American Institute of Mathematical Sciences

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

Perturbed elliptic equations with oscillatory nonlinearities

 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.
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.
 [1] Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922 [2] Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595 [3] Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1121-1147. doi: 10.3934/dcdsb.2021083 [4] Nikolaos S. Papageorgiou, Vicenţiu D. Rǎdulescu, Dušan D. Repovš. Nodal solutions for the Robin p-Laplacian plus an indefinite potential and a general reaction term. Communications on Pure and Applied Analysis, 2018, 17 (1) : 231-241. doi: 10.3934/cpaa.2018014 [5] Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069 [6] Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593 [7] Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2117-2138. doi: 10.3934/cpaa.2021060 [8] Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure and Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761 [9] Liang Zhang, X. H. Tang, Yi Chen. Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators. Communications on Pure and Applied Analysis, 2017, 16 (3) : 823-842. doi: 10.3934/cpaa.2017039 [10] Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 [11] Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 933-982. doi: 10.3934/dcds.2020067 [12] Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Dead cores and bursts for p-Laplacian elliptic equations with weights. Conference Publications, 2007, 2007 (Special) : 191-200. doi: 10.3934/proc.2007.2007.191 [13] Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033 [14] Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure and Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019 [15] Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063 [16] Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447 [17] Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51 [18] Petru Jebelean. Infinitely many solutions for ordinary $p$-Laplacian systems with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2008, 7 (2) : 267-275. doi: 10.3934/cpaa.2008.7.267 [19] Mohamed Karim Hamdani, Lamine Mbarki, Mostafa Allaoui, Omar Darhouche, Dušan D. Repovš. Existence and multiplicity of solutions involving the $p(x)$-Laplacian equations: On the effect of two nonlocal terms. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022129 [20] Xiying Sun, Qihuai Liu, Dingbian Qian, Na Zhao. Infinitely many subharmonic solutions for nonlinear equations with singular $\phi$-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (1) : 279-292. doi: 10.3934/cpaa.20200015

2020 Impact Factor: 1.392