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

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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

Abstract
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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 and 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), 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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