Existence and asymptotic behavior of solutions for nonlinear Schrödinger-Poisson systems with steep potential well (Q2798663)

The paper is concerned with the Schrödinger-Poisson system NEWLINE\[NEWLINE \begin{cases} -\Delta u+\lambda V(x)u+K(x)\phi u= a(x)f(u)\\ -\Delta \phi = K(x)u^2 \end{cases} NEWLINE\]NEWLINE on \(\mathbb R^3\). Under various conditions on \(V\), \(K\), \(a\), \(f\) the existence of a solution is obtained using variational methods. If \(V\geq 0\), \(\Omega=\operatorname{int} V^{-1}(0)\neq\emptyset\), and if the measure of \(\{x\in\mathbb R^3: V(x)<b\}\) is finite for some \(b>0\) then the behaviour of the solutions as \(\lambda\to\infty\) is investigated.