Agradecimentos: The author wishes to thank the anonymous referee for her/his useful suggestions in order to improve the manuscript. The author is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The author realized the manuscript within the auspices of the GNAMPA project titled Equazioni alle derivate parziali: problemi e modelli (Grant No. Prot_20191219-143223-545), of the FAPESP Project titled Operators with non standard growth (Grant No. 2019/23917-3), of the FAPESP Thematic Project titled Systems and partial differential equations (Grant No. 2019/02512-5) and of the CNPq Project titled Variational methods for singular fractional problems (Grant No. 3787749185990982) Ver menos

Abstract: In this paper, we deal with the following double phase problem {- div (vertical bar del u vertical bar(p-2) del u + a (x) vertical bar del u vertical bar(q-2) del u) = gamma + f (x, u) (vertical bar u vertical bar(p-2)u/vertical bar x vertical bar(p) + a(x) vertical bar u vertical bar(q-2) u/vertical bar x vertical bar(q)) in Omega, u = 0 in partial derivative partial derivative Omega, where Omega subset of R-N is an open, bounded set with Lipschitz boundary, 0 is an element of Omega, N >= 2, 1 < p < q < N, weight a(.) >= 0, gamma is areal parameter and f is a subcritical function. By variational method, we provide the existence of a non-trivial weak solution on the Musielak-Orlicz-Sobolev space W-0(1,H) (Omega), with modular functionH(t, x) = t(p) + a(x)t(q). For this, we first introduce the Hardy inequalities for space W-0(1,H) (Omega), under suitable assumptions on a(.) Ver menos