holds for any \(u\in H^1({\mathbb R}^N)\) and some \(\alpha_1>0,\) \(\alpha_2\geq 0.\)