Research ArticleOpen AccessGoogle Scholar indexed

Multiplicity of Solutions for a Class of Noncooperative Elliptic Systems

¹ School of Mathematical Sciences, Liaocheng University, Liaocheng, China
² School of Mathematical Sciences, Liaocheng University, Liaocheng, China

Advances in Pure Mathematics·Volume 13 (2023)·Pages 610–619·Published 31 August 2023·DOI10.4236/apm.2023.139040

Abstract

In this paper, we consider the following noncooperative elliptic systems where Ω is a bounded domain in R N with smooth boundary ∂ Ω, λ , δ , γ are real parameters, and . We assume that F is subquadratic at zero with respect to the variables u , v . By using a variant Clark’s theorem, we obtain infinitely many nontrivial solutions ( u k , v k ) with as k → ∞ . Compared with the existing literature, we do not need to assume the behavior of the nonlinearity ∇ F at infinity.

KeywordsNoncooperative Elliptic Systems(&lti&gtPS&lt/i&gt)&ltsup&gt*&lt/sup&gtConditionClark’s Theorem

Costa, D.G. and Magalhaes, C.A. (1994) A Variational Approach to Subquadratic Perturbations of Elliptic Systems. Journal of Differential Equations, 111, 103-122. https://doi.org/10.1006/jdeq.1994.1077
Costa, D.G. and Magalhaes, C.A. (1995) A Variational Approach to Noncooperative Elliptic Systems. Nonlinear Analysis: Theory, Methods & Applications, 25, 699-715. https://doi.org/10.1016/0362-546X(94)00180-P
De Figueredo, D.G. and Mitidieri, E. (1986) A Maximum Principle for an Elliptic system and Applications to Semilinear Problem. SIAM Journal on Mathematical Analysis, 17, 836-849. https://doi.org/10.1137/0517060
De Figueiredo, D.G. and Felmer, P.L. (1994) On Superquadratic Elliptic Systems. Transactions of the American Mathematical Society, 343, 99-116. https://doi.org/10.1090/S0002-9947-1994-1214781-2
Bartsch, T. and De Figueiredo, D.G. (1999) Infinitely Many Solutions of Nonlinear Elliptic Systems. In: Escher, J. and Simonett, G., Eds., Topics in Nonlinear Analysis. Progress in Nonlinear Differential Equations and Their Applications, Vol. 35, Birkhäuser, Basel, 51-67. https://doi.org/10.1007/978-3-0348-8765-6_4
Guo, Y.X. (2000) Nontrivial Solutions for Resonant Noncooperative Elliptic Systems. Communications on Pure and Applied Mathematics, 53, 1335-1349. https://doi.org/10.1002/1097-0312(200011)53:11 3.0.CO;2-3
Ke, X.-F. and Tang, C.-L. (2010) Existence of Solutions for a Class of Noncooperative Elliptic Systems. Journal of Mathematical Analysis and Applications, 370, 18-29. https://doi.org/10.1016/j.jmaa.2010.04.043
Clément, P., De Figueiredo, D.G. and Mitidieri, E. (1992) Positive Splutions of Semilinear Elliptic Systems. Communications in Partial Differential Equations, 17, 923-940. https://doi.org/10.1080/03605309208820869
Guo, Y.X. and Liu, J.Q. (2009) Bifurcation for Strongly Indefinite Functional and Applications to Hamiltonian System and Noncooperative Elliptic System. Journal of Mathematical Analysis and Applications, 359, 28-38. https://doi.org/10.1016/j.jmaa.2009.05.003
Hulshof, J. and Vandervorst, R. (1993) Differential Systems with Strongly Indefinite Variational Structure. Journal of Functional Analysis, 114, 32-58. https://doi.org/10.1006/jfan.1993.1062
Ke, X.-F. and Tang, C.-L. (2011) Existence and Multiplicity of Solutions for Asymptotically Linear Noncooperative Elliptic Systems. Journal of Mathematical Analysis and Applications, 375, 631-647. https://doi.org/10.1016/j.jmaa.2010.09.041

Menu

Multiplicity of Solutions for a Class of Noncooperative Elliptic Systems

Abstract

References (18)