Existence of positive solution for a class of generalised Schrödinger equation with supercritical growth

Abstract

We are concerned on the generalized quasilinear Schrödinger problems $(P)-\mathrm{div}(\vartheta (u)\mathrm{\nabla u})+\frac{1}{2}{\vartheta }^{\mathrm{\prime }}(u)|\mathrm{\nabla u}{|}^2+V(x)u=p(u)\text{in}{\mathbb{R}}^N,$ with $N\ge 3$, $V:{\mathbb{R}}^N\to \mathbb{R}$ is a function called potential and $p:\mathbb{R}\to \mathbb{R}$ is a continuous function. We prove the existence of positive solutions for the problem (P), for both the periodic case and the nonperiodic case, where depending on the choice of ϑ we get solutions that are related to the solitary wave solutions for quasilinear Schrödinger equations.

Keywords:

• Quasilinear Schrödinger equations
• variational methods
• supercritical exponent

2020 Mathematics Subject Classifications:

• Primary: 35J10
• 35J20
• 35J60
• Secondary: 35Q55

Disclosure statement

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

Additional information

Funding

Giovany M. Figueiredo supported in part by CNPQ, FAPDF and CAPES. Sandra I. Moreira supported in part by CNPQ, Processo 310825/2022-9. Ricardo Ruviaro supported in part by CNPQ, FAPDF and CAPES.