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

Previous Article
Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions
CPAA Home
This Issue
Next Article
Tug-of-war games and the infinity Laplacian with spatial dependence

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
Related Papers

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.

[1]	Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745
[2]	Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[3]	Giovanni Covi, Keijo Mönkkönen, Jesse Railo. Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems. Inverse Problems and Imaging, 2021, 15 (4) : 641-681. doi: 10.3934/ipi.2021009
[4]	Muriel Boulakia. Quantification of the unique continuation property for the nonstationary Stokes problem. Mathematical Control and Related Fields, 2016, 6 (1) : 27-52. doi: 10.3934/mcrf.2016.6.27
[5]	Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control and Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012
[6]	Gunther Uhlmann, Jenn-Nan Wang. Unique continuation property for the elasticity with general residual stress. Inverse Problems and Imaging, 2009, 3 (2) : 309-317. doi: 10.3934/ipi.2009.3.309
[7]	Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[8]	Jan Boman. Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform. Inverse Problems and Imaging, 2010, 4 (4) : 619-630. doi: 10.3934/ipi.2010.4.619
[9]	Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure and Applied Analysis, 2021, 20 (2) : 547-558. doi: 10.3934/cpaa.2020280
[10]	Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623
[11]	Taige Wang, Dihong Xu. A quantitative strong unique continuation property of a diffusive SIS model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1599-1614. doi: 10.3934/dcdss.2022024
[12]	Nikolaos S. Papageorgiou, Patrick Winkert. Double resonance for Robin problems with indefinite and unbounded potential. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 323-344. doi: 10.3934/dcdss.2018018
[13]	Mouhamed Moustapha Fall, Veronica Felli. Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5827-5867. doi: 10.3934/dcds.2015.35.5827
[14]	Shouchuan Hu, Nikolaos S. Papageorgiou. Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2005-2021. doi: 10.3934/cpaa.2012.11.2005
[15]	Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[16]	Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345
[17]	Peng Gao. Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2493-2510. doi: 10.3934/dcdsb.2018262
[18]	Peng Gao. Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 169-188. doi: 10.3934/dcds.2017007
[19]	Wen Wang, Dapeng Xie, Hui Zhou. Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1957-1974. doi: 10.3934/cpaa.2018093
[20]	José G. Llorente. Mean value properties and unique continuation. Communications on Pure and Applied Analysis, 2015, 14 (1) : 185-199. doi: 10.3934/cpaa.2015.14.185

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Leszek Gasiński Nikolaos S. Papageorgiou
on Google Scholar
Leszek Gasiński Nikolaos S. Papageorgiou

[Back to Top]