Quasilinear Schrödinger equations with a positive parameter and involving unbounded or decaying potentials

Abstract

We establish the existence of nonnegative and nonzero solutions for the following class of quasilinear Schrödinger equations: $\begin{array}{rl}& -\mathrm{\Delta }u+V\left(|x|\right)u+\frac{\kappa }{2}\left[\mathrm{\Delta }\right(u^2\left)\right]u=Q\left(|x|\right)h\left(u\right),x\in {\mathbb{R}}^N,\\ & u\left(x\right)\to 0\mathrm{a}\mathrm{s}|x|\to \mathrm{\infty },\end{array}$ where $N\ge 3$, κ is a positive parameter, V and Q are potentials that can be singular at the origin, unbounded or vanishing at infinity and the nonlinearity $h\left(u\right)$ is superlinear at infinity. In order to prove our main result, we have applied minimax methods together with careful $L^{\mathrm{\infty }}$-estimates and we need to use an argument of symmetric criticality principle type.

Keywords:

• Quasilinear Schrödinger equation
• unbounded or decaying potentials
• Moser iteration

2010 Mathematics Subject Classifications:

• 35J20
• 35J62
• 35J10
• 35J75

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Research partially supported by the National Institute of Science and Technology of Mathematics INCT-Mat, CAPES, CNPq grants 480746/2013-3 and 308735/2016-1.