Electron. J. Diff. Equ., Vol. 2013 (2013), No. 158, pp. 1-12.

Existence of solutions for a Neumann problem involving the p(x)-Laplacian

Giuseppina Barletta, Antonia Chinni

We study the existence and multiplicity of weak solutions for a parametric Neumann problem driven by the p(x)-Laplacian. Under a suitable condition on the behavior of the potential at $0^+$, we obtain an interval such that when a parameter $\lambda$ is in this interval, our problem admits at least one nontrivial weak solution. We show the multiplicity of solutions for potentials satisfying also the Ambrosetti-Rabinowitz condition. Moreover, if the right-hand side f satisfies the Ambrosetti-Rabinowitz condition, then we obtain the existence of two nontrivial weak solutions.

Submitted March 29, 2013. Published July 10, 2013.
Math Subject Classifications: 35J60, 35J20.
Key Words: p(x)-Laplacian; variable exponent Sobolev spaces.

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Giuseppina Barletta
Università degli Studi Mediterranea di Reggio Calabria
MECMAT-Dipartimento di Meccanica e Materiali, Via Graziella
Località Feo di Vito, 89100 Reggio Calabria, Italy
email: giuseppina.barletta@unirc.it
Antonia Chinnì
Department of Civil, Information Technology, Construction,
Environmental Engineering and Applied Mathematics
University of Messina, 98166 Messina, Italy
email: achinni@unime.it

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