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Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum
Multiple solutions for a class of nonlinear Neumann eigenvalue problems
1. | Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków |
2. | Department of Mathematics, National Technical University, Zografou Campus, Athens 15780 |
References:
[1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu, The spectrum and an index formula for the Neumann $p$-Laplacian and multiple solutions for problems with crossing nonlinearity, Discrete Contin. Dyn. Syst., 25 (2009), 431-456.
doi: 10.3934/dcds.2009.25.431. |
[2] |
A. Ambrosetti and D. Lupo, On a class of nonlinear Dirichlet problems with multiple solutions, Nonlinear Anal., 8 (1984), 1145-1150.
doi: 10.1016/0362-546X(84)90116-0. |
[3] |
A. Ambrosetti and G. Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal., 3 (1979), 635-645.
doi: 10.1016/0362-546X(79)90092-0. |
[4] |
T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal., 186 (2001), 117-152.
doi: 10.1006/jfan.2001.3789. |
[5] |
H. Brézis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465-472. |
[6] |
A. Castro, J. Cossio and C. Vélez, Existence and qualitative properties of solutions for nonlinear Dirichlet problems, Discrete Contin. Dyn. Syst., 33 (2013), 123-140.
doi: 10.3934/dcds.2013.33.123. |
[7] |
K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser Verlag, Boston, MA, 1993. |
[8] |
G. M. Coclite and M. M. Coclite, On a Dirichlet problem in bounded domains with singular nonlinearity, Discrete Contin. Dyn. Syst., 33 (2013), 4923-4944.
doi: 10.3934/dcds.2013.33.4923. |
[9] |
J. García Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math., 2 (2000), 385-404.
doi: 10.1142/S0219199700000190. |
[10] |
L. Gasiński and N. S. Papageorgiou, Existence of solutions and of multiple solutions for eigenvalue problems of hemivariational inequalities, Adv. Math. Sci. Appl., 11 (2001), 437-464. |
[11] |
L. Gasiński and N. S. Papageorgiou, Multiple solutions for semilinear hemivariational inequalities at resonance, Publ. Math. Debrecen, 59 (2001), 121-146. |
[12] |
L. Gasiński and N. S. Papageorgiou, Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance, Proc. Royal Soc. Edinburgh Section A, Mathematics, 131A (2001), 1091-1111.
doi: 10.1017/S0308210500001281. |
[13] |
L. Gasiński and N. S. Papageorgiou, A multiplicity result for nonlinear second order periodic equations with nonsmooth potential, Bull. Belg. Math. Soc. Simon Stevin, 9 (2002), 245-258. |
[14] |
L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Chapman and Hall/ CRC Press, Boca Raton, FL, 2006. |
[15] |
L. Gasiński and N. S. Papageorgiou, Nodal and multiple constant sign solutions for resonant $p$-Laplacian equations with a nonsmooth potential, Nonlinear Anal., 71 (2009), 5747-5772.
doi: 10.1016/j.na.2009.04.063. |
[16] |
L. Gasiński and N. S. Papageorgiou, Dirichlet (p,q)-equations at resonance, Discrete Contin. Dyn. Syst., 34 (2014), 2037-2060. 85.
doi: 10.3934/dcds.2014.34.2037. |
[17] |
L. Gasiński and N. S. Papageorgiou, A pair of positive solutions for (p,q)-equations with combined nonlinearities, Commun. Pure Appl. Anal., 13 (2014), 203-215.
doi: 10.3934/cpaa.2014.13.203. |
[18] |
T. Godoy, J.-P. Gossez and S. Paczka, On the antimaximum principle for the $p$-Laplacian with indefinite weight, Nonlinear Anal., 51 (2002), 449-467.
doi: 10.1016/S0362-546X(01)00839-2. |
[19] |
S. Th. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term, Discrete Contin. Dyn. Syst., 33 (2013), 2469-2494.
doi: 10.3934/dcds.2013.33.2469. |
[20] |
G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.
