In this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation- div\,\,a\left( {x,u,\nabla u} \right) = b\left( x \right){\left| u \right|^{p - 2}}u + \lambda H\left( {x,u,\nabla u} \right),where Ω is a bounded smooth domain of N.
This paper is devoted to study the following nonlinear anisotropic elliptic unilateral problem\begin{equation*}\begin{cases}A\,u -\mbox{div}\,\phi(u)=\mu \quad \mbox{in} \qquad \Omega \\\;u=0 \qquad \mbox{on} \quad \partial \Omega ,\end{cases}\end{equation*}where the right hand side $\,\mu\;$ belongs to $\; L^1(\Omega)+ W_{0}^{-1,\overrightarrow{p}'} (\Omega,\ \overrightarrow{\omega}^*)$. The operator $\displaystyle A\,u=-\sum_{i=1}^{N}\partial_{i}\,a_{i}(x,\ u,\ \nabla u)$ is a Leray-Lions anisotropic operator acting from $\; W_{0}^{1,\overrightarrow{p}} (\Omega,\ \overrightarrow{\omega})\;$ into its dual $\; W_{0}^{-1,\overrightarrow{p}'} (\Omega,\ \overrightarrow{\omega}^*)$ and $\phi_{i}\in C^{0}(\mathbb{R},\mathbb{R})$.
In this article, we use the topological degree based on the abstract Hammerstein equation to investigate the existence of weak solutions for a class of elliptic Dirichlet boundary value problems involving the fractional p(x)-Laplacian operator with discontinuous nonlinearities. The appropriate functional framework for this problems is the fractional Sobolev space with variable exponent.
RESUMENEn este artículo, usamos el grado topológico basado en la ecuación abstracta de Hammerstein para investigar la existencia de soluciones débiles para una clase de problemas elípticos de valor en la frontera de Dirichlet que involucran el operador p(x)-Laplaciano fraccional con no linealidades discontinuas. El marco funcional apropiado para estos problemas es el espacio de Sobolev fraccional con exponente variable.
The main crux of this manuscript is to establish the existence and uniqueness of solutions for nonlocal fractional evolution equations involving ψ−Caputo fractional derivatives of an arbitrary order α ∈ (0, 1) with nondense domain. The mild solutions of our proposed model are constructed by employing generalized ψ−Laplace transform and some new density functions. The proofs are based on Krasnoselskii fixed point theorem and some basic techniques of ψ−fractional calculus. As application, a nontrivial example is given to illustrate our theoritical results.
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