IJPAM: Volume 80, No. 5 (2012)

FOR THE $(p,q)$-LAPLACIAN IN $\mathbb{R}^{N}$

Nawel Benouhiba$^1$, Zahia Belyacine$^2$
$^{1,2}$Department of Mathematics
LAM, Badji Mokhtar-Annaba University
P.O. Box 12, El Hadjar, 23000, Annaba, ALGERIA

Abstract. This paper concerns the study of a nonlinear eigenvalue problem for the $(p,q)-$Laplacian with a positive weight \begin{equation*}
-\Delta_{p}u-\Delta_{q}u=\lambda g(x) \vert u\vert^{p-2}u\text{ in } \Real^{N}.
\end{equation*} Using the Mountain-Pass Theorem, we show the existence of a continuous set of positive eigenvalues.

Received: August 28, 2012

AMS Subject Classification: 35B33, 35B45, 35J60, 35J70

Key Words and Phrases: $(p,q)$-Laplacian, eigenvalue, Weak solution, Palais-Smale condition

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 80
Issue: 5