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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 |
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,, , ().
|
[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,, , ().
|
[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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