A nonlinear Schrödinger equation resonating at an essential spectrum
Summary: We consider the nonlinear Schrödinger equation \(-\Delta u + f(u) = V(x) u\) in \(\mathbb{R}^N\). The potential function \(V\) satisfies that the essential spectrum of the Schrödinger operator \(-\Delta - V\) is \([0, + \infty)\) and this Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. The nonlinearity \(f\) satisfies the resonance type condition \(\operatorname{\lim}_{\left|t\right| \rightarrow \infty} f(t) / t = 0\). Under some additional conditions on \(V\) and \(f\), we prove that this equation has infinitely many solutions.
scientific article; zbMATH DE number 6957367
scientific article