Multiplicity of positive solutions for a quasilinear Schrödinger equation with an almost critical nonlinearity

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} -Δu - Δ(u^2)u = |u|^{p-2}u & \mbox{ in } Ωu= 0 &\mbox{ on } \partialΩ, \end{array} \right. \end{equation*} where $\Omega$ is a smooth and bounded domain in ${\mathbb{R}}^N,N\ge 3$. More specifically we prove that, for $p$ near the critical exponent $22^{\ast }=4N/(N-2)$, the number of positive solutions is estimated below by topological invariants of the domain $\Omega$: the Ljusternick-Schnirelmann category and the Poincaré polynomial.