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September  2012, 11(5): 2005-2021. doi: 10.3934/cpaa.2012.11.2005

## Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities

 1 Department of Mathematics, Missouri State University, Spring eld, MO 65804, United States 2 Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780

Received  May 2011 Revised  December 2011 Published  March 2012

We study a semilinear parametric Dirichlet equation with an indefinite and unbounded potential. The reaction is the sum of a sublinear (concave) term and of an asymptotically linear resonant term. The resonance is with respect to any nonprincipal nonnegative eigenvalue of the differential operator. Using variational methods based on the critical point theory and Morse theory (critical groups), we show that when the parameter $\lambda>0$ is small, the problem has at least three nontrivial smooth solutions.
Citation: Shouchuan Hu, Nikolaos S. Papageorgiou. Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2005-2021. doi: 10.3934/cpaa.2012.11.2005
##### References:
 [1] S. Aizicovici, N. S. Papageorgiou and V. Staicu, "Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equation with Inequality Constraints," Memoirs of AMS, 196, No. 915, 2008.  Google Scholar [2] H. Berestycki and D. G. deFigueiredo, Double resonance is semilinear elliptic problems, Comm. Partial Diff. Equas., 6 (1981), 91-120. doi: 10.1080/03605308108820172.  Google Scholar [3] N. P. Cac, On an elliptic boundary value problem at double resonance, J. Math. Anal. Appl., 132 (1988), 473-483. doi: 10.1016/0022-247X(88)90075-3.  Google Scholar [4] K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems," Birkhauser, Boston, 1993.  Google Scholar [5] F. Clarke, "Optimization and Nonsmooth Analysis," Wiley, New York, 1983.  Google Scholar [6] Y. Deng, S. Peng and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic equatio involving critical exponent, Discrete Contin. Dynam. Systems, 32 (2012), 795-826.  Google Scholar [7] D. G. deFigueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Diff. Equas., 17 (1992), 339-346. doi: 10.1080/03605309208820844.  Google Scholar [8] M. Filippakis, D. O'Regan and N. S. Papageorgiou, Positive solutions and bifurcation phenomena for nonlinear elliptic equations of Logistic type:The superdiffusive case, Comm. Pure Appl. Anal., 9 (2010), 1507-1527. doi: 10.3934/cpaa.2010.9.1507.  Google Scholar [9] M. Filippakis and N. S. Papageorgiou, Multiplicity of solutions for doubly resonant Neumann problem, Bull. Belgian Math. Soc., 18 (2011), 135-156.  Google Scholar [10] Z. Guo, Z. Liu, J. Wei and F. Zhou, Bifurcations of some elliptic problems with a singular nonlinearity via Morse index, Comm. Pure. Appl. Anal., 10 (2011), 507-525. doi: 10.3934/cpaa.2011.10.507.  Google Scholar [11] N. Garofalo and F. H. Lin, Unique continuation for elliptic operators: A geometric variatoinal approach, Comm. Pure Appl. Math., 40 (1987), 347-366. Google Scholar [12] L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar [13] J. Garcia Melian, J. Rossi and J. Sabina de Lis, A convex-concave problem with a parameter on the boundary condition, Discrete Contin. Dynam. Systems, 32 (2012), 1095-1124.  Google Scholar [14] Q. Jiu and J. Su, Existence and multiplicity results for perturbations of the $p$-Laplacian, J. Math. Anal. Appl., 281 (2003), 587-601. doi: 10.1016/S0022-247X(03)00165-3.  Google Scholar [15] Z. Liang and J. Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158. doi: 10.1016/j.jmaa.2008.12.053.  Google Scholar [16] M. L. Miotto, Multiple solutions for elliptic problems in $R^N$ with critical Spbolev exponent and weight function, Comm. Pure Appl. Anal., 9 (2010), 233-248. doi: 10.3934/cpaa.2010.9.233.  Google Scholar [17] 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, Comm. Pure Appl. Anal., 10 (2011), 1791-1816. doi: 10.3934/cpaa.2011.10.1791.  Google Scholar [18] N. S. Papageorgiou and S. Th. Kyritsi, "Handbook of Applied Analysis," Springer, New York, 2009.  Google Scholar [19] J-M. Rakotoson, Generalized eigenvalue problem for totally discontinuous operator, Discrete Contin. Dynam. Systems, 28 (2010), 343-373. doi: 10.3934/dcds.2010.28.343.  Google Scholar [20] P. Pucci and J. Serrin, "The Maximum Principle," Birkhauser, Basel, 2007.  Google Scholar [21] S. Robinson, Double resonance in semilinear elliptic boundary value problem over bounded and unbounded domain, Nonlin. Anal., 21 (1993), 407-424. doi: 10.1016/0362-546X(93)90125-C.  Google Scholar [22] R. Showalter, "Hilbert Space Methods for Partial Differential Equations," Pitman, London, 1977.  Google Scholar [23] M. Struwe, "Variational Methods," Springer, Berlin, 1990.  Google Scholar [24] 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 [25] W. Zou, Multiple solutions for elliptic equations with resonance, Nonlinear Anal., 48 (2002), 363-376. doi: 10.1016/S0362-546X(00)00190-5.  Google Scholar

