Asymptotic bifurcation problems for quasilinear equations existence and multiplicity results - Drábek, Pavel (Topological Methods in Nonlinear Analysis, 2005)
In this paper we address the existence and multiplicityresults for $$\cases -\Delta_p u -\lambda |u|^{p-2} u = h (x,u) &\text{in }\Omega,\\ u = 0 &\text{on } \partial \Omega, \endcases$$where $p> 1$, $\Delta_p u = \text{\rm div}(| abla u|^{p-2} abla u)$, $h$ is abounded function and the spectral parameter $\lambda$ stays ``near'' theprincipal eigenvalue of the $p$-Laplacian.We show how the bifurcation theory combined with certain asymptoticestimates yield desired results.