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June  2013, 33(6): 2469-2494. doi: 10.3934/dcds.2013.33.2469

## Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term

 1 Department of Mathematics, Hellenic Naval Academy, Piraeus 18539 2 Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780

Received  January 2012 Revised  September 2012 Published  December 2012

We consider a nonlinear Dirichlet problem driven by the $p$-Laplace differential operator. We assume that the Carathéodory reaction term $f(z,x)$ exhibits an asymmetric behavior on the two semiaxes of $\mathbb{R}$. Namely, $f(z,\cdot)$ is $(p-1)$-linear near $-\infty$ and $(p-1)$-superlinear near $+\infty$, but without satisfying the well-known Ambrosetti--Rabinowitz condition (AR-condition). Combining variational methods based on critical point theory, with suitable truncation techniques and Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative).
Citation: Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2469-2494. doi: 10.3934/dcds.2013.33.2469
##### References:
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Differential Eqns, 198 (2004), 149.  doi: 10.1016/j.jde.2003.08.001.  Google Scholar [6] S. Carl and K. Perera, Sign changing amd multiple solutions for the $p$-Laplacian,, Abstr. Appl. Anal., 7 (2003), 613.  doi: 10.1155/S1085337502207010.  Google Scholar [7] J.-N. Corvellec, On the second deformation lemma,, Topol. Methods Nonlin. Anal., 17 (2001), 55.   Google Scholar [8] D. Costa and C. Magalhaes, Existence results for perturbation of the $p$-Laplacian,, Nonlin. Anal., 24 (1995), 409.  doi: 10.1016/0362-546X(94)E0046-J.  Google Scholar [9] N. Dancer and K. Perera, Some remarks on the Fučik spectrum of the $p$-Laplacian and critical groups,, J. Math. Anal. Appl., 254 (2001), 164.  doi: 10.1006/jmaa.2000.7228.  Google Scholar [10] Y. Deng, S. Peng and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic system involving critical exponent,, Discrete and Continuous Dynamical Systems, 32 (2012), 795.   Google Scholar [11] G. 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Uraltseva, "Linear and Quasilinear Elliptic Equations,", Academic Press, (1968).   Google Scholar [18] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlin. Anal., 12 (1988), 1203.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar [19] S. Liu, Multiple solutions for coercive $p$-Laplacian equations,, J. Math. Anal. Appl., 316 (2006), 229.  doi: 10.1016/j.jmaa.2005.04.034.  Google Scholar [20] J. Liu and S. Liu, The existence of multiple solutions to quasilinear elliptic equations,, Bull. London Math. Soc., 37 (2005), 592.  doi: 10.1112/S0024609304004023.  Google Scholar [21] J. Mawhin, Multiplicity of solutions for variational systems involving $\varphi$-Laplacians with singular $\varphi$ and periodic potentials,, Discrete and Continuous Dynamical Systems, 32 (2012), 4015.  doi: 10.3934/dcds.2012.32.4015.  Google Scholar [22] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential,, Communications on Pure and Applied Analysis, 10 (2011), 1401.  doi: 10.3934/cpaa.2011.10.1401.  Google Scholar [23] D. Motreanu, Donal 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,, Communications on Pure and Applied Analysis, 10 (2011), 1791.  doi: 10.3934/cpaa.2011.10.1791.  Google Scholar [24] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer-Verlag, (1989).   Google Scholar [25] O. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition,, J. Diff. Eqns, 245 (2008), 3628.  doi: 10.1016/j.jde.2008.02.035.  Google Scholar [26] D. Motreanu, 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] F. O. de Paiva, Multiple solutions for a class of quasilinear problems,, Discrete Cont. Dynam. Systems, 15 (2006), 669.  doi: 10.3934/dcds.2006.15.669.  Google Scholar [28] R. Palais, Homotopy theory of finite dimensional manifolds,, Topology, 5 (1966), 1.   Google Scholar [29] E. Papageorgiou and N. S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian,, J. Funct. Anal., 244 (2007), 63.  doi: 10.1016/j.jfa.2006.11.015.  Google Scholar [30] N. S. Papageorgiou, E. Rocha and V. Staicu, Multiplicity theorems for superlinear elliptic problems,, Calc. Var., 33 (2008), 199.  doi: 10.1007/s00526-008-0172-7.  Google Scholar [31] K. Perera, Existence and multiplicity results for a Sturm-Liouville equation asymptotically linear at $-\infty$ and superlinear at $+\infty$,, Nonlin. Anal., 39 (2000), 669.  doi: 10.1016/S0362-546X(98)00228-4.  Google Scholar [32] M. Schechter and W. Zou, Superlinear problems,, Pacific J. Math., 214 (2004), 145.  doi: 10.2140/pjm.2004.214.145.  Google Scholar [33] M. Tanaka, Existence of a nontrivial solution for a $p$-Laplacian equation with Fu\vcik type resonance at infinity,, Nonlin. Anal., 72 (2010), 507.  doi: 10.1016$|$j. na. 2008.02.117.  Google Scholar [34] J. Vazquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optim., 12 (1984), 191.  doi: 10.1007/BF01449041.  Google Scholar [35] Z. Q. Wang, On a superlinear elliptic equation,, Annales IHP, 8 (1991), 43.   Google Scholar [36] M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero,, Indiana Univ. Math. Jour., 52 (2003), 109.  doi: 10.1512/iumj.2003.52.2273.  Google Scholar [37] Z. Zhang, J. Chen and S. Li, Construction of pseudogradient vector field and sign-changing multiple solutions involving the $p$-Laplacian,, Jour. Diff. Eqns, 201 (2004), 287.  doi: 10.1016/j.jde.2004.03.019.  Google Scholar [38] Z. Zhang and S. Li, On sign-changing and multiple solutions of the $p$-Laplacian,, J. Funct. Anal., 197 (2003), 447.  doi: 10.1016/S0022-1236(02)00103-9.  Google Scholar [39] Z. Zhang, S. Li, S. Liu and W. Feng, On an asymptotically linear elliptic Dirichlet problem,, Abstract Appl. Anal., 7 (2002), 509.  doi: 10.1155/S1085337502207046.  Google Scholar

