Positive solutions for nonlinear Schrödinger equations with deepening potential well (Q1024258)

The paper aims to analyze the existence of positive real solutions to the following version of the multidimensional (with dimension \(D\geq 3\)) stationary nonlinear Schrödinger equation: \[ -\Delta u + (1+ \lambda g(x))u = f(u), \] where function \(g(x)\) describes a potential well, and nonlinear term \(f(u)\) is asymptotically linear, i.e., \(f(u)=u\phi(u)\), where \(\phi(u)\) is a nondecreasing function of \(u\) which takes a constant asymptotic value at infinity, \(\lim_{u\to\infty} \phi(u)=1 + \alpha\), with some positive \(\alpha\). By means of the mountain-pass theorem, the paper presents a rigorous proof of the existence of the solution in a certain region in the plane of \((\lambda,\alpha)\). Roughly, the solution tends to exist at large \(\alpha\), and it fails to exist at large \(\lambda\).