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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 and Continuous Dynamical Systems, 2013, 33 (6) : 2469-2494. doi: 10.3934/dcds.2013.33.2469
References:
[1]

W. Allegretto and Y. H. Huang, A Picone's identity for the $p$-Laplacian and applications, Nonlin. Anal., 32 (1998), 819-830. doi: 10.1016/S0362-546X(97)00530-0.

[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), 499-513. doi: 10.1007/BF02572378.

[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), 981-1012. doi: 10.1016/0362-546X(83)90115-3.

[4]

T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlin. Anal., 28 (1997), 419-441. doi: 10.1016/0362-546X(95)00167-T.

[5]

T. Bartsch and Z. Liu, On a superlinear elliptic $p$-Laplacian equation, J. Differential Eqns, 198 (2004), 149-175. doi: 10.1016/j.jde.2003.08.001.

[6]

S. Carl and K. Perera, Sign changing amd multiple solutions for the $p$-Laplacian, Abstr. Appl. Anal., 7 (2003), 613-625. doi: 10.1155/S1085337502207010.

[7]

J.-N. Corvellec, On the second deformation lemma, Topol. Methods Nonlin. Anal., 17 (2001), 55-66.

[8]

D. Costa and C. Magalhaes, Existence results for perturbation of the $p$-Laplacian, Nonlin. Anal., 24 (1995), 409-418. doi: 10.1016/0362-546X(94)E0046-J.

[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), 164-177. doi: 10.1006/jmaa.2000.7228.

[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), 795-826.

[11]

G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electr. Jour. Diff. Eqns, 8 (2002), pp. 12.

[12]

S. Th. Kyritsi-Yiallourou and N. S. Papageorgiou, "Handbook of Applied Analysis, Advances in Mechanics and Mathematics 19," Springer, New York, 2009. doi: 10.1007/b120946.

[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), 385-404. doi: 10.1142/S0219199700000190.

[14]

L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC Press, Boca Raton, Fl, 2006.

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A. Granas and J. Dugundji, "Fixed Point Theory," Springer-Verlag, New York, 2003.

[16]

Z. Guo and Z. Liu, Perturbed elliptic equations with oscillatory nonlinearities, Discrete and Continuous Dynamical Systems, 32 (2012), 3567-3585. doi: 10.3934/dcds.2012.32.3567.

[17]

O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York, 1968.

[18]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.

[19]

S. Liu, Multiple solutions for coercive $p$-Laplacian equations, J. Math. Anal. Appl., 316 (2006), 229-236. doi: 10.1016/j.jmaa.2005.04.034.

[20]

J. Liu and S. Liu, The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc., 37 (2005), 592-600. doi: 10.1112/S0024609304004023.

[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), 4015-4026. doi: 10.3934/dcds.2012.32.4015.

[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), 1401-1414. doi: 10.3934/cpaa.2011.10.1401.

[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), 1791-1816. doi: 10.3934/cpaa.2011.10.1791.

[24]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989.

[25]

O. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Diff. Eqns, 245 (2008), 3628-3638. doi: 10.1016/j.jde.2008.02.035.

[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), 507-531. doi: 10.1007/s00229-007-0127-x.

[27]

F. O. de Paiva, Multiple solutions for a class of quasilinear problems, Discrete Cont. Dynam. Systems, 15 (2006), 669-680. doi: 10.3934/dcds.2006.15.669.

[28]

R. Palais, Homotopy theory of finite dimensional manifolds, Topology, 5 (1966), 1-16.

[29]

E. Papageorgiou and N. S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian, J. Funct. Anal., 244 (2007), 63-77. doi: 10.1016/j.jfa.2006.11.015.

[30]

N. S. Papageorgiou, E. Rocha and V. Staicu, Multiplicity theorems for superlinear elliptic problems, Calc. Var., 33 (2008), 199-230. doi: 10.1007/s00526-008-0172-7.

[31]

K. Perera, Existence and multiplicity results for a Sturm-Liouville equation asymptotically linear at $-\infty$ and superlinear at $+\infty$, Nonlin. Anal., 39 (2000), 669-684. doi: 10.1016/S0362-546X(98)00228-4.

[32]

M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160. doi: 10.2140/pjm.2004.214.145.

[33]

M. Tanaka, Existence of a nontrivial solution for a $p$-Laplacian equation with Fu\vcik type resonance at infinity, Nonlin. Anal., 72 (2010), 507-526. doi: 10.1016$|$j. na. 2008.02.117.

[34]

J. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.

[35]

Z. Q. Wang, On a superlinear elliptic equation, Annales IHP, Section C, Analyse Nonlineaire, 8 (1991), 43-57.

[36]

M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. Jour., 52 (2003), 109-132. doi: 10.1512/iumj.2003.52.2273.

[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), 287-303. doi: 10.1016/j.jde.2004.03.019.

[38]

Z. Zhang and S. Li, On sign-changing and multiple solutions of the $p$-Laplacian, J. Funct. Anal., 197 (2003), 447-468. doi: 10.1016/S0022-1236(02)00103-9.

