Perturbed elliptic equations with oscillatory nonlinearities

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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

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

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

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