Multiplicity of solutions to a potential operator equation and its applications. (English) Zbl 1474.35686

Huang, Jincheng

Abstr. Appl. Anal. 2014, Article ID 947139, 11 p. (2014).

Summary: We consider the multiplicity of solutions for operator equation involving homogeneous potential operators. With the help of Nehari manifold and fibering maps, we prove that such equation has at least two nontrivial solutions. Furthermore, we apply this result to prove the existence of two nonnegative solutions for three types of quasilinear elliptic systems involving \((p, q)\)-Laplacian operator and concave-convex nonlinearities.

MSC:

35R20	Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
35A15	Variational methods applied to PDEs
35J57	Boundary value problems for second-order elliptic systems
35J62	Quasilinear elliptic equations

References:

[1]	Aghajani, A.; Shamshiri, J., Multiplicity of positive solutions for quasilinear elliptic \(p\)-Laplacian systems, Electronic Journal of Differential Equations, 2012, 111, 1-16 (2012) · Zbl 1258.35107
[2]	Bozhkov, Y.; Mitidieri, E., Existence of multiple solutions for quasilinear systems via fibering method, Journal of Differential Equations, 190, 1, 239-267 (2003) · Zbl 1021.35034
[3]	Afrouzi, G. A.; Rasouli, S. H., A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system, Nonlinear Analysis: Theory, Methods & Applications, 71, 1-2, 445-455 (2009) · Zbl 1173.35444
[4]	Brown, K. J.; Wu, T.-F., A fibering map approach to a potential operator equation and its applications, Differential and Integral Equations, 22, 11-12, 1097-1114 (2009) · Zbl 1240.35569
[5]	Brown, K. J.; Wu, T.-F., A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function, Journal of Mathematical Analysis and Applications, 337, 2, 1326-1336 (2008) · Zbl 1132.35361
[6]	Li, H.; Wu, X.-P.; Tang, C.-L., Multiple positive solutions for a class of semilinear elliptic systems with nonlinear boundary condition, Journal of Applied Mathematics and Computing, 38, 1-2, 617-630 (2012) · Zbl 1298.35058
[7]	Ishiwata, M., Effect of topology on the multiplicity of solutions for some semilinear elliptic systems with critical Sobolev exponent, Nonlinear Differential Equations and Applications, 16, 3, 283-296 (2009) · Zbl 1173.35440
[8]	Drábek, P.; Pohozaev, S. I., Positive solutions for the \(p\)-Laplacian: application of the fibering method, Proceedings of the Royal Society of Edinburgh Section A, 127, 4, 703-726 (1997) · Zbl 0880.35045