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November  2011, 10(6): 1791-1816. doi: 10.3934/cpaa.2011.10.1791

## A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems

 1 Département de Mathématiques, Université de Perpignan, Avenue de Villeneuve 52, 66860 Perpignan Cedex 2 Department of Mathematics, National University of Ireland, University Road, Galway 3 Department of Mathematics, National Technical University, Zografou Campus, Athens 15780

Received  May 2010 Revised  March 2011 Published  May 2011

In this paper we present a framework which permits the unified treatment of the existence of multiple solutions for superlinear and sublinear Neumann problems. Using critical point theory, truncation techniques, the method of upper-lower solutions, Morse theory and the invariance properties of the negative gradient flow, we show that the problem can have seven nontrivial smooth solutions, four of which have constant sign and three are nodal.
Citation: D. Motreanu, Donal O'Regan, Nikolaos S. Papageorgiou. A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1791-1816. doi: 10.3934/cpaa.2011.10.1791
##### References:
 [1] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong'' resonance at infinity,, Nonlinear Anal., 7 (1983), 981.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar [2] T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems,, Math. Z., 233 (2000), 655.  doi: 10.1007/s002090050492.  Google Scholar [3] T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance,, Nonlinear Anal., 28 (1997), 419.  doi: doi:10.1016/0362-546X(95)00167-T.  Google Scholar [4] H. Brezis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers,, C. R. Acad. Sci. Paris Ser. I Math., 317 (1993), 465.   Google Scholar [5] K.-C. Chang, "Infinite-dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and Their Applications,", 6. Birkh\, (1993).   Google Scholar [6] K.-C. Chang, Morse theory in nonlinear analysis. Nonlinear functional analysis and applications to differential equations,, (Trieste, (1997), 60.   Google Scholar [7] K.-C. Chang and M.-Y Jiang, Morse theory for indefinite nonlinear elliptic problems,, Ann. Inst. H. Poincar\'e Anal. Non Lin閍ire, 26 (2009), 139.  doi: doi:10.1016/j.anihpc.2007.08.004.  Google Scholar [8] D. Costa and C. Magalhaes, Variational elliptic problems which are nonquadratic at infinity,, Nonlinear Anal., 23 (1994), 1401.  doi: doi:10.1016/0362-546X(94)90135-X.  Google Scholar [9] D. G. de Figueiredo and J.-P. Gossez, Strict monotonicity of eigenvalues and unique continuation,, Comm. Partial Differential Equations, 17 (1992), 339.  doi: 10.1080/03605309208820844.  Google Scholar [10] N. Dunford and J. Schwartz, "Linear Operators. I. General Theory,", Pure and Applied Mathematics, (1958).   Google Scholar [11] G. Fei, On periodic solutions of superquadratic Hamiltonian systems,, Electron. J. Differential Equations, 8 (2002).   Google Scholar [12] J. P. Garcia Azorero, J. Peral Alonso and J. Manfredi, Sobolev versus Holder local minimizers and global multiplicity for some quasilinear elliptic equations,, Commun. Contemp. Math., 2 (2000), 385.  doi: 10.1142/S0219199700000190.  Google Scholar [13] N. Garofalo and F.-H. Lin, Unique continuation for elliptic operators: a geometric-variational approach,, Comm. Pure Appl. Math., 40 (1987), 347.  