doi: 10.1016/0362-546X(88)90053-3. |
[21] |
S. Li and Z. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. Anal. Math., 81 (2000), 373-396.
doi: 10.1007/BF02788997. |
[22] |
S. A. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal., 12 (2013), 815-829.
doi: 10.3934/cpaa.2013.12.815. |
[23] |
A. Mercaldo, J. D. Rossi and S. Segura de León, C. Trombetti, Behaviour of $p$-Laplacian problems with Neumann boundary conditions when $p$ goes to 1, Commun. Pure Appl. Anal., 12 (2013), 253-267.
doi: 10.3934/cpaa.2013.12.253. |
[24] |
D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems, J. Differential Equations, 232 (2007), 1-35.
doi: 10.1016/j.jde.2006.09.008. |
[25] |
D. Motreanu and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operators, Proc. Amer. Math. Soc., 139 (2011), 3527-3535.
doi: 10.1090/S0002-9939-2011-10884-0. |
[26] |
R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1-16. |
[27] |
E. H. Papageorgiou and N. S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian, J. Funct. Anal., 244 (2007), 63-77.
doi: 10.1016/j.jfa.2006.11.015. |
[28] |
M. Struwe, A note on a result of Ambrosetti and Mancini, Ann. Mat. Pura Appl., 131 (1982), 107-115.
doi: 10.1007/BF01765148. |
[29] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 2008. |
[30] |
J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895.
doi: 10.1016/S0362-546X(00)00221-2. |
[31] |
J. Tyagi, Multiple solutions for singular $N$-Laplace equations with a sign changing nonlinearity, Commun. Pure Appl. Anal., 12 (2013), 2381-2391.
doi: 10.3934/cpaa.2013.12.2381. |
[32] |
J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[33] |
P. Winkert, Multiplicity results for a class of elliptic problems with nonlinear boundary condition, Commun. Pure Appl. Anal., 12 (2013), 785-802.
doi: 10.3934/cpaa.2013.12.785. |
show all references
References:
[1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu, The spectrum and an index formula for the Neumann $p$-Laplacian and multiple solutions for problems with crossing nonlinearity, Discrete Contin. Dyn. Syst., 25 (2009), 431-456.
doi: 10.3934/dcds.2009.25.431. |
[2] |
A. Ambrosetti and D. Lupo, On a class of nonlinear Dirichlet problems with multiple solutions, Nonlinear Anal., 8 (1984), 1145-1150.
doi: 10.1016/0362-546X(84)90116-0. |
[3] |
A. Ambrosetti and G. Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal., 3 (1979), 635-645.
doi: 10.1016/0362-546X(79)90092-0. |
[4] |
T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal., 186 (2001), 117-152.
doi: 10.1006/jfan.2001.3789. |
[5] |
H. Brézis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465-472. |
[6] |
A. Castro, J. Cossio and C. Vélez, Existence and qualitative properties of solutions for nonlinear Dirichlet problems, Discrete Contin. Dyn. Syst., 33 (2013), 123-140.
doi: 10.3934/dcds.2013.33.123. |
[7] |
K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser Verlag, Boston, MA, 1993. |
[8] |
G. M. Coclite and M. M. Coclite, On a Dirichlet problem in bounded domains with singular nonlinearity, Discrete Contin. Dyn. Syst., 33 (2013), 4923-4944.
doi: 10.3934/dcds.2013.33.4923. |
[9] |
J. García Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math., 2 (2000), 385-404.
doi: 10.1142/S0219199700000190. |
[10] |
L. Gasiński and N. S. Papageorgiou, Existence of solutions and of multiple solutions for eigenvalue problems of hemivariational inequalities, Adv. Math. Sci. Appl., 11 (2001), 437-464. |
[11] |
L. Gasiński and N. S. Papageorgiou, Multiple solutions for semilinear hemivariational inequalities at resonance, Publ. Math. Debrecen, 59 (2001), 121-146. |
[12] |
L. Gasiński and N. S. Papageorgiou, Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance, Proc. Royal Soc. Edinburgh Section A, Mathematics, 131A (2001), 1091-1111.
