# American Institute of Mathematical Sciences

July  2015, 35(7): 3087-3102. doi: 10.3934/dcds.2015.35.3087

## Multiple solutions to elliptic inclusions via critical point theory on closed convex sets

 1 Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy

Received  August 2014 Revised  October 2014 Published  January 2015

The existence of multiple solutions $u\in H^1_0(\Omega)$ to a differential inclusion of the type $-\Delta u\in \partial J(x,u)$ in $\Omega$, where $\partial J(x,\cdot)$ denotes the generalized sub-differential of $J(x,\cdot)$, is investigated through critical point theorems for locally Lipschitz continuous functionals on closed convex sets of a Hilbert space.
Citation: Salvatore A. Marano, Sunra J. N. Mosconi. Multiple solutions to elliptic inclusions via critical point theory on closed convex sets. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3087-3102. doi: 10.3934/dcds.2015.35.3087
##### References:
 [1] G. Alberti, S. Bianchini and G. Crippa, On the $L^p$-differentiability of certain classes of functions, Rev. Mat. Iberoam., 30 (2014), 349-367. doi: 10.4171/RMI/782.  Google Scholar [2] D. Averna, S. A. Marano and D. Motreanu, Multiple solutions for a Dirichlet problem with $p$-Laplacian and set-valued nonlinearity, Bull. Austral. Math. Soc., 77 (2008), 285-303. doi: 10.1017/S0004972708000282.  Google Scholar [3] S. Carl and S. Heikkilä, Fixed Point Theory in Ordered Sets and Applications, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-7585-0.  Google Scholar [4] S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and their Inequalities, Springer Monogr. Math., Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-46252-3.  Google Scholar [5] S. Carl and D. Motreanu, Sub-supersolution method for multi-valued elliptic and evolution problems, in Handbook of Nonconvex Analysis and Applications (eds D. Y. Gao and D. Motreanu), International Press, 2010, 45-98.  Google Scholar [6] F. H. Clarke, Optimization and Nonsmooth Analysis, Classics Appl. Math., 5, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971309.  Google Scholar [7] L. Gasiński and N. S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance, Discrete Contin. Dyn. Syst., 34 (2014), 2037-2060.  Google Scholar [8] L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Ser. Math. Anal. Appl., 8, Chapman and Hall/CRC Press, Boca Raton, 2005.  Google Scholar [9] S. Hu and N. S. Papageorgiou, Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems, Comm. Pure Appl. Anal., 12 (2013), 2889-2922. doi: 10.3934/cpaa.2013.12.2889.  Google Scholar [10] A. Iannizzotto, S. A. Marano and D. Motreanu, Positive, negative, and nodal solutions to elliptic differential inclusions depending on a parameter, Adv. Nonlinear Stud., 13 (2013), 431-445.  Google Scholar [11] S. Th. Kyritsi and N. S. Papageorgiou, An obstacle problem for nonlinear hemivariational inequalities at resonance, J. Math. Anal. Appl., 276 (2002), 292-313. doi: 10.1016/S0022-247X(02)00443-2.  Google Scholar [12] S. Th. Kyritsi and N. S. Papageorgiou, Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities, Nonlinear Anal., 61 (2005), 373-403. doi: 10.1016/j.na.2004.12.001.  Google Scholar [13] V. K. Le, Existence and enclosure of solutions to noncoercive systems of inequalities with multivalued mappings and non-power growths, Discrete Contin. Dyn. Syst., 33 (2013), 255-276. doi: 10.3934/dcds.2013.33.255.  Google Scholar [14] S. A. Marano and S. J. N. Mosconi, Non-smooth critical point theory on closed convex sets, Comm. Pure Appl. Anal., 13 (2014), 1187-1202. doi: 10.3934/cpaa.2014.13.1187.  Google Scholar [15] S. A. Marano and S. J. N. Mosconi, Critical points on closed convex sets vs. critical points and applications,, submitted for publication., ().   Google Scholar [16] S. A. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Comm. Pure Appl. Anal., 12 (2013), 815-829. doi: 10.3934/cpaa.2013.12.815.  Google Scholar [17] A. C. Ponce, Selected problems on elliptic equations involving measures,, , ().   Google Scholar [18] P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.  Google Scholar [19] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1997.  Google Scholar [20] W. Rudin, Real and Complex Analysis, $3^{rd}$dition, Mac-Graw-Hill, New York, 1987.  Google Scholar [21] J. S. Raymond, On the multiplicity of the solutions of the equation $-\Delta u=\lambda\cdot f(u)$, J. Differential Equations, 180 (2002), 65-88. doi: 10.1006/jdeq.2001.4057.  Google Scholar [22] W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Grad. Texts in Math., {120}, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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##### References:
