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Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term
1. | Department of Mathematics, Hellenic Naval Academy, Piraeus 18539 |
2. | Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780 |
References:
[1] |
W. Allegretto and Y. H. Huang, A Picone's identity for the $p$-Laplacian and applications, Nonlin. Anal., 32 (1998), 819-830.
doi: 10.1016/S0362-546X(97)00530-0. |
[2] |
D. Arcoya and S. Villegas, Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at $-\infty$ and superlinear at $+\infty$, Math. Z., 219 (1995), 499-513.
doi: 10.1007/BF02572378. |
[3] |
P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlin. Anal., 7 (1983), 981-1012.
doi: 10.1016/0362-546X(83)90115-3. |
[4] |
T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlin. Anal., 28 (1997), 419-441.
doi: 10.1016/0362-546X(95)00167-T. |
[5] |
T. Bartsch and Z. Liu, On a superlinear elliptic $p$-Laplacian equation, J. Differential Eqns, 198 (2004), 149-175.
doi: 10.1016/j.jde.2003.08.001. |
[6] |
S. Carl and K. Perera, Sign changing amd multiple solutions for the $p$-Laplacian, Abstr. Appl. Anal., 7 (2003), 613-625.
doi: 10.1155/S1085337502207010. |
[7] |
J.-N. Corvellec, On the second deformation lemma, Topol. Methods Nonlin. Anal., 17 (2001), 55-66. |
[8] |
D. Costa and C. Magalhaes, Existence results for perturbation of the $p$-Laplacian, Nonlin. Anal., 24 (1995), 409-418.
doi: 10.1016/0362-546X(94)E0046-J. |
[9] |
N. Dancer and K. Perera, Some remarks on the Fučik spectrum of the $p$-Laplacian and critical groups, J. Math. Anal. Appl., 254 (2001), 164-177.
doi: 10.1006/jmaa.2000.7228. |
[10] |
Y. Deng, S. Peng and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic system involving critical exponent, Discrete and Continuous Dynamical Systems, 32 (2012), 795-826. |
[11] |
G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electr. Jour. Diff. Eqns, 8 (2002), pp. 12. |
[12] |
S. Th. Kyritsi-Yiallourou and N. S. Papageorgiou, "Handbook of Applied Analysis, Advances in Mechanics and Mathematics 19," Springer, New York, 2009.
doi: 10.1007/b120946. |
[13] |
J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404.
doi: 10.1142/S0219199700000190. |
[14] |
L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC Press, Boca Raton, Fl, 2006. |
[15] |
A. Granas and J. Dugundji, "Fixed Point Theory," Springer-Verlag, New York, 2003. |
[16] |
Z. Guo and Z. Liu, Perturbed elliptic equations with oscillatory nonlinearities, Discrete and Continuous Dynamical Systems, 32 (2012), 3567-3585.
doi: 10.3934/dcds.2012.32.3567. |
[17] |
O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York, 1968. |
[18] |
G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal., 12 (1988), 1203-1219.
doi: 10.1016/0362-546X(88)90053-3. |
[19] |
S. Liu, Multiple solutions for coercive $p$-Laplacian equations, J. Math. Anal. Appl., 316 (2006), 229-236.
doi: 10.1016/j.jmaa.2005.04.034. |
[20] |
J. Liu and S. Liu, The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc., 37 (2005), 592-600.
doi: 10.1112/S0024609304004023. |
[21] |
J. Mawhin, Multiplicity of solutions for variational systems involving $\varphi$-Laplacians with singular $\varphi$ and periodic potentials, Discrete and Continuous Dynamical Systems, 32 (2012), 4015-4026.
doi: 10.3934/dcds.2012.32.4015. |
[22] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential, Communications on Pure and Applied Analysis, 10 (2011), 1401-1414.
doi: 10.3934/cpaa.2011.10.1401. |
[23] |
D. Motreanu, Donal 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, Communications on Pure and Applied Analysis, 10 (2011), 1791-1816.
doi: 10.3934/cpaa.2011.10.1791. |
[24] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989. |
[25] |
O. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Diff. Eqns, 245 (2008), 3628-3638.
doi: 10.1016/j.jde.2008.02.035. |
[26] |
D. Motreanu, V. Motreanu and N. S. Papageorgiou, A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations, Manuscripta Math., 124 (2007), 507-531.