show all references

##### References:
 [1] W. Allegretto and Y. H. Huang, A Picone's identity for the $p$-Laplacian and applications,, Nonlin. Anal., 32 (1998), 819.  doi: 10.1016/S0362-546X(97)00530-0.  Google Scholar [2] D. Arcoya and S. Villegas, Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at $-\infty$ and superlinear at $+\infty$,, Math. Z., 219 (1995), 499.  doi: 10.1007/BF02572378.  Google Scholar [3] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity,, Nonlin. Anal., 7 (1983), 981.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar [4] T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance,, Nonlin. Anal., 28 (1997), 419.  doi: 10.1016/0362-546X(95)00167-T.  Google Scholar [5] T. Bartsch and Z. Liu, On a superlinear elliptic $p$-Laplacian equation,, J. Differential Eqns, 198 (2004), 149.  doi: 10.1016/j.jde.2003.08.001.  Google Scholar [6] S. Carl and K. Perera, Sign changing amd multiple solutions for the $p$-Laplacian,, Abstr. Appl. Anal., 7 (2003), 613.  doi: 10.1155/S1085337502207010.  Google Scholar [7] J.-N. Corvellec, On the second deformation lemma,, Topol. Methods Nonlin. Anal., 17 (2001), 55.   Google Scholar [8] D. Costa and C. Magalhaes, Existence results for perturbation of the $p$-Laplacian,, Nonlin. Anal., 24 (1995), 409.  doi: 10.1016/0362-546X(94)E0046-J.  Google Scholar [9] N. Dancer and K. Perera, Some remarks on the Fučik spectrum of the $p$-Laplacian and critical groups,, J. Math. Anal. Appl., 254 (2001), 164.  doi: 10.1006/jmaa.2000.7228.  Google Scholar [10] Y. Deng, S. Peng and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic system involving critical exponent,, Discrete and Continuous Dynamical Systems, 32 (2012), 795.   Google Scholar [11] G. Fei, On periodic solutions of superquadratic Hamiltonian systems,, Electr. Jour. Diff. Eqns, 8 (2002).   Google Scholar [12] S. Th. Kyritsi-Yiallourou and N. S. Papageorgiou, "Handbook of Applied Analysis, Advances in Mechanics and Mathematics 19,", Springer, (2009).  doi: 10.1007/b120946.  Google Scholar [13] J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations,, Comm. Contemp. Math., 2 (2000), 385.  doi: 10.1142/S0219199700000190.  Google Scholar [14] L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,", Chapman & Hall/CRC Press, (2006).   Google Scholar [15] A. Granas and J. Dugundji, "Fixed Point Theory,", Springer-Verlag, (2003).   Google Scholar [16] Z. Guo and Z. Liu, Perturbed elliptic equations with oscillatory nonlinearities,, Discrete and Continuous Dynamical Systems, 32 (2012), 3567.  doi: 10.3934/dcds.2012.32.3567.  Google Scholar [17] O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations,", Academic Press, (1968).   Google Scholar [18] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlin. Anal., 12 (1988), 1203.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar [19] S. Liu, Multiple solutions for coercive $p$-Laplacian equations,, J. Math. Anal. Appl., 316 (2006), 229.  doi: 10.1016/j.jmaa.2005.04.034.  Google Scholar [20] J. Liu and S. Liu, The existence of multiple solutions to quasilinear elliptic equations,, Bull. London Math. Soc., 37 (2005), 592.  doi: 10.1112/S0024609304004023.  