Existence result for hemivariational inequality involving p(x)-Laplacian

• Autorzy: Barnaś S.
• Języki publikacji: EN

Abstrakty

EN

In this paper we study the nonlinear elliptic problem with p(x)-Laplacian (hemivariational inequality). We prove the existence of a nontrivial solution. Our approach is based on critical point theory for locally Lipschitz functionals due to Chang [J. Math. Anal. Appl. 80 (1981), 102-129].

Słowa kluczowe

EN

p(x)-Laplacian; Palais-Smale condition; mountain pass theorem; variable exponent Sobolev space

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2012
• Tom: Vol. 32, no. 3
• Strony: 439--454
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Barnaś S.

• Cracow University of Technology, Institute of Mathematics, ul. Warszawska 24, 31-155 Kraków, Poland,

Bibliografia

• [1] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
• [2] K.C. Chang, Critical Point Theory and Applications, Shanghai Scientific and Technology Press, Shanghai, 1996.
• [3] K.C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102–129.
• [4] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1993.
• [5] G. Dai, Infinitely many solutions for a hemivariational inequality involving the p(x)-Laplacian, Nonlinear Anal. 71 (2009), 186–195.
• [6] X. Fan, Eigenvalues of the p(x)-Laplacian Neumann problems, Nonlinear Anal. 67 (2007), 2982–2992.
• [7] X. Fan, Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl. 312 (2005), 464–477.
• [8] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), 1843–1853.
• [9] X.L. Fan, Q.H. Zhang, D. Zhao Eigenvalues of p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), 1843–1853.
• [10] X. Fan, Q. Zhang, D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), 306–317.
• [11] X.L. Fan, D. Zhao, On the generalized Orlicz - Sobolev space Wk,p(x)(Ω), J. Gansu Educ. College 12 (1998) 1, 1–6.
• [12] X. Fan, D. Zhao, On the spaces Lp(x)(Ω) and Wm;p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446.
• [13] L. Gasinski, N.S. Papageorgiou, Nonlinear hemivariational inequalities at resonance, Bull. Austr. Math. Soc. 60 (1999) 3, 353–364.
• [14] L. Gasinski, N.S. Papageorgiou, Solutions and Multiple Solutions for Quasilinear Hemivariational Inequalities at Resonance, Proc. Roy. Soc. Edinb. 131A (2001) 5, 1091–1111.
• [15] L. Gasinski, N.S. Papageorgiou, An existence theorem for nonlinear hemivariational inequalities at resonance, Bull. Austr. Math. Soc. 63 (2001) 1, 1–14.
• [16] L. Gasinski, N.S. Papageorgiou, Nonlinear Analysis, Chapman and Hall/ CRC Press, Boca Raton, FL, 2006.
• [17] B. Ge, X. Xue, Multiple solutions for inequality Dirichlet problems by the p(x)-Laplacian, Nonlinear Anal. Real World Appl. 11 (2010), 3198–3210.
• [18] B. Ge, X. Xue, Q. Zhou, The existence of radial solutions for differential inclusion problems in RN involving the p(x)-Laplacian, Nonlinear Anal. 73 (2010), 622–633.
• [19] S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluver, Dordrecht, The Netherlands, 1997.
• [20] N. Kourogenic, N.S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance, J. Aust. Math. Soc. 69 (2000), 245–271.
• [21] Z. Naniewicz, P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel-Dekker, New York, 1995.
• [22] Ch. Qian, Z. Shen, Existence and multiplicity of solutions for p(x)-Laplacian equation with nonsmooth potential, Nonlinear Anal. Real World Appl. 11 (2010), 106–116.
• [23] Ch. Qian, Z. Shen, M. Yang, Existence of solutions for p(x)-Laplacian nonhomogeneous Neumann problems with indefinite weight, Nonlinear Anal. 11 (2010), 446–458.