 [1] G. Alberti, S. Bianchini and G. Crippa, On the $L^p$-differentiability of certain classes of functions, Rev. Mat. Iberoam., 30 (2014), 349-367. doi: 10.4171/RMI/782.  Google Scholar [2] D. Averna, S. A. Marano and D. Motreanu, Multiple solutions for a Dirichlet problem with $p$-Laplacian and set-valued nonlinearity, Bull. Austral. Math. Soc., 77 (2008), 285-303. doi: 10.1017/S0004972708000282.  Google Scholar [3] S. Carl and S. Heikkilä, Fixed Point Theory in Ordered Sets and Applications, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-7585-0.  Google Scholar [4] S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and their Inequalities, Springer Monogr. Math., Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-46252-3.  Google Scholar [5] S. Carl and D. Motreanu, Sub-supersolution method for multi-valued elliptic and evolution problems, in Handbook of Nonconvex Analysis and Applications (eds D. Y. Gao and D. Motreanu), International Press, 2010, 45-98.  Google Scholar [6] F. H. Clarke, Optimization and Nonsmooth Analysis, Classics Appl. Math., 5, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971309.  Google Scholar [7] L. Gasiński and N. S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance, Discrete Contin. Dyn. Syst., 34 (2014), 2037-2060.  Google Scholar [8] L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Ser. Math. Anal. Appl., 8, Chapman and Hall/CRC Press, Boca Raton, 2005.  Google Scholar [9] S. Hu and N. S. Papageorgiou, Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems, Comm. Pure Appl. Anal., 12 (2013), 2889-2922. doi: 10.3934/cpaa.2013.12.2889.  Google Scholar [10] A. Iannizzotto, S. A. Marano and D. Motreanu, Positive, negative, and nodal solutions to elliptic differential inclusions depending on a parameter, Adv. Nonlinear Stud., 13 (2013), 431-445.  Google Scholar [11] S. Th. Kyritsi and N. S. Papageorgiou, An obstacle problem for nonlinear hemivariational inequalities at resonance, J. Math. Anal. Appl., 276 (2002), 292-313. doi: 10.1016/S0022-247X(02)00443-2.  Google Scholar [12] S. Th. Kyritsi and N. S. Papageorgiou, Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities, Nonlinear Anal., 61 (2005), 373-403. doi: 10.1016/j.na.2004.12.001.  Google Scholar [13] V. K. Le, Existence and enclosure of solutions to noncoercive systems of inequalities with multivalued mappings and non-power growths, Discrete Contin. Dyn. Syst., 33 (2013), 255-276. doi: 10.3934/dcds.2013.33.255.  Google Scholar [14] S. A. Marano and S. J. N. Mosconi, Non-smooth critical point theory on closed convex sets, Comm. Pure Appl. Anal., 13 (2014), 1187-1202. doi: 10.3934/cpaa.2014.13.1187.  Google Scholar [15] S. A. Marano and S. J. N. Mosconi, Critical points on closed convex sets vs. critical points and applications,, submitted for publication., ().   Google Scholar [16] S. A. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Comm. Pure Appl. Anal., 12 (2013), 815-829. doi: 10.3934/cpaa.2013.12.815.  Google Scholar [17] A. C. Ponce, Selected problems on elliptic equations involving measures,, , ().   Google Scholar [18] P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.  Google Scholar [19] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1997.  Google Scholar [20] W. Rudin, Real and Complex Analysis, $3^{rd}$dition, Mac-Graw-Hill, New York, 1987.  Google Scholar [21] J. S. Raymond, On the multiplicity of the solutions of the equation $-\Delta u=\lambda\cdot f(u)$, J. Differential Equations, 180 (2002), 65-88. doi: 10.1006/jdeq.2001.4057.  Google Scholar [22] W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Grad. Texts in Math., {120}, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar
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