Ground states for strongly indefinite Schrödinger equations with competing nonlinearities

Abstract

We survey recent variational methods for strongly indefinite Schrödinger equations with sign-changing nonlinearities. The main object is an energy functional of the form \[ J(u)=\frac12\|u^+\|^2-\frac12\|u^-\|^2 -\int_{\mathbb{R}^N}F(u)\,dx+λ\int_{\mathbb{R}^N}G(u)\,dx, \] where the splitting $X=X^+\oplus X^-$ is induced by a spectral gap of the linear Schrödinger operator, and where the nonlinear part \[ I(u)=\int_{\mathbb{R}^N}F(u)\,dx-λ\int_{\mathbb{R}^N}G(u)\,dx \] is allowed to change sign. We discuss the generalized linking theorem developed for such functionals, and the abstract multiplicity theory for critical orbits in dislocation spaces. In the final part, we prove a new ground state result for the competing pure-power case \[ f(u)=|u|^{p-2}u,\qquad g(u)=|u|^{q-2}u,\qquad 2<q<p<2^*. \] More precisely, for $λ>0$ sufficiently small, the corresponding strongly indefinite functional possesses a nontrivial critical point of least energy among all nontrivial critical points.