Electron. J. Diff. Equ., Vol. 2010(2010), No. 162, pp. 1-23.

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

Nikolaos E. Sidiropoulos

Abstract:
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.

Submitted July 7, 2010. Published November 12, 2010.
Math Subject Classifications: 35J60, 35J62, 35J92.
Key Words: Indefinite quasilinear elliptic problems; subcritical nonlinearities; p-Laplacian; p-q-Laplacian; fibering method; mountain pass theorem.

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Nikolaos E. Sidiropoulos
Department of Sciences, Technical University of Crete
73100 Chania, Greece
email: niksidirop@gmail.com

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