doi: 10.1007/s00229-007-0127-x. |
[27] |
F. O. de Paiva, Multiple solutions for a class of quasilinear problems, Discrete Cont. Dynam. Systems, 15 (2006), 669-680.
doi: 10.3934/dcds.2006.15.669. |
[28] |
R. Palais, Homotopy theory of finite dimensional manifolds, Topology, 5 (1966), 1-16. |
[29] |
E. 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. |
[30] |
N. S. Papageorgiou, E. Rocha and V. Staicu, Multiplicity theorems for superlinear elliptic problems, Calc. Var., 33 (2008), 199-230.
doi: 10.1007/s00526-008-0172-7. |
[31] |
K. Perera, Existence and multiplicity results for a Sturm-Liouville equation asymptotically linear at $-\infty$ and superlinear at $+\infty$, Nonlin. Anal., 39 (2000), 669-684.
doi: 10.1016/S0362-546X(98)00228-4. |
[32] |
M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160.
doi: 10.2140/pjm.2004.214.145. |
[33] |
M. Tanaka, Existence of a nontrivial solution for a $p$-Laplacian equation with Fu\vcik type resonance at infinity, Nonlin. Anal., 72 (2010), 507-526.
doi: 10.1016$|$j. na. 2008.02.117. |
[34] |
J. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[35] |
Z. Q. Wang, On a superlinear elliptic equation, Annales IHP, Section C, Analyse Nonlineaire, 8 (1991), 43-57. |
[36] |
M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. Jour., 52 (2003), 109-132.
doi: 10.1512/iumj.2003.52.2273. |
[37] |
Z. Zhang, J. Chen and S. Li, Construction of pseudogradient vector field and sign-changing multiple solutions involving the $p$-Laplacian, Jour. Diff. Eqns, 201 (2004), 287-303.
doi: 10.1016/j.jde.2004.03.019. |
[38] |
Z. Zhang and S. Li, On sign-changing and multiple solutions of the $p$-Laplacian, J. Funct. Anal., 197 (2003), 447-468.
doi: 10.1016/S0022-1236(02)00103-9. |
[39] |
Z. Zhang, S. Li, S. Liu and W. Feng, On an asymptotically linear elliptic Dirichlet problem, Abstract Appl. Anal., 7 (2002), 509-516.
doi: 10.1155/S1085337502207046. |
show all references
References:
[1] |
W. Allegretto and Y. H. Huang, A Picone's identity for the $p$-Laplacian and applications, Nonlin. Anal., 32 (1998), 819-830.
doi: 10.1016/S0362-546X(97)00530-0. |
[2] |
D. Arcoya and S. Villegas, Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at $-\infty$ and superlinear at $+\infty$, Math. Z., 219 (1995), 499-513.
doi: 10.1007/BF02572378. |
[3] |
P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlin. Anal., 7 (1983), 981-1012.
doi: 10.1016/0362-546X(83)90115-3. |
[4] |
T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlin. Anal., 28 (1997), 419-441.
doi: 10.1016/0362-546X(95)00167-T. |
[5] |
T. Bartsch and Z. Liu, On a superlinear elliptic $p$-Laplacian equation, J. Differential Eqns, 198 (2004), 149-175.
doi: 10.1016/j.jde.2003.08.001. |
[6] |
S. Carl and K. Perera, Sign changing amd multiple solutions for the $p$-Laplacian, Abstr. Appl. Anal., 7 (2003), 613-625.
doi: 10.1155/S1085337502207010. |
[7] |
J.-N. Corvellec, On the second deformation lemma, Topol. Methods Nonlin. Anal., 17 (2001), 55-66. |
[8] |
D. Costa and C. Magalhaes, Existence results for perturbation of the $p$-Laplacian, Nonlin. Anal., 24 (1995), 409-418.
doi: 10.1016/0362-546X(94)E0046-J. |
[9] |
N. Dancer and K. Perera, Some remarks on the Fučik spectrum of the $p$-Laplacian and critical groups, J. Math. Anal. Appl., 254 (2001), 164-177.
doi: 10.1006/jmaa.2000.7228. |
[10] |
Y. Deng, S. Peng and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic system involving critical exponent, Discrete and Continuous Dynamical Systems, 32 (2012), 795-826. |
[11] |
G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electr. Jour. Diff. Eqns, 8 (2002), pp. 12. |
[12] |
S. Th. Kyritsi-Yiallourou and N. S. Papageorgiou, "Handbook of Applied Analysis, Advances in Mechanics and Mathematics 19," Springer, New York, 2009.
