American Institute of Mathematical Sciences

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July  2014, 13(4): 1491-1512. doi: 10.3934/cpaa.2014.13.1491

Multiple solutions for a class of nonlinear Neumann eigenvalue problems

Leszek Gasiński 1, and  Nikolaos S. Papageorgiou 2,

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

Received  June 2013 Revised  December 2014 Published  February 2014

We consider a parametric nonlinear equation driven by the Neumann $p$-Laplacian. Using variational methods we show that when the parameter $\lambda > \widehat{\lambda}_1$ (where $\widehat{\lambda}_1$ is the first nonzero eigenvalue of the negative Neumann $p$-Laplacian), then the problem has at least three nontrivial smooth solutions, two of constant sign (one positive and one negative) and the third nodal. In the semilinear case (i.e., $p=2$), using Morse theory and flow invariance argument, we show that the problem has three nodal solutions.
Keywords: flow invariance., critical groups, Smooth solutions of constant sign, first nonzero eigenvalue, negative gradient flow, mountain pass theorem, nodal solutions.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35J60, 35J92, 58E0.
Citation: Leszek Gasiński, Nikolaos S. Papageorgiou. Multiple solutions for a class of nonlinear Neumann eigenvalue problems. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1491-1512. doi: 10.3934/cpaa.2014.13.1491
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal., 186 (2001), 117-152. doi: 10.1006/jfan.2001.3789.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser Verlag, Boston, MA, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. Gasiński and N. S. Papageorgiou, Multiple solutions for semilinear hemivariational inequalities at resonance, Publ. Math. Debrecen, 59 (2001), 121-146.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Chapman and Hall/ CRC Press, Boca Raton, FL, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1-16.  Google Scholar

[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.  Google Scholar

[28]

M. Struwe, A note on a result of Ambrosetti and Mancini, Ann. Mat. Pura Appl., 131 (1982), 107-115. doi: 10.1007/BF01765148.  Google Scholar

[29]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal., 186 (2001), 117-152. doi: 10.1006/jfan.2001.3789.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser Verlag, Boston, MA, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. Gasiński and N. S. Papageorgiou, Multiple solutions for semilinear hemivariational inequalities at resonance, Publ. Math. Debrecen, 59 (2001), 121-146.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Chapman and Hall/ CRC Press, Boca Raton, FL, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1-16.  Google Scholar

[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.  Google Scholar

[28]

M. Struwe, A note on a result of Ambrosetti and Mancini, Ann. Mat. Pura Appl., 131 (1982), 107-115. doi: 10.1007/BF01765148.  Google Scholar

[29]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.  Google Scholar

[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.  Google Scholar

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