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##### References:
 [1] S. Aizicovici, N. S. Papageorgiou and V. Staicu, "Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equation with Inequality Constraints," Memoirs of AMS, 196, No. 915, 2008.  Google Scholar [2] H. Berestycki and D. G. deFigueiredo, Double resonance is semilinear elliptic problems, Comm. Partial Diff. Equas., 6 (1981), 91-120. doi: 10.1080/03605308108820172.  Google Scholar [3] N. P. Cac, On an elliptic boundary value problem at double resonance, J. Math. Anal. Appl., 132 (1988), 473-483. doi: 10.1016/0022-247X(88)90075-3.  Google Scholar [4] K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems," Birkhauser, Boston, 1993.  Google Scholar [5] F. Clarke, "Optimization and Nonsmooth Analysis," Wiley, New York, 1983.  Google Scholar [6] Y. Deng, S. Peng and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic equatio involving critical exponent, Discrete Contin. Dynam. Systems, 32 (2012), 795-826.  Google Scholar [7] D. G. deFigueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Diff. Equas., 17 (1992), 339-346. doi: 10.1080/03605309208820844.  Google Scholar [8] M. Filippakis, D. O'Regan and N. S. Papageorgiou, Positive solutions and bifurcation phenomena for nonlinear elliptic equations of Logistic type:The superdiffusive case, Comm. Pure Appl. Anal., 9 (2010), 1507-1527. doi: 10.3934/cpaa.2010.9.1507.  Google Scholar [9] M. Filippakis and N. S. Papageorgiou, Multiplicity of solutions for doubly resonant Neumann problem, Bull. Belgian Math. Soc., 18 (2011), 135-156.  Google Scholar [10] Z. Guo, Z. Liu, J. Wei and F. Zhou, Bifurcations of some elliptic problems with a singular nonlinearity via Morse index, Comm. Pure. Appl. Anal., 10 (2011), 507-525. doi: 10.3934/cpaa.2011.10.507.  Google Scholar [11] N. Garofalo and F. H. Lin, Unique continuation for elliptic operators: A geometric variatoinal approach, Comm. Pure Appl. Math., 40 (1987), 347-366. Google Scholar [12] L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar [13] J. Garcia Melian, J. Rossi and J. Sabina de Lis, A convex-concave problem with a parameter on the boundary condition, Discrete Contin. Dynam. Systems, 32 (2012), 1095-1124.  Google Scholar [14] Q. Jiu and J. Su, Existence and multiplicity results for perturbations of the $p$-Laplacian, J. Math. Anal. Appl., 281 (2003), 587-601. doi: 10.1016/S0022-247X(03)00165-3.  Google Scholar [15] Z. Liang and J. Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158. doi: 10.1016/j.jmaa.2008.12.053.  Google Scholar [16] M. L. Miotto, Multiple solutions for elliptic problems in $R^N$ with critical Spbolev exponent and weight function, Comm. Pure Appl. Anal., 9 (2010), 233-248. doi: 10.3934/cpaa.2010.9.233.  Google Scholar [17] 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, Comm. Pure Appl. Anal., 10 (2011), 1791-1816. doi: 10.3934/cpaa.2011.10.1791.  Google Scholar [18] N. S. Papageorgiou and S. Th. Kyritsi, "Handbook of Applied Analysis," Springer, New York, 2009.  Google Scholar [19] J-M. Rakotoson, Generalized eigenvalue problem for totally discontinuous operator, Discrete Contin. Dynam. Systems, 28 (2010), 343-373. doi: 10.3934/dcds.2010.28.343.  Google Scholar [20] P. Pucci and J. Serrin, "The Maximum Principle," Birkhauser, Basel, 2007.  Google Scholar [21] S. Robinson, Double resonance in semilinear elliptic boundary value problem over bounded and unbounded domain, Nonlin. Anal., 21 (1993), 407-424. doi: 10.1016/0362-546X(93)90125-C.  Google Scholar [22] R. Showalter, "Hilbert Space Methods for Partial Differential Equations," Pitman, London, 1977.  Google Scholar [23] M. Struwe, "Variational Methods," Springer, Berlin, 1990.  Google Scholar [24] 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 [25] W. Zou, Multiple solutions for elliptic equations with resonance, Nonlinear Anal., 48 (2002), 363-376. doi: 10.1016/S0362-546X(00)00190-5.  Google Scholar
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