Generalized linking theorem with an application to a semilinear Schrödinger equation. (English) Zbl 0947.35061

Summary: Consider the semilinear Schrödinger equation \[ -\Delta u+ V(x)u= f(x,u),\quad u\in H^1(\mathbb{R}^N).\tag{\(*\)} \] It is shown that if \(f\), \(V\) are periodic in the \(x\)-variables, \(f\) is superlinear at \(u= 0\) and \(\pm\infty\) and 0 lies in a spectral gap of \(-\Delta+V\), then \((*)\) has at least one nontrivial solution. If in addition \(f\) is odd in \(u\), then \((*)\) has infinitely many (geometrically distinct) solutions. The proofs rely on a degree-theory and a linking-type argument developed in this paper.


35J60 Nonlinear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J10 Schrödinger operator, Schrödinger equation
47H11 Degree theory for nonlinear operators