# American Institute of Mathematical Sciences

August  2012, 5(4): 715-727. doi: 10.3934/dcdss.2012.5.715

## Multiplicity results to elliptic problems in $\mathbb{R}^N$

 1 Dipartimento Patrimonio Architettonico e Urbanistico, Facoltà di Architettura, Università di Reggio Calabria, Salita Melissari, 89124 Reggio Calabria, Italy 2 Department of Science for Engineering and Architecture (Mathematics Section), Engineering Faculty, University of Messina, 98166 - Messina, Italy

Received  January 2011 Revised  June 2011 Published  November 2011

The aim of this paper is to investigate elliptic variational-hemivariational inequalities on unbounded domains. In particular, by using a recent critical point theorem, existence results of at least two nontrivial solutions are established.
Citation: Giuseppina Barletta, Gabriele Bonanno. Multiplicity results to elliptic problems in $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 715-727. doi: 10.3934/dcdss.2012.5.715
##### References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. [2] G. Barletta, Existence of solutions for some discontinuous problems involving the p-Laplacian, J. Nonlinear Funct. Anal. Differ. Equ., 2 (2008), 95-119. [3] G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differential Equations, 244 (2008), 3031-3059. doi: 10.1016/j.jde.2008.02.025. [4] H. Brézis, "Analyse Fonctionnelle - Théorie et Applications," Collection Mathématiques Appliquées pur la Maîtrise, Masson, Paris, 1983. [5] F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics Appl. Math., 5, SIAM, Philadelphia, 1990. [6] J. V. Gonçalves and C. O. Alves, Existence of positive solutions for $m-$laplacian equations in $\mathbb{R}^N2$ involving critical Sobolev exponents, Nonlinear Anal., 32 (1998), 53-70. doi: 10.1016/S0362-546X(97)00452-5. [7] A. Kristály, C. Varga and V. Varga, A nonsmooth principle of symmetric criticality and variational–hemivariational inequalities, J. Math. Anal. Appl., 325 (2007), 975-986. doi: 10.1016/j.jmaa.2006.02.062. [8] A. Kristály, A double eigenvalue problem for Schrödinger equations involving sublinear nonlinearities at infinity, Electron. J. Differential Equations, 2007, 1-11. [9] S. A. Marano and D. Motreanu, On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems, Nonlinear Anal., 48 (2002), 37-52. doi: 10.1016/S0362-546X(00)00171-1. [10] B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226. [11] G. Zhang and S. Liu, Three symmetric solutions for a class of elliptic equations involving the p-Laplacian with discontinuous nonlinearities in $\mathbb{R}^N2$, Nonlinear Anal., 67 (2007), 2232-2239. doi: 10.1016/j.na.2006.09.013.

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##### References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. [2] G. Barletta, Existence of solutions for some discontinuous problems involving the p-Laplacian, J. Nonlinear Funct. Anal. Differ. Equ., 2 (2008), 95-119. [3] G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differential Equations, 244 (2008), 3031-3059. doi: 10.1016/j.jde.2008.02.025. [4] H. Brézis, "Analyse Fonctionnelle - Théorie et Applications," Collection Mathématiques Appliquées pur la Maîtrise, Masson, Paris, 1983. [5] F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics Appl. Math., 5, SIAM, Philadelphia, 1990. [6] J. V. Gonçalves and C. O. Alves, Existence of positive solutions for $m-$laplacian equations in $\mathbb{R}^N2$ involving critical Sobolev exponents, Nonlinear Anal., 32 (1998), 53-70. doi: 10.1016/S0362-546X(97)00452-5. [7] A. Kristály, C. Varga and V. Varga, A nonsmooth principle of symmetric criticality and variational–hemivariational inequalities, J. Math. Anal. Appl., 325 (2007), 975-986. doi: 10.1016/j.jmaa.2006.02.062. [8] A. Kristály, A double eigenvalue problem for Schrödinger equations involving sublinear nonlinearities at infinity, Electron. J. Differential Equations, 2007, 1-11. [9] S. A. Marano and D. Motreanu, On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems, Nonlinear Anal., 48 (2002), 37-52. doi: 10.1016/S0362-546X(00)00171-1. [10] B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226. [11] G. Zhang and S. Liu, Three symmetric solutions for a class of elliptic equations involving the p-Laplacian with discontinuous nonlinearities in $\mathbb{R}^N2$, Nonlinear Anal., 67 (2007), 2232-2239. doi: 10.1016/j.na.2006.09.013.
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