THE NEHARI MANIFOLD APPROACH FOR DIRICHLET PROBLEM INVOLVING THE p(x)-LAPLACIAN EQUATION

Abstract

In this paper, using the Nehari manifold approach and some variational techniques, we discuss the multiplicity of positive solutions for the p(x)-Laplacian problems with non-negative weight functions and prove that an elliptic equation has at least two positive solutions.

Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth

“…Proposition 3. (see [9])Assume that the boundary of D × Ω possesses the cone property and p (., .) ∈ C D × Ω .…”

Existence of Solutions for an Elliptic Equation With Nonstandard Growth

“…On the other hand, problems involving nonstandard growth conditions are extremely attractive because they can model phenomenons which arise from the study of electrorheological fluids or elastic mechanics, stationary thermo-rheological viscous flows of non-Newtonian fluids and they also appear in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [1,14,15]. We refer the reader to [2,3,4,9,10,12,13] and the references therein for the study of p(x)-Laplacian equations.…”

Existence of Weak Solutions for a Nonlocal Problem Involving the p(x)-Laplace Operator

“…However, in case p (x) ̸ = 2 the situation is very crucial, as for example, one encounters the lack of the homogeneity, and a result of this, some classical theories, such as the theory of Sobolev spaces, is not applicable. For the papers involving the p(x)-Laplacian operator we refer the readers to [3,5,11,13,16,17,18] and references therein. Moreover, the nonlinear problems involving the p (x)-Laplacian extremely attractive because they can be used to model dynamical phenomena which arise from the study of electrorheological fluids or elastic mechanics.…”