Existence Results for Double Phase Problem in Sobolev–Orlicz Spaces with Variable Exponents in Complete Manifold

Abstract

In this paper, we study the existence of non-negative non-trivial solutions for a class of double-phase problems where the source term is a Caratheodory function that satisfies the Ambrosetti–Rabinowitz type condition in the framework of Sobolev–Orlicz spaces with variable exponents in complete manifold. Our approach is based on the Nehari manifold and some variational techniques. Furthermore, the Hölder ine-quality, continuous and compact embedding results are proved.

Nonnegative solution of a class of double phase problems with logarithmic nonlinearity

“…Author details 1 Laboratoy LAMA, National School of Applied Sciences, Sidi Mohamed Ben Abdellah University, Fez, Morocco. 2 Laboratory LAMA, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Fez, Morocco. 3 Dipartimento di Matematica e Informatica, Università di Catania, Catania, Italy.…”

“…A nice overview of the recent work on such equations with variable exponents can be found in [32] by Ragusa et al, [34] by Shi et al, [20,21] by Gaczkowski,et al,[2][3][4] by Aberqi et al in the context of Sobolev spaces on complete manifolds. We refer to Benslimane et al [11,12] for more results.…”

Onpz–Laplacian System Involving Critical Nonlinearities

“…This section is devoted to recalling some definitions and properties which will be used in the next sections (see [22][23][24][25][26][27]).…”

“…Here, ðM, gÞ is a complete compact Riemannian Nmanifold, α, β ∈ ℝ + * to be specified later, and p, q, a, b ∈ Cð MÞ satisfying the assumptions (23) and (24) in Section 3. − Δ g,pðzÞ is the Laplacian operator on ðM, gÞ.…”

Infinitely many solutions to Kirchhoff double phase problems with variable exponents

“…In this case, only few and very recent results exist. We refer to the papers of Aberqi-Bennouna-Benslimane-Ragusa [1], Bahrouni-Rȃdulescu-Winkert [5], Crespo-Blanco-Gasiński-Harjulehto-Winkert [10], Kim-Kim-Oh-Zeng [20] , Leonardi-Papageorgiou [22], Vetro-Winkert [26] and Zeng-Rȃdulescu-Winkert [28].…”