Quasilinear elliptic equations on $$\mathbb{R}^N $$with infinitely many solutions

Abstract

We give verifiable conditions ensuring that second order quasilinear elliptic equations on \(\mathbb{R}^N \) have infinitely many solutions in the Sobolev space \(W^{2,p} (\mathbb{R}^N )\) for generic right-hand sides. This amounts to translating in concrete terms the more elusive hypotheses of an abstract theorem. Salient points include the proof that a key denseness property is equivalent to the existence of nontrivial solutions to an auxiliary problem, and an estimate of the size of the set of critical points of nonlinear Schrodinger operators. Conditions for the real-analyticity of Nemytskii operators are also discussed.