Asymptotic bifurcation problems for quasilinear equations existence and multiplicity results

Pavel Drábek

DOI: http://dx.doi.org/10.12775/TMNA.2005.009

Abstract


In this paper we address the existence and multiplicity
results for
$$
\cases
-\Delta_p u -\lambda |u|^{p-2} u = h (x,u) &\text{in }\Omega,
\\
u = 0 &\text{on } \partial \Omega,
\endcases
$$
where $p> 1$, $\Delta_p u = \text{\rm div}(|\nabla u|^{p-2}\nabla u)$,
$h$ 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.

Keywords


p-Laplacian; Fredholm alternative; bifurcation; asymptotic estimates

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