K. Kefi, M. K. Hamdani, N. T. Chung, J. Liu

Multiple Solutions for a Class of $(p(x),q(x))$-Biharmonic Operator with Hardy Potential and Nonlocal Source Term

abstract:
We study a class of nonhomogeneous $(p(x),q(x))$-biharmonic problems which is seldom studied because the nonlinearity has nonstandard growth and contains a nonlocal term and a Hardy potential. Based on variational methods, especially the abstract critical point result of Bonanno-Candito-D'Aguí [Adv. Nonlinear Stud. 14 (2014), no. 4, 915-939] and a recent three critical points theorem of Bonanno--Marano [Appl. Anal. 89 (2010), 1-10], we prove the existence of at least one non-zero critical point and the existence of at least three distinct critical points without assuming the classical Ambrosetti-Rabinowitz condition. Our results generalize and extend several existing results.

Mathematics Subject Classification: 35J60, 35G30, 35J35, 46E35

Key words and phrases: $p(x)$-biharmonic, critical theorem, generalized Sobolev space, Palais-Smale