×

A quasilinear Neumann problem involving the \(p(x)\)-Laplacian. (English) Zbl 1177.35096

Summary: We study the following Neumann problem:
\[ \begin{cases} -\Delta_{p(x)}u+ \alpha(x)|u|^{p(x)-2}u= \alpha(x)f(u)+\lambda g(x,u) &\text{in }\Omega,\\ \frac{\partial u}{\partial v}=0 &\text{on }\partial\Omega, \end{cases} \]
and we prove that, under suitable assumptions on functions \(\alpha\), \(f\), \(p\) and \(g\), the Ricceri two-local-minima theorem, together with the Palais-Smale property, ensures the existence of at least three solutions of it. This work could be considered as a possible extension of some results by Cammaroto, Chinnì and Di Bella who handled the case where \(p(x)\) is constant.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations
35J20 Variational methods for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ružička, M., Electrorheological Fluids: Modeling and Mathematical Theory (2000), Springer-Verlag: Springer-Verlag Berlin · Zbl 0968.76531
[2] Zhikov, V. V., Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv., 9, 33-66 (1987) · Zbl 0599.49031
[3] Cammaroto, F.; Chinnì, A.; Di Bella, B., Some multiplicity results for quasilinear Neumann problems, Arch. Math., 86, 154-162 (2006) · Zbl 1206.35127
[4] Ricceri, B., Sublevel sets and global minima of coercive functionals and local minima of their perturbations, J. Nonlinear Convex Anal., 5, 157-168 (2004) · Zbl 1083.49004
[5] Fan, X. L.; Zhao, D., On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m, p(x)}(\Omega)\), J. Math. Appl., 263, 424-446 (2001) · Zbl 1028.46041
[6] Musielak, J., Orlicz spaces and modular spaces, (Dold, A.; Eckmann, B., Lecture Notes in Math., vol. 1034 (1983), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0557.46020
[7] Kováčik, O.; Rákosník, J., On spaces \(L^{p(x)}(\Omega)\) and \(W^{k, p(x)}(\Omega)\), Czechoslovak Math. J., 41, 592-618 (1991) · Zbl 0784.46029
[8] Fan, X. L.; Deng, Shao-Gao, Remarks on Ricceri’s variational principle and applications to the \(p(x)\)-Laplacian equations, Nonlinear Anal., 67, 3064-3075 (2007) · Zbl 1134.35035
[9] Naselli, O., A class of functionals on a Banach spaces for which strong and weak local minima do coincide, Optimization, 50, 407-411 (2001) · Zbl 1009.49015
[10] Brézis, H., Analyse fonctionelle, (Théorie et Applications (1983), Masson)
[11] Fan, X. L.; Zhang, Q. H., Existence of solutions for \(p(x)\)-Laplacian Dirichlet problems, Nonlinear Anal., 52, 1843-1852 (2003) · Zbl 1146.35353
[12] E. Zeidler, Nonlinear Functional Analysis and Applications. Vol. III, New York, 1985; E. Zeidler, Nonlinear Functional Analysis and Applications. Vol. III, New York, 1985 · Zbl 0583.47051
[13] Pucci, P.; Serrin, J., A mountain pass theorem, J. Differential Equations, 60, 142-149 (1985) · Zbl 0585.58006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.