doi: 10.1002/cpa.3160400305.  Google Scholar [14] L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,", Chapman & Hall/CRC Press, (2006).   Google Scholar [15] Z. Guo and Z. Zhang, $W^{1, p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations,, J. Math. Anal. Appl., 286 (2003), 32.  doi: 10.1016/S0022-247X(03)00282-8.  Google Scholar [16] S. Heikkila and V. Lakshmikantham, "Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations,", Monographs and Textbooks in Pure and Applied Mathematics, (1994).   Google Scholar [17] R. Iannacci and M. Nkashama, Nonlinear boundary value problems at resonance,, Nonlinear Anal., 11 (1987), 455.  doi: doi:10.1016/0362-546X(87)90064-2.  Google Scholar [18] R. Iannacci and M. Nkashama, Nonlinear two point boundary value problem without Landesman-Lazer condition,, Proc. Amer. Math. Soc., 106 (1989), 943.   Google Scholar [19] C. C. Kuo, On the solvability of a nonlinear second-order elliptic equation at resonance,, Proc. Amer. Math. Soc., 124 (1996), 83.   Google Scholar [20] 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 [21] S. Li and Z. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems,, J. Anal. Math., 81 (2000), 373.  doi: 10.1007/BF02788997.  Google Scholar [22] J. Q. Liu and S. Wu, Calculating critical groups of solutions for elliptic problem with jumping nonlinearity,, Nonlinear Anal., 49 (2002), 779.  doi: 10.1016/S0362-546X(01)00139-0.  Google Scholar [23] J. Mawhin, Semicoercive monotone variational problems,, Acad. Roy. Belg. Bull. Cl. Sci., 73 (1987), 118.   Google Scholar [24] J. Mawhin, J. Ward and M. Willem, Variational methods and semilinear elliptic equations,, Arch. Rational Mech. Anal., 95 (1986), 269.  doi: 10.1007/BF00251362.  Google Scholar [25] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Applied Mathematical Sciences, (1989).   Google Scholar [26] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations,, Manuscripta Math., 124 (2007), 507.  doi: 10.1007/s00229-007-0127-x.  Google Scholar [27] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A unified approach for multiple constant sign and nodal solutions,, Adv. Differential Equations, 12 (2007), 1363.   Google Scholar [28] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonlinear Neumann problems near resonance,, Indiana Univ. Math. J., 58 (2009), 1257.  doi: 10.1512/iumj.2009.58.3565.  Google Scholar [29] D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems,, J. Differential Equations, 232 (2007), 1.  doi: 10.1016/j.jde.2006.09.008.  Google Scholar [30] A. Qian, Existence of infinitely many nodal solutions for a superlinear Neumann boundary value problem,, Bound. Value Probl., (2005), 329.  doi: 10.1155/BVP.2005.329.  Google Scholar [31] C. I. Tang and X. P. Wu, Existence and multiplicity for solutions of Neumann problem for semilinear elliptic equations,, J. Math. Anal. Appl., 288 (2003), 660.  doi: 10.1016/j.jmaa.2003.09.034.  Google Scholar [32] J. Wang, J. Xu and F. Zhang, Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition,, Discrete and Continuous Dynamical Systems, 27 (2010), 1241.  doi: 10.3934/dcds.2010.27.1241.  Google Scholar [33] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optim., 12 (1984), 191.  doi: 10.1007/BF01449041.  Google Scholar

