Existence of solution for quasilinear Schrödinger equations using a linking structure

Abstract

In this work, we establish existence of weak solutions for quasilinear Schrödinger equations where the potential is bounded from below and above by positive constants. The nonlinearity has an iteration with higher eigenvalues for the associated linear problem. Hence, the energy functional associated with our main problem admits a linking structure. The main difficulty here comes from the fact that zero is not anymore a local minimum for the energy functional. Hence, we apply a linking theorem proving existence of weak solutions for quasilinear Schrödinger equations provided that the nonlinear term is an asymptotically-superlinear function. Due to the lack of compactness for the Sobolev embeddings we need to recover some kind of compactness required in variational methods. In order to do that we apply some fine estimates together with Lions' Lemma.

Keywords:

• Variational methods
• quasilinear Schrödinger equations
• linking theorems
• superlinear elliptic equations
• bounded potentials

AMS Subject Classifications:

• 35J20
• 35J25
• 35J60
• 35J92

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was partially supported by CNPq grants 429955/2018-9.