Opuscula Math. 35, no. 6 (2015), 853-866
http://dx.doi.org/10.7494/OpMath.2015.35.6.853

Opuscula Mathematica

# Continuous spectrum of Steklov nonhomogeneous elliptic problem

Mostafa Allaoui

Abstract. By applying two versions of the mountain pass theorem and Ekeland's variational principle, we prove three different situations of the existence of solutions for the following Steklov problem: \begin{aligned}\Delta_{p(x)} u&=|u|^{p(x)-2}u \phantom{\lambda} \quad\text{in}\;\Omega, \\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}&= \lambda|u|^{q(x)-2}u \quad\text{on}\;\partial\Omega,\end{aligned} where $$\Omega \subset \mathbb{R}^N$$ $$(N\geq 2)$$ is a bounded smooth domain and $$p,q: \overline{\Omega}\rightarrow(1,+\infty)$$ are continuous functions.

Keywords: $$p(x)$$-Laplacian, Steklov problem, critical point theorem.

Mathematics Subject Classification: 35J48, 35J66.

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1. M. Allaoui, A. El Amrouss, A. Ourraoui, Existence and multiplicity of solutions for a Steklov problem involving the $$p(x)$$-Laplace operator, Electron. J. Diff. Equ. 132 (2012), 1-12.
2. J. Chabrowski, Y. Fu, Existence of solutions for $$p(x)$$-Laplacian problems on a bounded domain, J. Math. Anal. Appl. 306 (2005), 604-618.
3. Y.M. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383-1406.
4. S.G. Dend, Eigenvalues of the $$p(x)$$-Laplacian Steklov problem, J. Math. Anal. Appl. 339 (2008), 925-937.
5. S.G. Deng, A local mountain pass theorem and applications to a double perturbed $$p(x)$$-Laplacian equations, Appl. Math. Comput. 211 (2009), 234-241.
6. X. Ding, X. Shi, Existence and multiplicity of solutions for a general $$p(x)$$-Laplacian Neumann problem, Nonlinear Anal. 70 (2009), 3713-3720.
7. X.L. Fan, J.S. Shen, D. Zhao, Sobolev embedding theorems for spaces $$W^{k,p(x)}$$, J. Math. Anal. Appl. 262 (2001), 749-760.
8. X.L. Fan, D. Zhao, On the spaces $$L^{p(x)}$$ and $$W^{m,p(x)}$$, J. Math. Anal. Appl. 263 (2001), 424-446.
9. X.L. Fan, S.G. Deng, Remarks on Ricceri's variational principle and applications to the $$p(x)$$-Laplacian equations, Nonlinear Anal. 67 (2007), 3064-3075.
10. R. Filippucci, P. Pucci, V. Radulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Comm. Partial Differential Equations 33 (2008), 706-717.
11. P. Harjulehto, P. Hästö, Ú.V. Lê, M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), 4551-4574.
12. R. Kajikia, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal. 225 (2005), 352-370.
13. N. Mavinga, M.N. Nkashama, Steklov spectrum and nonresonance for elliptic equations with nonlinear boundary conditions, Electron. J. Diff. Equ. Conf. 19 (2010), 197-205.
14. M. Mihailescu, V. Radulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc. 135 (2007), 2929-2937.
15. M. Mihăilescu, V. Rădulescu, Eigenvalue problems associated with nonhomogenenous differential operators in Orlicz-Sobolev spaces, Anal. Appl. 6 (2008), 83-98.
16. T.G. Myers, Thin films with high surface tension, SIAM Review 40 (1998), 441-462.
17. V. Radulescu, I. Stancut, Combined concave-convex effects in anisotropic elliptic equations with variable exponent, Nonlinear Differential Equations Appl., in press (DOI 10.1007/s00030-014-0288-8). http://dx.doi.org/10.1007/s00030-014-0288-8.
18. M. Rǔžicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.
19. L.L. Wang, Y.H. Fan, W.G. Ge, Existence and multiplicity of solutions for a Neumann problem involving the $$p(x)$$-Laplace operator, Nonlinear Anal. 71 (2009), 4259-4270.
20. Q.H. Zhang, Existence of solutions for $$p(x)$$-Laplacian equations with singular coefficients in $$R^N$$, J. Math. Anal. Appl. 348 (2008), 38-50.
21. V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. 29 (1987), 33-66.
22. V.V. Zhikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, translated from Russian by G.A. Yosifian, Springer-Verlag, Berlin, 1994.
• Mostafa Allaoui
• University Mohamed I, Faculty of Sciences, Department of Mathematics, Oujda, Morocco
• Communicated by Vicentiu D. Radulescu.
• Revised: 2014-11-10.
• Accepted: 2014-11-13.
• Published online: 2015-06-06.