Asymptotic bifurcation problems for quasilinear equations existence and multiplicity results

Abstract

In this paper we address the existence and multiplicity results for $$ \begin{cases} -\Delta_p u -\lambda |u|^{p-2} u = h (x,u) &\text{in }\Omega, \\ u = 0 &\text{on } \partial \Omega, \end{cases} $$ where $p> 1$, $\Delta_p u = {\rm div}(|\nabla u|^{p-2}\nabla u)$, $h$ is a bounded function and the spectral parameter $\lambda$ stays "near" the principal eigenvalue of the $p$-Laplacian.

We show how the bifurcation theory combined with certain asymptotic estimates yield desired results.