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September  2013, 12(5): 1985-1999. doi: 10.3934/cpaa.2013.12.1985

Multiplicity of solutions for Neumann problems with an indefinite and unbounded potential

Leszek Gasiński 1, and  Nikolaos S. Papageorgiou 2,

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

Received  January 2012 Revised  November 2012 Published  January 2013

We consider a semilinear Neumann problem driven by the negative Laplacian plus an indefinite and unbounded potential and with a Carathéodory reaction term. Using variational methods based on the critical point theory, combined with Morse theory (critical groups), we prove two multiplicity theorems.
Keywords: local minimizer, multiplicity theorems., unique continuation property, Indefinite and unbounded potential, Harnack inequality, mountain pass theorem, critical groups.
Mathematics Subject Classification: Primary: 35J80, 35J85; Secondary: 58E0.
Citation: Leszek Gasiński, Nikolaos S. Papageorgiou. Multiplicity of solutions for Neumann problems with an indefinite and unbounded potential. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1985-1999. doi: 10.3934/cpaa.2013.12.1985
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).   Google Scholar

[2]

T. Bartsch, Critical point theory on partially ordered Hilbert spaces,, J. Funct. Anal., 186 (2001), 117.  doi: doi:10.1080/03605309208820844.  Google Scholar

[3]

K.-C. Chang, "Infinite-Dimensional Morse Theory and Multiple Solution Problems,'', Birkh{\, (1993).   Google Scholar

[4]

K.-C. Chang, "Methods in Nonlinear Analysis,'', Springer-Verlag, (2005).   Google Scholar

[5]

D. de Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation,, Comm. Partial Differential Equations, 17 (1992), 339.  doi: doi:10.1080/03605309208820844.  Google Scholar

[6]

N. Garofalo and F.-H. Lin, Unique continuation for elliptic operators: A geometric variational approach,, Comm. Pure Appl. Math., 40 (1987), 347.  doi: doi:10.1002/cpa.3160400305.  Google Scholar

[7]

L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,'', Chapman and Hall/ CRC Press, (2006).   Google Scholar

[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.  doi: doi:10.1016/j.na.2009.04.063.  Google Scholar

[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.  doi: doi:10.4171/ZAA/1415.  Google Scholar

[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.   Google Scholar

[11]

L. Gasiński and N. S. Papageorgiou, Anisotropic nonlinear Neumann problems,, Calc. Var. Partial Differential Equations, 42 (2011), 323.  doi: doi:10.1007/s00526-011-0390-2.  Google Scholar

[12]

L. Gasiński and N. S. Papageorgiou, Neumann problems resonant at zero and infinity,, Ann. Mat. Pura Appl., 191 (2012), 395.  doi: doi:10.1007/s10231-011-0188-z.  Google Scholar

[13]

L. Gasiński and N. S. Papageorgiou, Dirichlet problems with double resonance and an indefinite potential,, Nonlinear Anal., 75 (2012), 4560.  doi: doi:10.1016/j.na.2011.09.014.  Google Scholar

[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.  doi: doi:10.1016/S0362-546X(03)00100-7.  Google Scholar

[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.  doi: doi:10.3934/cpaa.2011.10.1791.  Google Scholar

[16]

N. S. Papageorgiou and S. Kyritsi, "Handbook of Applied Analysis,'', Springer-Verlag, (2009).   Google Scholar

[17]

P. Pucci and J. Serrin, "The Maximum Principle,'', Birkh{\, (2007).   Google Scholar

[18]

A. Qian, Existence of infinitely many solutions for a superlinear Neumann boundary value problem,, Boundary Value Problems, 2005 (2005), 329.  doi: doi:10.1155/BVP.2005.329.  Google Scholar

[19]

R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations,'', Pitman, (1977).   Google Scholar

[20]

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

[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.  doi: doi:10.1016/j.jmaa.2003.09.034.  Google Scholar

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).   Google Scholar

[2]

T. Bartsch, Critical point theory on partially ordered Hilbert spaces,, J. Funct. Anal., 186 (2001), 117.  doi: doi:10.1080/03605309208820844.  Google Scholar

[3]

K.-C. Chang, "Infinite-Dimensional Morse Theory and Multiple Solution Problems,'', Birkh{\, (1993).   Google Scholar

[4]

K.-C. Chang, "Methods in Nonlinear Analysis,'', Springer-Verlag, (2005).   Google Scholar

[5]

D. de Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation,, Comm. Partial Differential Equations, 17 (1992), 339.  doi: doi:10.1080/03605309208820844.  Google Scholar

[6]

N. Garofalo and F.-H. Lin, Unique continuation for elliptic operators: A geometric variational approach,, Comm. Pure Appl. Math., 40 (1987), 347.  doi: doi:10.1002/cpa.3160400305.  Google Scholar

[7]

L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,'', Chapman and Hall/ CRC Press, (2006).   Google Scholar

[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.  doi: doi:10.1016/j.na.2009.04.063.  Google Scholar

[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.  doi: doi:10.4171/ZAA/1415.  Google Scholar

[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.   Google Scholar

[11]

L. Gasiński and N. S. Papageorgiou, Anisotropic nonlinear Neumann problems,, Calc. Var. Partial Differential Equations, 42 (2011), 323.  doi: doi:10.1007/s00526-011-0390-2.  Google Scholar

[12]

L. Gasiński and N. S. Papageorgiou, Neumann problems resonant at zero and infinity,, Ann. Mat. Pura Appl., 191 (2012), 395.  doi: doi:10.1007/s10231-011-0188-z.  Google Scholar

[13]

L. Gasiński and N. S. Papageorgiou, Dirichlet problems with double resonance and an indefinite potential,, Nonlinear Anal., 75 (2012), 4560.  doi: doi:10.1016/j.na.2011.09.014.  Google Scholar

[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.  doi: doi:10.1016/S0362-546X(03)00100-7.  Google Scholar

[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.  doi: doi:10.3934/cpaa.2011.10.1791.  Google Scholar

[16]

N. S. Papageorgiou and S. Kyritsi, "Handbook of Applied Analysis,'', Springer-Verlag, (2009).   Google Scholar

[17]

P. Pucci and J. Serrin, "The Maximum Principle,'', Birkh{\, (2007).   Google Scholar

[18]

A. Qian, Existence of infinitely many solutions for a superlinear Neumann boundary value problem,, Boundary Value Problems, 2005 (2005), 329.  doi: doi:10.1155/BVP.2005.329.  Google Scholar

[19]

R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations,'', Pitman, (1977).   Google Scholar

[20]

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

[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.  doi: doi:10.1016/j.jmaa.2003.09.034.  Google Scholar

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