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

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

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.
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).

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[19]

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

[20]

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

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[19]

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

[20]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,'', Springer-Verlag, (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. doi: doi:10.1016/j.jmaa.2003.09.034.

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