Opuscula Math. 36, no. 1 (2016), 81-101
http://dx.doi.org/10.7494/OpMath.2016.36.1.81

 
Opuscula Mathematica

Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces

Ionela-Loredana Stăncuţ
Iulia Dorotheea Stîrcu

Abstract. In this paper we consider an eigenvalue problem that involves a nonhomogeneous elliptic operator, variable growth conditions and a potential \(V\) on a bounded domain in \(\mathbb{R}^N\) (\(N\geq 3\)) with a smooth boundary. We establish three main results with various assumptions. The first one asserts that any \(\lambda\gt 0\) is an eigenvalue of our problem. The second theorem states the existence of a constant \(\lambda_{*}\gt 0\) such that any \(\lambda\in(0,\lambda_{*}]\) is an eigenvalue, while the third theorem claims the existence of a constant \(\lambda^{*}\gt 0\) such that every \(\lambda\in[\lambda^{*}, \infty)\) is an eigenvalue of the problem.

Keywords: anisotropic Orlicz-Sobolev space, potential, critical point, weak solution, eigenvalue.

Mathematics Subject Classification: 35D30, 35J60, 58E05.

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  1. E. Acerbi, G. Mingione, Gradient estimates for the \(p(x)\)-Laplacean system, J. Reine Angew. Math. 584 (2005), 117-148.
  2. R. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  3. D.R. Adams, L.I. Hedberg, Function Spaces and Potential Theory, [in:] Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 314, Springer-Verlag, Berlin, 1996.
  4. A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory, J. Funct. Anal. 14 (1973), 349-381.
  5. J. Chabrowski, Y. Fu, Existence of solutions for \(p(x)\)-Laplacian problems on a bounded domain, J. Math. Anal. Appl. 306 (2005), 604-618.
  6. Y. Chen, S. Levine, R. Rao, Functionals with p(x)-growth in image processing, Duquesne University, Department of Mathematics and Computer Science Technical Report 2004-01, available at http://www.mathcs.duq.edu/~sel/CLR05SIAPfinal.pdf.
  7. Ph. Clément, M. García-Huidobro, R. Manásevich, K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations 11 (2000), 33-62.
  8. Ph. Clément, B. de Pagter, G. Sweers, F. de Thélin, Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterr. J. Math. 1 (2004), 241-267.
  9. G. Dankert, Sobolev embedding theorems in Orlicz spaces, PhD Thesis, University of Köln, 1966.
  10. T.K. Donaldson, N.S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Funct. Anal. 8 (1971), 52-75.
  11. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353.
  12. X. Fan, Solutions for \(p(x)\)-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl. 312 (2005), 464-477.
  13. X. Fan, Q. Zhang, D. Zhao, Eigenvalues of \(p(x)\)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), 306-317.
  14. M. García-Huidobro, V.K. Le, R. Manásevich, K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting, NoDEA Nonlinear Differential Equations Appl. 6 (1999), 207-225.
  15. J.P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc. 190 (1974), 163-205.
  16. O. Kováčik, J. Rákosník, On spaces \(L^{p(x)}\) and \(W^{k,p(x)}\), Czechoslovak Math. J. 41 (1991) 4, 592-618.
  17. M. Mihăilescu, G. Moroşanu, V. Rădulescu, Eigenvalue problems for anisotropic elliptic equations: an Orlicz-Sobolev space setting, Nonlinear Anal. 73 (2010), 3239-3252.
  18. M. Mihăilescu, V. Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. London Ser. A 462 (2006), 2625-2641.
  19. M. Mihăilescu, V. Rădulescu, Eigenvalue problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces, Analysis and Applications 6 (2008) 1, 1-16.
  20. M. Mihăilescu, V. Rădulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl. 330 (2007) 1, 416-432.
  21. M. Mihăilescu, V. Rădulescu, Neumann problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces, Ann. Inst. Fourier 58 (2008) 6, 2087-2111.
  22. J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034, Springer, Berlin, 1983.
  23. H. Nakano, Modulared Semi-ordered Linear Spaces, Maruzen Co., Ltd, Tokyo, 1950.
  24. R. O'Neill, Fractional integration in Orlicz spaces, Trans. Amer. Math. Soc. 115 (1965), 300-328.
  25. W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), 200-212.
  26. M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991.
  27. V. Rădulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal., in press, DOI 10.1016/j.na.2014.11.007. http://dx.doi.org/10.1016/j.na.2014.11.007.
  28. V. Rădulescu, D. Repovš, Partial Differential Equations with Variable Exponents: Variational methods and Quantitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015.
  29. V. Rădulescu, I. Stăncuţ, Combined concave-convex effects in anisotropic elliptic equations with variable exponent, NoDEA Nonlinear Differential Equations Appl., in press, DOI 10.1007/s00030-014-0288-8. http://dx.doi.org/10.1007/s00030-014-0288-8.
  30. V. Rădulescu, B. Zhang, Morse theory and local linking for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Nonlinear Anal. Real World Appl. 17 (2014), 311-321.
  31. M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin, 2000.
  32. M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, Heidelberg, 1996.
  33. M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
  • Ionela-Loredana Stăncuţ
  • University of Craiova, Department of Mathematics, 200585 Craiova, Romania
  • Iulia Dorotheea Stîrcu
  • University of Craiova, Department of Mathematics, 200585 Craiova, Romania
  • Communicated by Vicentiu D. Radulescu.
  • Received: 2015-01-09.
  • Accepted: 2015-03-23.
  • Published online: 2015-09-19.
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Cite this article as:
Ionela-Loredana Stăncuţ, Iulia Dorotheea Stîrcu, Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces, Opuscula Math. 36, no. 1 (2016), 81-101, http://dx.doi.org/10.7494/OpMath.2016.36.1.81

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