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Multiplicity of solutions for Neumann problems with an indefinite and unbounded potential
1. | Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków, Poland |
2. | Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780 |
References:
[1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc., 196 (2008), 915. |
[2] |
T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal., 186 (2001), 117-152.
doi: doi:10.1080/03605309208820844. |
[3] |
K.-C. Chang, "Infinite-Dimensional Morse Theory and Multiple Solution Problems,'' Birkhäuser Verlag, Boston, MA, 1993. |
[4] |
K.-C. Chang, "Methods in Nonlinear Analysis,'' Springer-Verlag, Berlin, 2005. |
[5] |
D. de Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations, 17 (1992), 339-346.
doi: doi:10.1080/03605309208820844. |
[6] |
N. Garofalo and F.-H. Lin, Unique continuation for elliptic operators: A geometric variational approach, Comm. Pure Appl. Math., 40 (1987), 347-366.
doi: doi:10.1002/cpa.3160400305. |
[7] |
L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,'' Chapman and Hall/ CRC Press, Boca Raton, FL, 2006. |
[8] |
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: doi:10.1016/j.na.2009.04.063. |
[9] |
L. Gasiński and N. S. Papageorgiou, Existence of three nontrivial smooth solutions for nonlinear resonant neumann problems driven by the $p$-Laplacian, J. Anal. Appl., 29 (2010), 413-428.
doi: doi:10.4171/ZAA/1415. |
[10] |
L. Gasiński and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems with asymmetric reaction, via Morse theory, Adv. Nonlinear Stud., 11 (2011), 781-808. |
[11] |
L. Gasiński and N. S. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. Partial Differential Equations, 42 (2011), 323-354.
doi: doi:10.1007/s00526-011-0390-2. |
[12] |
L. Gasiński and N. S. Papageorgiou, Neumann problems resonant at zero and infinity, Ann. Mat. Pura Appl., 191 (2012), 395-430.
doi: doi:10.1007/s10231-011-0188-z. |
[13] |
L. Gasiński and N. S. Papageorgiou, Dirichlet problems with double resonance and an indefinite potential, Nonlinear Anal., 75 (2012), 4560-4595.
doi: doi:10.1016/j.na.2011.09.014. |
[14] |
C. Li, The existence of infinitely many solutions of a class of nonlinear elliptic equations with Neumann boundary condition for both resonance and oscillation problems, Nonlinear Anal., 54 (2003), 431-443.
doi: doi:10.1016/S0362-546X(03)00100-7. |
[15] |
D. Motreanu, D. O'Regan and N. S. Papageorgiou, A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems, Commun. Pure Appl. Anal., 10 (2011), 1791-1816.
doi: doi:10.3934/cpaa.2011.10.1791. |
[16] |
N. S. Papageorgiou and S. Kyritsi, "Handbook of Applied Analysis,'' Springer-Verlag, New York, 2009. |
[17] |
P. Pucci and J. Serrin, "The Maximum Principle,'' Birkhäuser Verlag, Basel, 2007. |
[18] |
A. Qian, Existence of infinitely many solutions for a superlinear Neumann boundary value problem, Boundary Value Problems, 2005 (2005), 329-335.
doi: doi:10.1155/BVP.2005.329. |
[19] |
R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations,'' Pitman, London, 1977. |
[20] |
M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,'' Springer-Verlag, Berlin, 2008. |
[21] |
C.-L. Tang and X.-P. Wu, Existence and multiplicity for solutions of Neumann problems for semilinear elliptic equations, J. Math. Anal. Appl., 288 (2003), 660-670.
doi: doi:10.1016/j.jmaa.2003.09.034. |
show all references
References:
[1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc., 196 (2008), 915. |
[2] |
T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal., 186 (2001), 117-152.
doi: doi:10.1080/03605309208820844. |
[3] |
K.-C. Chang, "Infinite-Dimensional Morse Theory and Multiple Solution Problems,'' Birkhäuser Verlag, Boston, MA, 1993. |
[4] |
K.-C. Chang, "Methods in Nonlinear Analysis,'' Springer-Verlag, Berlin, 2005. |
[5] |
D. de Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations, 17 (1992), 339-346.
doi: doi:10.1080/03605309208820844. |
[6] |
N. Garofalo and F.-H. Lin, Unique continuation for elliptic operators: A geometric variational approach, Comm. Pure Appl. Math., 40 (1987), 347-366.
doi: doi:10.1002/cpa.3160400305. |
[7] |
L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,'' Chapman and Hall/ CRC Press, Boca Raton, FL, 2006. |
[8] |
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: doi:10.1016/j.na.2009.04.063. |
[9] |
L. Gasiński and N. S. Papageorgiou, Existence of three nontrivial smooth solutions for nonlinear resonant neumann problems driven by the $p$-Laplacian, J. Anal. Appl., 29 (2010), 413-428.
doi: doi:10.4171/ZAA/1415. |
[10] |
L. Gasiński and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems with asymmetric reaction, via Morse theory, Adv. Nonlinear Stud., 11 (2011), 781-808. |
[11] |
L. Gasiński and N. S. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. Partial Differential Equations, 42 (2011), 323-354.
doi: doi:10.1007/s00526-011-0390-2. |
[12] |
L. Gasiński and N. S. Papageorgiou, Neumann problems resonant at zero and infinity, Ann. Mat. Pura Appl., 191 (2012), 395-430.
doi: doi:10.1007/s10231-011-0188-z. |
[13] |
L. Gasiński and N. S. Papageorgiou, Dirichlet problems with double resonance and an indefinite potential, Nonlinear Anal., 75 (2012), 4560-4595.
doi: doi:10.1016/j.na.2011.09.014. |
[14] |
C. Li, The existence of infinitely many solutions of a class of nonlinear elliptic equations with Neumann boundary condition for both resonance and oscillation problems, Nonlinear Anal., 54 (2003), 431-443.
doi: doi:10.1016/S0362-546X(03)00100-7. |
[15] |
D. Motreanu, D. O'Regan and N. S. Papageorgiou, A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems, Commun. Pure Appl. Anal., 10 (2011), 1791-1816.
doi: doi:10.3934/cpaa.2011.10.1791. |
[16] |
N. S. Papageorgiou and S. Kyritsi, "Handbook of Applied Analysis,'' Springer-Verlag, New York, 2009. |
[17] |
P. Pucci and J. Serrin, "The Maximum Principle,'' Birkhäuser Verlag, Basel, 2007. |
[18] |
A. Qian, Existence of infinitely many solutions for a superlinear Neumann boundary value problem, Boundary Value Problems, 2005 (2005), 329-335.
doi: doi:10.1155/BVP.2005.329. |
[19] |
R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations,'' Pitman, London, 1977. |
[20] |
M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,'' Springer-Verlag, Berlin, 2008. |
[21] |
C.-L. Tang and X.-P. Wu, Existence and multiplicity for solutions of Neumann problems for semilinear elliptic equations, J. Math. Anal. Appl., 288 (2003), 660-670.
doi: doi:10.1016/j.jmaa.2003.09.034. |
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