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 - A, 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. 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. doi: 10.1017/S0004972708000282.

[3]

S. Carl and S. Heikkilä, Fixed Point Theory in Ordered Sets and Applications,, Springer-Verlag, (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., (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), (2010), 45.

[6]

F. H. Clarke, Optimization and Nonsmooth Analysis,, Classics Appl. Math., 5 (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.

[8]

L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,, Ser. Math. Anal. Appl., (2005).

[9]

S. Hu and N. S. Papageorgiou, Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems,, Comm. Pure Appl. Anal., 12 (2013), 2889. 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.

[11]

S. Th. Kyritsi and N. S. Papageorgiou, An obstacle problem for nonlinear hemivariational inequalities at resonance,, J. Math. Anal. Appl., 276 (2002), 292. 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. 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. 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. 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. doi: 10.3934/cpaa.2013.12.815.

[17]

A. C. Ponce, Selected problems on elliptic equations involving measures,, , ().

[18]

P. Pucci and J. Serrin, The Maximum Principle,, Birkhäuser, (2007).

[19]

R. T. Rockafellar, Convex Analysis,, Princeton Univ. Press, (1997).

[20]

W. Rudin, Real and Complex Analysis,, $3^{rd}$dition, (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. doi: 10.1006/jdeq.2001.4057.

[22]

W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation,, Grad. Texts in Math., (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. 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. doi: 10.1017/S0004972708000282.

[3]

S. Carl and S. Heikkilä, Fixed Point Theory in Ordered Sets and Applications,, Springer-Verlag, (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., (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), (2010), 45.

[6]

F. H. Clarke, Optimization and Nonsmooth Analysis,, Classics Appl. Math., 5 (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.

[8]

L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,, Ser. Math. Anal. Appl., (2005).

[9]

S. Hu and N. S. Papageorgiou, Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems,, Comm. Pure Appl. Anal., 12 (2013), 2889. 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.

[11]

S. Th. Kyritsi and N. S. Papageorgiou, An obstacle problem for nonlinear hemivariational inequalities at resonance,, J. Math. Anal. Appl., 276 (2002), 292. 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. 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. 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. 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. doi: 10.3934/cpaa.2013.12.815.

[17]

A. C. Ponce, Selected problems on elliptic equations involving measures,, , ().

[18]

P. Pucci and J. Serrin, The Maximum Principle,, Birkhäuser, (2007).

[19]

R. T. Rockafellar, Convex Analysis,, Princeton Univ. Press, (1997).

[20]

W. Rudin, Real and Complex Analysis,, $3^{rd}$dition, (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. doi: 10.1006/jdeq.2001.4057.

[22]

W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation,, Grad. Texts in Math., (1989). doi: 10.1007/978-1-4612-1015-3.

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