doi: 10.1007/b120946. |
[13] |
J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404.
doi: 10.1142/S0219199700000190. |
[14] |
L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC Press, Boca Raton, Fl, 2006. |
[15] |
A. Granas and J. Dugundji, "Fixed Point Theory," Springer-Verlag, New York, 2003. |
[16] |
Z. Guo and Z. Liu, Perturbed elliptic equations with oscillatory nonlinearities, Discrete and Continuous Dynamical Systems, 32 (2012), 3567-3585.
doi: 10.3934/dcds.2012.32.3567. |
[17] |
O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York, 1968. |
[18] |
G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal., 12 (1988), 1203-1219.
doi: 10.1016/0362-546X(88)90053-3. |
[19] |
S. Liu, Multiple solutions for coercive $p$-Laplacian equations, J. Math. Anal. Appl., 316 (2006), 229-236.
doi: 10.1016/j.jmaa.2005.04.034. |
[20] |
J. Liu and S. Liu, The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc., 37 (2005), 592-600.
doi: 10.1112/S0024609304004023. |
[21] |
J. Mawhin, Multiplicity of solutions for variational systems involving $\varphi$-Laplacians with singular $\varphi$ and periodic potentials, Discrete and Continuous Dynamical Systems, 32 (2012), 4015-4026.
doi: 10.3934/dcds.2012.32.4015. |
[22] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential, Communications on Pure and Applied Analysis, 10 (2011), 1401-1414.
doi: 10.3934/cpaa.2011.10.1401. |
[23] |
D. Motreanu, Donal 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, Communications on Pure and Applied Analysis, 10 (2011), 1791-1816.
doi: 10.3934/cpaa.2011.10.1791. |
[24] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989. |
[25] |
O. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Diff. Eqns, 245 (2008), 3628-3638.
doi: 10.1016/j.jde.2008.02.035. |
[26] |
D. Motreanu, V. Motreanu and N. S. Papageorgiou, A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations, Manuscripta Math., 124 (2007), 507-531.
doi: 10.1007/s00229-007-0127-x. |
[27] |
F. O. de Paiva, Multiple solutions for a class of quasilinear problems, Discrete Cont. Dynam. Systems, 15 (2006), 669-680.
doi: 10.3934/dcds.2006.15.669. |
[28] |
R. Palais, Homotopy theory of finite dimensional manifolds, Topology, 5 (1966), 1-16. |
[29] |
E. 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. |
[30] |
N. S. Papageorgiou, E. Rocha and V. Staicu, Multiplicity theorems for superlinear elliptic problems, Calc. Var., 33 (2008), 199-230.
doi: 10.1007/s00526-008-0172-7. |
[31] |
K. Perera, Existence and multiplicity results for a Sturm-Liouville equation asymptotically linear at $-\infty$ and superlinear at $+\infty$, Nonlin. Anal., 39 (2000), 669-684.
doi: 10.1016/S0362-546X(98)00228-4. |
[32] |
M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160.
doi: 10.2140/pjm.2004.214.145. |
[33] |
M. Tanaka, Existence of a nontrivial solution for a $p$-Laplacian equation with Fu\vcik type resonance at infinity, Nonlin. Anal., 72 (2010), 507-526.
doi: 10.1016$|$j. na. 2008.02.117. |
[34] |
J. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[35] |
Z. Q. Wang, On a superlinear elliptic equation, Annales IHP, Section C, Analyse Nonlineaire, 8 (1991), 43-57. |
[36] |
M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. Jour., 52 (2003), 109-132.
doi: 10.1512/iumj.2003.52.2273. |
[37] |
Z. Zhang, J. Chen and S. Li, Construction of pseudogradient vector field and sign-changing multiple solutions involving the $p$-Laplacian, Jour. Diff. Eqns, 201 (2004), 287-303.
doi: 10.1016/j.jde.2004.03.019. |
[38] |
Z. Zhang and S. Li, On sign-changing and multiple solutions of the $p$-Laplacian, J. Funct. Anal., 197 (2003), 447-468.
doi: 10.1016/S0022-1236(02)00103-9. |
[39] |
Z. Zhang, S. Li, S. Liu and W. Feng, On an asymptotically linear elliptic Dirichlet problem, Abstract Appl. Anal., 7 (2002), 509-516.
doi: 10.1155/S1085337502207046. |
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