American Institute of Mathematical Sciences

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October  2012, 32(10): 3567-3585. doi: 10.3934/dcds.2012.32.3567

Perturbed elliptic equations with oscillatory nonlinearities

Zuji Guo 1, and  Zhaoli Liu 2,

1. 

School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China
2. 

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received  August 2010 Revised  February 2012 Published  May 2012

In this paper, arbitrarily many solutions, in particular arbitrarily many nodal solutions, are proved to exist for perturbed elliptic equations of the form \begin{equation*}\label{} \left\{ \begin{array}{ll} \displaystyle -\Delta_p u+|u|^{p-2}u = Q(x)(f(u)+\varepsilon g(u)),\ \ \ x\in \mathbb R^N, \\ u\in W^{1,p}(\mathbb R^N), \end{array} \right. (P_\varepsilon) \end{equation*} where $\Delta_p$ is the $p$-Laplacian operator defined by $\Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, $Q\in \mathcal{C}(\mathbb R^N,\mathbb R)$ is a positive function, $f\in\mathcal{C}(\mathbb R, \mathbb R)$ oscillates either near the origin or near the infinity, and $\epsilon$ is a real number. For $g$ it is only required that $g\in\mathcal{C}(\mathbb R, \mathbb R)$. Under appropriate assumptions on $Q$ and $f$ the following results which are special cases of more general ones are proved: the unperturbed problem $(P_0)$ has infinitely many nodal solutions, and for any $n\in\mathbb N$ the perturbed problem $(P_\varepsilon)$ has at least $n$ nodal solutions provided that $|\epsilon|$ is sufficiently small.
Keywords: invariant sets of desending flow, Arbitrarily many solutions, perturbed elliptic equations, nodal solutions, p-Laplacian operator., osillatory terms.
Mathematics Subject Classification: 35J20, 35J6.
Citation: Zuji Guo, Zhaoli Liu. Perturbed elliptic equations with oscillatory nonlinearities. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3567-3585. doi: 10.3934/dcds.2012.32.3567
References:
[1]

G. Anello and G. Cordaro, Perturbation from Dirichlet problem involving oscillating nonlinearities, J. Differential Equations, 234 (2007), 80-90. doi: 10.1016/j.jde.2006.11.011.  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), 655-677. doi: 10.1007/s002090050492.  Google Scholar

[3]

T. Bartsch and Z. Liu, Location and critical groups of critical points in Banach spaces with an application to nonlinear eigenvalue problems, Adv. Differential Equations, 9 (2004), 645-676.  Google Scholar

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T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the $p$-Laplacian, Commun. Contemp. Math., 6 (2004), 245-258.  Google Scholar

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

[6]

T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a $p$-Laplacian equation, Proc. London Math. Soc. (3), 91 (2005), 129-152. doi: 10.1112/S0024611504015187.  Google Scholar

[7]

S. Chen and S. Li, Splitting lemma at infinity and a strongly resonant problem with periodic nonlinearity, Calc. Var. Partial Differential Equations, 27 (2006), 105-123. doi: 10.1007/s00526-006-0025-1.  Google Scholar

[8]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and montonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493-516. doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

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[10]

A. Kristály, Detection of arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms, J. Differential Equations, 245 (2008), 3849-3868. doi: 10.1016/j.jde.2008.05.014.  Google Scholar

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S. Li and Z. Liu, Perturbations from symmeric elliptic boundary value problems, J. Differential Equations, 185 (2002), 271-280. doi: 10.1006/jdeq.2001.4160.  Google Scholar

[12]

Y. Li, Z. Liu and C. Zhao, Nodal solutions of a perturbed elliptic problem, Topol. Methods Nonlinear Anal., 32 (2008), 49-68.  Google Scholar

[13]

P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Comm. Partial Differential Equations, 21 (1996), 721-733. doi: 10.1080/03605309608821205.  Google Scholar

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J. Saint Raymond, On the multiplicity of the solutions of equations $-\Delta u=\lambda f(u)$, J. Differential Equations, 180 (2002), 65-88. doi: 10.1006/jdeq.2001.4057.  Google Scholar

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R. Schaaf and K. Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions, Trans. Amer. Math. Soc., 306 (1988), 853-859. doi: 10.1090/S0002-9947-1988-0933322-5.  Google Scholar

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P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.  Google Scholar

show all references

References:
[1]

G. Anello and G. Cordaro, Perturbation from Dirichlet problem involving oscillating nonlinearities, J. Differential Equations, 234 (2007), 80-90. doi: 10.1016/j.jde.2006.11.011.  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), 655-677. doi: 10.1007/s002090050492.  Google Scholar

[3]

T. Bartsch and Z. Liu, Location and critical groups of critical points in Banach spaces with an application to nonlinear eigenvalue problems, Adv. Differential Equations, 9 (2004), 645-676.  Google Scholar

[4]

T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the $p$-Laplacian, Commun. Contemp. Math., 6 (2004), 245-258.  Google Scholar

[5]

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

[6]

T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a $p$-Laplacian equation, Proc. London Math. Soc. (3), 91 (2005), 129-152. doi: 10.1112/S0024611504015187.  Google Scholar

[7]

S. Chen and S. Li, Splitting lemma at infinity and a strongly resonant problem with periodic nonlinearity, Calc. Var. Partial Differential Equations, 27 (2006), 105-123. doi: 10.1007/s00526-006-0025-1.  Google Scholar

[8]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and montonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493-516. doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[9]

P. Habets, E. Serra and M. Tarallo, Multiplicity results for boundary value problems with potentials oscillating around resonance, J. Differential Equations, 138 (1997), 133-156. doi: 10.1006/jdeq.1997.3267.  Google Scholar

[10]

A. Kristály, Detection of arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms, J. Differential Equations, 245 (2008), 3849-3868. doi: 10.1016/j.jde.2008.05.014.  Google Scholar

[11]

S. Li and Z. Liu, Perturbations from symmeric elliptic boundary value problems, J. Differential Equations, 185 (2002), 271-280. doi: 10.1006/jdeq.2001.4160.  Google Scholar

[12]

Y. Li, Z. Liu and C. Zhao, Nodal solutions of a perturbed elliptic problem, Topol. Methods Nonlinear Anal., 32 (2008), 49-68.  Google Scholar

[13]

P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Comm. Partial Differential Equations, 21 (1996), 721-733. doi: 10.1080/03605309608821205.  Google Scholar

[14]

J. Saint Raymond, On the multiplicity of the solutions of equations $-\Delta u=\lambda f(u)$, J. Differential Equations, 180 (2002), 65-88. doi: 10.1006/jdeq.2001.4057.  Google Scholar

[15]

R. Schaaf and K. Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions, Trans. Amer. Math. Soc., 306 (1988), 853-859. doi: 10.1090/S0002-9947-1988-0933322-5.  Google Scholar

[16]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.  Google Scholar

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