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The aim of this article is to study nonlinear problem driven by thep(t)p\left(t)-Laplacian with nonsmooth potential. We establish the existence of homoclinic solutions by using variational principle for locally Lipschitz functions and the properties of the generalized Lebesgue-Sobolev space under two cases of the nonsmooth potential: periodic and nonperiodic, respectively. The resulting problem engages two major difficulties: first, due to the appearance of the variable exponent, commonly known methods and techniques for studying constant exponent equations fail in the setting of problems involving variable exponents. Another difficulty we must overcome is verifying the link geometry and certifying boundedness of the Palais-Smale sequence. To our best knowledge, our theorems appear to be the first such result about homoclinic solution for differential inclusion system involving thep(t)p\left(t)-Laplacian.

Homoclinic solutions for a differential inclusion system involving the p ( t )-Laplacian