Existence and multiplicity of solutions for a Dirichlet problem involving the discrete p(x)-Laplacian operator

In the present paper, using the three critical points theorem and variational method, we study the existence and multiplicity of solutions for a Dirichlet problem involving the discrete p(x)-Laplacian operator

Existence of solutions for p(x)p(x)-Laplacian equations

We discuss the problem \begin{equation*} \left\{ \begin{array}{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p(x)-2}\nabla u\right) =\lambda (a\left( x\right) \left\vert u\right\vert ^{q(x)-2}u+b(x)\left\vert u\right\vert ^{h(x)-2}u)\text{,} & \text{for }x\in \Omega , \\ u=0\text{,} & \text{for }x\in \partial \Omega . \end{array} \right. \end{equation*} where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^{N}$ $\left( N\geq 2\right)$ and $p$ is Lipschitz continuous, $q$ and $h$ are continuous functions on $\overline{\Omega }$ such that $1<q(x)<p(x)<h(x)<p^{\ast }(x)$ and $p(x)<N$. We show the existence of at least one nontrivial weak solution. Our approach relies on the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem

Electrorheological Fluids Equations Involving Variable Exponent with Dependence on the Gradient via Mountain Pass Techniques

This article deals with a quasilinear elliptic equation with variable exponent under a homogenous Dirichlet boundary-value condition, where nonlinearity also depends on the gradient of the solution. By using an iterative method based on Mountain Pass techniques, the existence of a positive solution is obtained. © 2016, Copyright © Taylor & Francis Group, LLC

Existence of one weak solution for p(X)-biharmonic equations involving a concave-convex nonlinearity

In the present paper, using variational approach and the theory of the variable exponent Lebesgue spaces, the existence of nontrivial weak solutions to a fourth order elliptic equation involving a p(x)-biharmonic operator and a concave-convex nonlinearity the Navier boundary conditions is obtained. © 2017, Drustvo Matematicara Srbije. All rights reserved

Existence of solutions for an elliptic equation with nonstandard growth

This paper deals with the existence of solutions for some elliptic equations with nonstandard growth under zero Dirichlet boundary condition. Using a direct variational method and the theory of the variable exponent Sobolev spaces, we set some conditions that ensures the existence of nontrivial weak solutions. © 2013 Academic Publications, Ltd

Existence of one weak solution for p(X)-biharmonic equations involving a concave-convex nonlinearity

In the present paper, using variational approach and the theory of the variable exponent Lebesgue spaces, the existence of nontrivial weak solutions to a fourth order elliptic equation involving a p(x)-biharmonic operator and a concave-convex nonlinearity the Navier boundary conditions is obtained. © 2017, Drustvo Matematicara Srbije. All rights reserved