[39]

Z. Zhang, S. Li, S. Liu and W. Feng, On an asymptotically linear elliptic Dirichlet problem, Abstract Appl. Anal., 7 (2002), 509-516. doi: 10.1155/S1085337502207046.

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), 819-830. doi: 10.1016/S0362-546X(97)00530-0.

[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), 499-513. doi: 10.1007/BF02572378.

[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), 981-1012. doi: 10.1016/0362-546X(83)90115-3.

[4]

T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlin. Anal., 28 (1997), 419-441. doi: 10.1016/0362-546X(95)00167-T.

[5]

T. Bartsch and Z. Liu, On a superlinear elliptic $p$-Laplacian equation, J. Differential Eqns, 198 (2004), 149-175. doi: 10.1016/j.jde.2003.08.001.

[6]

S. Carl and K. Perera, Sign changing amd multiple solutions for the $p$-Laplacian, Abstr. Appl. Anal., 7 (2003), 613-625. doi: 10.1155/S1085337502207010.

[7]

J.-N. Corvellec, On the second deformation lemma, Topol. Methods Nonlin. Anal., 17 (2001), 55-66.

[8]

D. Costa and C. Magalhaes, Existence results for perturbation of the $p$-Laplacian, Nonlin. Anal., 24 (1995), 409-418. doi: 10.1016/0362-546X(94)E0046-J.

[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), 164-177. doi: 10.1006/jmaa.2000.7228.

[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), 795-826.

[11]

G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electr. Jour. Diff. Eqns, 8 (2002), pp. 12.

[12]

S. Th. Kyritsi-Yiallourou and N. S. Papageorgiou, "Handbook of Applied Analysis, Advances in Mechanics and Mathematics 19," Springer, New York, 2009. doi: 10.1007/b120946.

[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), 385-404. doi: 10.1142/S0219199700000190.

[14]

L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC Press, Boca Raton, Fl, 2006.

[15]

A. Granas and J. Dugundji, "Fixed Point Theory," Springer-Verlag, New York, 2003.

[16]

Z. Guo and Z. Liu, Perturbed elliptic equations with oscillatory nonlinearities, Discrete and Continuous Dynamical Systems, 32 (2012), 3567-3585. doi: 10.3934/dcds.2012.32.3567.

[17]

O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York, 1968.

[18]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.

[19]

S. Liu, Multiple solutions for coercive $p$-Laplacian equations, J. Math. Anal. Appl., 316 (2006), 229-236. doi: 10.1016/j.jmaa.2005.04.034.

[20]

J. Liu and S. Liu, The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc., 37 (2005), 592-600. doi: 10.1112/S0024609304004023.

[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), 4015-4026. doi: 10.3934/dcds.2012.32.4015.

[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), 1401-1414. doi: 10.3934/cpaa.2011.10.1401.

[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), 1791-1816. doi: 10.3934/cpaa.2011.10.1791.

[24]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989.

[25]

O. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Diff. Eqns, 245 (2008), 3628-3638. doi: 10.1016/j.jde.2008.02.035.

[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), 507-531. doi: 10.1007/s00229-007-0127-x.

[27]

F. O. de Paiva, Multiple solutions for a class of quasilinear problems, Discrete Cont. Dynam. Systems, 15 (2006), 669-680. doi: 10.3934/dcds.2006.15.669.

[28]

R. Palais, Homotopy theory of finite dimensional manifolds, Topology, 5 (1966), 1-16.

[29]

E. Papageorgiou and N. S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian, J. Funct. Anal., 244 (2007), 63-77. doi: 10.1016/j.jfa.2006.11.015.

[30]

N. S. Papageorgiou, E. Rocha and V. Staicu, Multiplicity theorems for superlinear elliptic problems, Calc. Var., 33 (2008), 199-230. doi: 10.1007/s00526-008-0172-7.

[31]

K. Perera, Existence and multiplicity results for a Sturm-Liouville equation asymptotically linear at $-\infty$ and superlinear at $+\infty$, Nonlin. Anal., 39 (2000), 669-684. doi: 10.1016/S0362-546X(98)00228-4.

[32]

M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160. doi: 10.2140/pjm.2004.214.145.

[33]

M. Tanaka, Existence of a nontrivial solution for a $p$-Laplacian equation with Fu\vcik type resonance at infinity, Nonlin. Anal., 72 (2010), 507-526. doi: 10.1016$|$j. na. 2008.02.117.

[34]

J. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.

[35]

Z. Q. Wang, On a superlinear elliptic equation, Annales IHP, Section C, Analyse Nonlineaire, 8 (1991), 43-57.

[36]

M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. Jour., 52 (2003), 109-132. doi: 10.1512/iumj.2003.52.2273.

[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), 287-303. doi: 10.1016/j.jde.2004.03.019.

[38]

Z. Zhang and S. Li, On sign-changing and multiple solutions of the $p$-Laplacian, J. Funct. Anal., 197 (2003), 447-468. doi: 10.1016/S0022-1236(02)00103-9.

[39]

Z. Zhang, S. Li, S. Liu and W. Feng, On an asymptotically linear elliptic Dirichlet problem, Abstract Appl. Anal., 7 (2002), 509-516. doi: 10.1155/S1085337502207046.

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