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

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

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

Giuseppina Barletta ^1, and Gabriele Bonanno ^2,

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

Abstract
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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.

Keywords: multiple solutions, elliptic problem, variational methods., Variational-hemivariational inequality.

Mathematics Subject Classification: Primary: 35J60, 35J87; Secondary: 58E3.

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.

show all references

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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