Multiplicity of positive solutions for a quasilinear Schr\"odinger equation with an almost critical nonlinearity

TL;DR

This paper establishes the existence of multiple positive solutions for a quasilinear Schrödinger equation with a nonlinearity near the critical exponent, linking the number of solutions to topological features of the domain.

Contribution

It provides new existence results for multiple positive solutions in a quasilinear Schrödinger problem with almost critical nonlinearity, relating solutions to domain topology.

Findings

Number of solutions estimated by topological invariants

Solutions exist near the critical exponent

Results apply to smooth bounded domains

Abstract

In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem \begin{equation*} \left\{ \begin{array}[c]{ll} -\Delta u - \Delta (u^2)u = |u|^{p-2}u & \mbox{ in } \Omega u= 0 &\mbox{ on } \partial\Omega, \end{array} \right. \end{equation*} where $\Omega$ is a smooth and bounded domain in $\mathbb R^{N},N\geq3$. More specifically we prove that, for $p$ near the critical exponent $22^{*}=4N/(N-2)$, the number of positive solutions is estimated below by topological invariants of the domain $\Omega$: the Ljusternick-Schnirelmann category and the Poincar\'e polynomial.