show all references

##### References:
 [1] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong'' resonance at infinity,, Nonlinear Anal., 7 (1983), 981.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar [2] T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems,, Math. Z., 233 (2000), 655.  doi: 10.1007/s002090050492.  Google Scholar [3] T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance,, Nonlinear Anal., 28 (1997), 419.  doi: doi:10.1016/0362-546X(95)00167-T.  Google Scholar [4] H. Brezis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers,, C. R. Acad. Sci. Paris Ser. I Math., 317 (1993), 465.   Google Scholar [5] K.-C. Chang, "Infinite-dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and Their Applications,", 6. Birkh\, (1993).   Google Scholar [6] K.-C. Chang, Morse theory in nonlinear analysis. Nonlinear functional analysis and applications to differential equations,, (Trieste, (1997), 60.   Google Scholar [7] K.-C. Chang and M.-Y Jiang, Morse theory for indefinite nonlinear elliptic problems,, Ann. Inst. H. Poincar\'e Anal. Non Lin閍ire, 26 (2009), 139.  doi: doi:10.1016/j.anihpc.2007.08.004.  Google Scholar [8] D. Costa and C. Magalhaes, Variational elliptic problems which are nonquadratic at infinity,, Nonlinear Anal., 23 (1994), 1401.  doi: doi:10.1016/0362-546X(94)90135-X.  Google Scholar [9] D. G. de Figueiredo and J.-P. Gossez, Strict monotonicity of eigenvalues and unique continuation,, Comm. Partial Differential Equations, 17 (1992), 339.  doi: 10.1080/03605309208820844.  Google Scholar [10] N. Dunford and J. Schwartz, "Linear Operators. I. General Theory,", Pure and Applied Mathematics, (1958).   Google Scholar [11] G. Fei, On periodic solutions of superquadratic Hamiltonian systems,, Electron. J. Differential Equations, 8 (2002).   Google Scholar [12] J. P. Garcia Azorero, J. Peral Alonso and J. Manfredi, Sobolev versus Holder local minimizers and global multiplicity for some quasilinear elliptic equations,, Commun. Contemp. Math., 2 (2000), 385.  doi: 10.1142/S0219199700000190.  Google Scholar [13] N. Garofalo and F.-H. Lin, Unique continuation for elliptic operators: a geometric-variational approach,, Comm. Pure Appl. Math., 40 (1987), 347.  doi: 10.1002/cpa.3160400305.  Google Scholar [14] L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,", Chapman & Hall/CRC Press, (2006).   Google Scholar [15] Z. Guo and Z. Zhang, $W^{1, p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations,, J. Math. Anal. Appl., 286 (2003), 32.  doi: 10.1016/S0022-247X(03)00282-8.  Google Scholar [16] S. Heikkila and V. Lakshmikantham, "Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations,", Monographs and Textbooks in Pure and Applied Mathematics, (1994).   Google Scholar [17] R. Iannacci and M. Nkashama, Nonlinear boundary value problems at resonance,, Nonlinear Anal., 11 (1987), 455.  doi: doi:10.1016/0362-546X(87)90064-2.  Google Scholar [18] R. Iannacci and M. Nkashama, Nonlinear two point boundary value problem without Landesman-Lazer condition,, Proc. Amer. Math. Soc., 106 (1989), 943.   Google Scholar [19] C. C. Kuo, On the solvability of a nonlinear second-order elliptic equation at resonance,, Proc. Amer. Math. Soc., 124 (1996), 83.   Google Scholar [20] 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 [21] S. Li and Z. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems,, J. Anal. Math., 81 (2000), 373.  doi: 10.1007/BF02788997.  Google Scholar [22] J. Q. Liu and S. Wu, Calculating critical groups of solutions for elliptic problem with jumping nonlinearity,, Nonlinear Anal., 49 (2002), 779.  doi: 10.1016/S0362-546X(01)00139-0.  Google Scholar [23] J. Mawhin, Semicoercive monotone variational problems,, Acad. Roy. Belg. Bull. Cl. Sci., 73 (1987), 118.   Google Scholar [24] J. Mawhin, J. Ward and M. Willem, Variational methods and semilinear elliptic equations,, Arch. Rational Mech. Anal., 95 (1986), 269.  doi: 10.1007/BF00251362.  Google Scholar [25] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Applied Mathematical Sciences, (1989).   Google Scholar [26] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations,, Manuscripta Math., 124 (2007), 507.  doi: 10.1007/s00229-007-0127-x.  Google Scholar [27] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A unified approach for multiple constant sign and nodal solutions,, Adv. Differential Equations, 12 (2007), 1363.   Google Scholar [28] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonlinear Neumann problems near resonance,, Indiana Univ. Math. J., 58 (2009), 1257.  doi: 10.1512/iumj.2009.58.3565.  Google Scholar [29] D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems,, J. Differential Equations, 232 (2007), 1.  doi: 10.1016/j.jde.2006.09.008.  Google Scholar [30] A. Qian, Existence of infinitely many nodal solutions for a superlinear Neumann boundary value problem,, Bound. Value Probl., (2005), 329.  doi: 10.1155/BVP.2005.329.  Google Scholar [31] C. I. Tang and X. P. Wu, Existence and multiplicity for solutions of Neumann problem for semilinear elliptic equations,, J. Math. Anal. Appl., 288 (2003), 660.  doi: 10.1016/j.jmaa.2003.09.034.  Google Scholar [32] J. Wang, J. Xu and F. Zhang, Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition,, Discrete and Continuous Dynamical Systems, 27 (2010), 1241.  doi: 10.3934/dcds.2010.27.1241.  Google Scholar [33] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optim., 12 (1984), 191.  doi: 10.1007/BF01449041.  Google Scholar
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