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The aim of this paper is to establish the existence of infinitely many solutions for a class of anisotropic p → -Laplacian problems with Dirichlet boundary conditions, depending on two real parameters. These problems have a variational structure and some critical point theorems are applied. We also give an example to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability-type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic problem involving the fractional p (· , ·) -Laplacian operator. As one of the main tools, topological degree theory is applied. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the following non-homogeneous Neumann-type problem 0.1 - div (α (| ∇ u |) ∇ u) + u = u | u | p - 2 , in B 1 , ∂ u ∂ ν = 0 , on ∂ B 1 , where B 1 is the unit ball in R n and p > 2 . We establish the existence of a non-constant, positive, radially non-decreasing weak solution for (0.1), under certain assumptions on α . Our approach relies on the theory of Orlicz spaces combined with a new variational method that allows one to deal with problems beyond the usual locally compactness structure and a variant of Mountain Pass Theorem. [ABSTRACT FROM AUTHOR]

We analyze the existence of a capacity solution to the following nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, φ , - A u = ρ (u) | ∇ φ | 2 in Ω , div (ρ (u) ∇ φ) = 0 in Ω , φ = φ 0 on ∂ Ω , u = 0 on ∂ Ω , where Ω ⊂ R d , d ≥ 2 and A u = - div a (x , u , ∇ u) is a Leray–Lions operator defined on W 0 1 L M (Ω) , M is a N-function which does not have to satisfy a Δ 2 condition. Therefore, we work with generalized Orlicz–Sobolev spaces which are not necessarily reflexive. The function φ 0 is given. The proof combines truncation methods, monotonicity techniques and regularizing methods in Orlicz spaces. We introduce a sequence of approximate problems which converges (up to a subsequence) in a certain sense to a capacity solution in the context of non-reflexive Orlicz–Sobolev spaces. [ABSTRACT FROM AUTHOR]

Using the Nehari manifold, we establish the existence of positive solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present. We show how the existence of solutions depends on the properties of the Nehari manifold. [ABSTRACT FROM AUTHOR]