Google Scholar [21] J. Mawhin, Multiplicity of solutions for variational systems involving $\varphi$-Laplacians with singular $\varphi$ and periodic potentials,, Discrete and Continuous Dynamical Systems, 32 (2012), 4015.  doi: 10.3934/dcds.2012.32.4015.  Google Scholar [22] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential,, Communications on Pure and Applied Analysis, 10 (2011), 1401.  doi: 10.3934/cpaa.2011.10.1401.  Google Scholar [23] D. Motreanu, Donal 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,, Communications on Pure and Applied Analysis, 10 (2011), 1791.  doi: 10.3934/cpaa.2011.10.1791.  Google Scholar [24] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer-Verlag, (1989).   Google Scholar [25] O. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition,, J. Diff. Eqns, 245 (2008), 3628.  doi: 10.1016/j.jde.2008.02.035.  Google Scholar [26] D. Motreanu, 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] F. O. de Paiva, Multiple solutions for a class of quasilinear problems,, Discrete Cont. Dynam. Systems, 15 (2006), 669.  doi: 10.3934/dcds.2006.15.669.  Google Scholar [28] R. Palais, Homotopy theory of finite dimensional manifolds,, Topology, 5 (1966), 1.   Google Scholar [29] E. Papageorgiou and N. S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian,, J. Funct. Anal., 244 (2007), 63.  doi: 10.1016/j.jfa.2006.11.015.  Google Scholar [30] N. S. Papageorgiou, E. Rocha and V. Staicu, Multiplicity theorems for superlinear elliptic problems,, Calc. Var., 33 (2008), 199.  doi: 10.1007/s00526-008-0172-7.  Google Scholar [31] K. Perera, Existence and multiplicity results for a Sturm-Liouville equation asymptotically linear at $-\infty$ and superlinear at $+\infty$,, Nonlin. Anal., 39 (2000), 669.  doi: 10.1016/S0362-546X(98)00228-4.  Google Scholar [32] M. Schechter and W. Zou, Superlinear problems,, Pacific J. Math., 214 (2004), 145.  doi: 10.2140/pjm.2004.214.145.  Google Scholar [33] M. Tanaka, Existence of a nontrivial solution for a $p$-Laplacian equation with Fu\vcik type resonance at infinity,, Nonlin. Anal., 72 (2010), 507.  doi: 10.1016$|$j. na. 2008.02.117.  Google Scholar [34] J. Vazquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optim., 12 (1984), 191.  doi: 10.1007/BF01449041.  Google Scholar [35] Z. Q. Wang, On a superlinear elliptic equation,, Annales IHP, 8 (1991), 43.   Google Scholar [36] M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero,, Indiana Univ. Math. Jour., 52 (2003), 109.  doi: 10.1512/iumj.2003.52.2273.  Google Scholar [37] Z. Zhang, J. Chen and S. Li, Construction of pseudogradient vector field and sign-changing multiple solutions involving the $p$-Laplacian,, Jour. Diff. Eqns, 201 (2004), 287.  doi: 10.1016/j.jde.2004.03.019.  Google Scholar [38] Z. Zhang and S. Li, On sign-changing and multiple solutions of the $p$-Laplacian,, J. Funct. Anal., 197 (2003), 447.  doi: 10.1016/S0022-1236(02)00103-9.  Google Scholar [39] Z. Zhang, S. Li, S. Liu and W. Feng, On an asymptotically linear elliptic Dirichlet problem,, Abstract Appl. Anal., 7 (2002), 509.  doi: 10.1155/S1085337502207046.  Google Scholar
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