Uberlandio B. Severo | Searcharxiv

arXiv2018

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

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 $Ω$ 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 $Ω$: the Ljusternick-Schnirelmann category and the Poincaré polynomial.

math.AP↗

arXiv2013

Ground state solution for a Kirchhoff problem with exponential critical growth

We establish the existence of a positive ground state solution for a Kirchhoff problem in $\mathbb{R}^2$ involving critical exponential growth, that is, the nonlinearity behaves like $\exp(α_{0}s^{2})$ as $|s| \to \infty$, for some $α_{0}>0$. In order to obtain our existence result we used minimax techniques combined with the Trudinger-Moser inequality.

math.AP↗