doi: 10.1017/S0308210500001281. |
[13] |
L. Gasiński and N. S. Papageorgiou, A multiplicity result for nonlinear second order periodic equations with nonsmooth potential, Bull. Belg. Math. Soc. Simon Stevin, 9 (2002), 245-258. |
[14] |
L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Chapman and Hall/ CRC Press, Boca Raton, FL, 2006. |
[15] |
L. Gasiński and N. S. Papageorgiou, Nodal and multiple constant sign solutions for resonant $p$-Laplacian equations with a nonsmooth potential, Nonlinear Anal., 71 (2009), 5747-5772.
doi: 10.1016/j.na.2009.04.063. |
[16] |
L. Gasiński and N. S. Papageorgiou, Dirichlet (p,q)-equations at resonance, Discrete Contin. Dyn. Syst., 34 (2014), 2037-2060. 85.
doi: 10.3934/dcds.2014.34.2037. |
[17] |
L. Gasiński and N. S. Papageorgiou, A pair of positive solutions for (p,q)-equations with combined nonlinearities, Commun. Pure Appl. Anal., 13 (2014), 203-215.
doi: 10.3934/cpaa.2014.13.203. |
[18] |
T. Godoy, J.-P. Gossez and S. Paczka, On the antimaximum principle for the $p$-Laplacian with indefinite weight, Nonlinear Anal., 51 (2002), 449-467.
doi: 10.1016/S0362-546X(01)00839-2. |
[19] |
S. Th. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term, Discrete Contin. Dyn. Syst., 33 (2013), 2469-2494.
doi: 10.3934/dcds.2013.33.2469. |
[20] |
G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.
doi: 10.1016/0362-546X(88)90053-3. |
[21] |
S. Li and Z. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. Anal. Math., 81 (2000), 373-396.
doi: 10.1007/BF02788997. |
[22] |
S. A. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal., 12 (2013), 815-829.
doi: 10.3934/cpaa.2013.12.815. |
[23] |
A. Mercaldo, J. D. Rossi and S. Segura de León, C. Trombetti, Behaviour of $p$-Laplacian problems with Neumann boundary conditions when $p$ goes to 1, Commun. Pure Appl. Anal., 12 (2013), 253-267.
doi: 10.3934/cpaa.2013.12.253. |
[24] |
D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems, J. Differential Equations, 232 (2007), 1-35.
doi: 10.1016/j.jde.2006.09.008. |
[25] |
D. Motreanu and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operators, Proc. Amer. Math. Soc., 139 (2011), 3527-3535.
doi: 10.1090/S0002-9939-2011-10884-0. |
[26] |
R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1-16. |
[27] |
E. H. Papageorgiou and N. S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian, J. Funct. Anal., 244 (2007), 63-77.
doi: 10.1016/j.jfa.2006.11.015. |
[28] |
M. Struwe, A note on a result of Ambrosetti and Mancini, Ann. Mat. Pura Appl., 131 (1982), 107-115.
doi: 10.1007/BF01765148. |
[29] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 2008. |
[30] |
J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895.
doi: 10.1016/S0362-546X(00)00221-2. |
[31] |
J. Tyagi, Multiple solutions for singular $N$-Laplace equations with a sign changing nonlinearity, Commun. Pure Appl. Anal., 12 (2013), 2381-2391.
doi: 10.3934/cpaa.2013.12.2381. |
[32] |
J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[33] |
P. Winkert, Multiplicity results for a class of elliptic problems with nonlinear boundary condition, Commun. Pure Appl. Anal., 12 (2013), 785-802.
doi: 10.3934/cpaa.2013.12.785. |
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