Existence of solutions to indefinite quasilinear elliptic problems of \(p\)-\(q\)-Laplacian type

Sidiropoulos, Nikolaos E.

Electronic journal of differential equations, Tome 2010 (2010)

Résumé

We study the indefinite quasilinear elliptic problem

$\displaylines{ -\Delta u-\Delta _{p}u=a(x)|u|^{q-2}u-b(x)|u|^{s-2}u \quad\hbox{in }\Omega , \cr u=0\quad\hbox{on }\partial \Omega , }$

where $\Omega $ is a bounded domain in $\mathbb{R}^{N}, N\geq 2$, with a sufficiently smooth boundary, $q,s$ are subcritical exponents, $a(\cdot)$ changes sign and $b(x)\geq 0$ a.e. in $\Omega $. Our proofs are variational in character and are based either on the fibering method or the mountain pass theorem.

Détail

Classification : 35J60, 35J62, 35J92
Keywords: indefinite quasilinear elliptic problems, subcritical nonlinearities, p-Laplacian, p-q-Laplacian, fibering method, mountain pass theorem