𝑝(𝑥)-Laplacian with indefinite weight

Kefi, Khaled

doi:10.1090/S0002-9939-2011-10850-5

$p(x)$-Laplacian with indefinite weight
HTML articles powered by AMS MathViewer

by Khaled Kefi

Proc. Amer. Math. Soc. 139 (2011), 4351-4360

DOI: https://doi.org/10.1090/S0002-9939-2011-10850-5

Published electronically: April 20, 2011

PDF | Request permission

Abstract:

We consider the eigenvalue problem $-\textrm {div}\big (|\nabla u|^{p(x)-2}\nabla u\big )=$ $\lambda V(x)|u|^{q(x)-2}u$, in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in $\mathbb {R}^{N}$, $\lambda >0$, $p,q$ are continuous functions on $\overline {\Omega }$ and $V$ is a given function in a generalized Lebesgue space $L^{s(x)}(\Omega )$ such that $V>0$ in an open set $\Omega _{0}\subset \Omega$, where $|\Omega _{0}| >0$. We prove under appropriate conditions on the functions $p,q$ and $s$ that any $\lambda >0$ sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland’s variational principle.

References

Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7
Olfa Allegue and Mounir Bezzarga, A multiplicity of solutions for a nonlinear degenerate problem involving a $p(x)$-Laplace-type operator, Complex Var. Elliptic Equ. 55 (2010), no. 5-6, 417–429. MR 2641985, DOI 10.1080/17476930903443775
Claudianor O. Alves and Marco A. S. Souto, Existence of solutions for a class of problems in $\Bbb R^N$ involving the $p(x)$-Laplacian, Contributions to nonlinear analysis, Progr. Nonlinear Differential Equations Appl., vol. 66, Birkhäuser, Basel, 2006, pp. 17–32. MR 2187792, DOI 10.1007/3-7643-7401-2_{2}
K. Benali and M. Bezzarga, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Potential Theory and Stochastics in Albac, Aurel Cornea Memorial Volume, Conference Proceedings Albac, September 4-8, 2007, Theta, 2008, pp. 21-34.
Khaled Benali and Khaled Kefi, Mountain pass and Ekeland’s principle for eigenvalue problem with variable exponent, Complex Var. Elliptic Equ. 54 (2009), no. 8, 795–809. MR 2544002, DOI 10.1080/17476930902999041
Jan Chabrowski and Yongqiang Fu, Existence of solutions for $p(x)$-Laplacian problems on a bounded domain, J. Math. Anal. Appl. 306 (2005), no. 2, 604–618. MR 2136336, DOI 10.1016/j.jmaa.2004.10.028
David E. Edmunds and Jiří Rákosník, Sobolev embeddings with variable exponent, Studia Math. 143 (2000), no. 3, 267–293. MR 1815935, DOI 10.4064/sm-143-3-267-293
Xianling Fan and Xiaoyou Han, Existence and multiplicity of solutions for $p(x)$-Laplacian equations in $\mathbf R^N$, Nonlinear Anal. 59 (2004), no. 1-2, 173–188. MR 2092084, DOI 10.1016/j.na.2004.07.009
Xianling Fan, Jishen Shen, and Dun Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega )$, J. Math. Anal. Appl. 262 (2001), no. 2, 749–760. MR 1859337, DOI 10.1006/jmaa.2001.7618
X. Fan and D. Zhao, On the generalized Orlicz-Sobolev space $W^{k,p(x)}(\Omega )$, J. Gancu Educ. College 12 (1) (1998), 1-6.
Xian-Ling Fan and Qi-Hu Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), no. 8, 1843–1852. MR 1954585, DOI 10.1016/S0362-546X(02)00150-5
Xianling Fan and Dun Zhao, On the spaces $L^{p(x)}(\Omega )$ and $W^{m,p(x)}(\Omega )$, J. Math. Anal. Appl. 263 (2001), no. 2, 424–446. MR 1866056, DOI 10.1006/jmaa.2000.7617
X. L. Fan, Q. Zhang and D. Zhao, Eigenvalues for $p(x)$-Laplacian Dirichlet problem, Non-linear Anal. 52 (2003), 1843-1852.
H. Le Dret, M2-Équations aux dérivées partielles elliptiques, Notes de cours, Université Pierre et Marie Curie (2005).
Ondrej Kováčik and Jiří Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41(116) (1991), no. 4, 592–618. MR 1134951
Julian Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. MR 724434, DOI 10.1007/BFb0072210
Mihai Mihăilescu and Vicenţiu Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006), no. 2073, 2625–2641. MR 2253555, DOI 10.1098/rspa.2005.1633
Mihai Mihăilescu and Vicenţiu Rădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2929–2937. MR 2317971, DOI 10.1090/S0002-9939-07-08815-6
Mihai Mihăilescu and Vicenţiu Rădulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl. 330 (2007), no. 1, 416–432. MR 2302933, DOI 10.1016/j.jmaa.2006.07.082
Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 845785, DOI 10.1090/cbms/065