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

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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

Salvatore A. Marano ^1, and Sunra J. N. Mosconi ^1,

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

Abstract
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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.

Keywords: Critical point, function defined on a closed convex set, elliptic differential inclusion, multiple solutions., Schauder condition.

Mathematics Subject Classification: Primary: 58E05, 49J35; Secondary: 49J5.

Citation: Salvatore A. Marano, Sunra J. N. Mosconi. Multiple solutions to elliptic inclusions via critical point theory on closed convex sets. Discrete and 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.
[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.
[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.
[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.
[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.
[6]	F. H. Clarke, Optimization and Nonsmooth Analysis, Classics Appl. Math., 5, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971309.
[7]	L. Gasiński and N. S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance, Discrete Contin. Dyn. Syst., 34 (2014), 2037-2060.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	S. A. Marano and S. J. N. Mosconi, Critical points on closed convex sets vs. critical points and applications, submitted for publication.
[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.
[17]	A. C. Ponce, Selected problems on elliptic equations involving measures, arXiv:1204.0668.
[18]	P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.
[19]	R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1997.
[20]	W. Rudin, Real and Complex Analysis, $3^{rd}$dition, Mac-Graw-Hill, New York, 1987.
[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.
[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.

show all references

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.
[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.
[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.
[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.
[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.
[6]	F. H. Clarke, Optimization and Nonsmooth Analysis, Classics Appl. Math., 5, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971309.
[7]	L. Gasiński and N. S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance, Discrete Contin. Dyn. Syst., 34 (2014), 2037-2060.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	S. A. Marano and S. J. N. Mosconi, Critical points on closed convex sets vs. critical points and applications, submitted for publication.
[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.
[17]	A. C. Ponce, Selected problems on elliptic equations involving measures, arXiv:1204.0668.
[18]	P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.
[19]	R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1997.
[20]	W. Rudin, Real and Complex Analysis, $3^{rd}$dition, Mac-Graw-Hill, New York, 1987.
[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.
[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.

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