Existence of renormalized solutions for some degenerate and non-coercive elliptic equations

summary:This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by \begin{aligned}t 2&-{\rm div}( b(|u|)|\nabla u|^{p-2}\nabla u) + d(|u|)|\nabla u|^{p} = f - {\rm div}(c(x)|u|^{\alpha }) &\quad &\mbox {in}\ \Omega ,\\ & u = 0 &\quad &\mbox {on}\ \partial \Omega , \end{aligned}t where $\Omega$ is a bounded open set of $\mathbb {R}^N$ ($N\geq 2$) with $1<p<N$ and $f \in L^{1}(\Omega ),$ under some growth conditions on the function $b(\cdot )$ and $d(\cdot ),$ where $c(\cdot )$ is assumed to be in $L^{\frac {N}{(p-1)}}(\Omega ).$ We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded

Existence of solutions in the sense of distributions of anisotropic nonlinear elliptic equations with variable exponent

The aim of this paper is to study the existence of solutions in the sense of distributions for a~strongly nonlinear elliptic problem where the second term of the equation $f$ is in $W^{-1,\overrightarrow{p}'(\,\cdot\,)}(\Omega)$ which is the dual space of the anisotropic Sobolev $W_{0}^{1,\overrightarrow{p}(\,\cdot\,)}(\Omega)$ and later $f$ will be in~$L^{1}(\Omega)$

Parabolic problems in non-standard Sobolev spaces of infinite order

This paper is devoted to the study of the existence of solutions for the strongly nonlinear $p(x)$-parabolic equation$$\frac{\partial u}{\partial t} + Au + g(x,t,u) = f(x,t),$$where $A$ is a Leray-Lions operator acted from $V^{\infty,p(x)}(a_\alpha,Q_{T})$ into its dual. The nonlinear term $\&gt;g(x,t,s)\&gt;$ satisfies growth and sign conditions and the datum $\&gt;f\&gt;$ is assumed to be in the dual space $V^{-\infty,p'(x)}(a_\alpha,Q_{T})\&gt;.

An existence result for two-dimensional parabolic integro-differential equations involving CEV model

In this paper, we present an existence result of weak solutions for some parabolic equations involving the so-called CEV model with jumps

Existence of infinitely many weak solutions for some quasilinear p⃗(x)\vec {p}(x)-elliptic Neumann problems

summary:We consider the following quasilinear Neumann boundary-value problem of the type $$\begin {cases} -\displaystyle \sum _{i=1}^{N}\frac {\partial }{\partial x_{i}}a_{i}\Big (x,\frac {\partial u}{\partial x_{i}}\Big ) + b(x)|u|^{p_{0}(x)-2}u = f(x,u)+ g(x,u) &\text {in} \ \Omega , \\ \quad \dfrac {\partial u}{\partial \gamma } = 0 &\text {on} \ \partial \Omega . \end {cases}$$ We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev spaces and we give some examples

Existence and regularity of entropy solutions for strongly nonlinear p(x)-elliptic equations

This article is devoted to study the existence of solutions for the strongly nonlinear p(x)-elliptic problem $$displaylines{ - operatorname{div} a(x,u,abla u) + g(x,u,abla u) = f- operatorname{div} phi(u) quad ext{in } Omega, cr u = 0 quad ext{on } partialOmega, }$$ with $fin L^1(Omega)$ and $phi in C^{0}(mathbb{R}^{N})$, also we will give some regularity results for these solutions

Quasilinear class of noncoercive parabolic problems with Hardy potential and L1-data

In this article, we study the following noncoercive quasilinear parabolic problem ∂u∂t−diva(x,t,u,∇u)+ν∣u∣s−1u=λ∣u∣p−2u∣x∣p+finQT,u=0onΣT,u(x,0)=u0inΩ,\left\{\begin{array}{ll}\frac{\partial u}{\partial t}-\hspace{0.1em}\text{div}\hspace{0.1em}a\left(x,t,u,\nabla u)+\nu {| u| }^{s-1}u=\lambda \frac{{| u| }^{p-2}u}{{| x| }^{p}}+f& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{Q}_{T},\\ u=0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{\Sigma }_{T},\\ u\left(x,0)={u}_{0}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\end{array}\right. with f∈L1(QT)f\in {L}^{1}\left({Q}_{T}) and u0∈L1(Ω){u}_{0}\in {L}^{1}\left(\Omega ) and show the existence of entropy solutions for this noncoercive parabolic problem with Hardy potential and L1-data

Existence of solutions for some quasilinear p⃗(x)\vec {p}(x)-elliptic problem with Hardy potential

summary:We consider the anisotropic quasilinear elliptic Dirichlet problem $$\begin {cases} \displaystyle -\sum _{i=1}^{N} D^{i} a_{i}(x,u,\nabla u) + |u|^{s(x)-1}u= f +\lambda \frac {|u|^{p_{0}(x)-2}u}{|x|^{p_{0}(x)}}&\text {in}\ \Omega ,\\ u = 0 & \text {on}\ \partial \Omega , \end {cases}$$ where $\Omega$ is an open bounded subset of $\Bbb R^N$ containing the origin. We show the existence of entropy solution for this equation where the data $f$ is assumed to be in $L^{1}(\Omega )$ and $\lambda$ is a positive constant

Existence of entropy solutions for some nonlinear elliptic problems involving variable exponent and measure data

In this paper, we study the existence of entropy solutions for some nonlinear $p(x)-$elliptic equation of the type Au - \mbox{div }\phi(u) + H(x,u,\nabla u) = \mu, where $A$ is an operator of Leray-Lions type acting from $W_{0}^{1,p(x)}(\Omega)$ into its dual, the strongly nonlinear term $H$ is assumed only to satisfy some nonstandard growth condition with respect to $|\nabla u|,$ here $\>\phi(\cdot)\in C^{0}(I\!\!R,I\!\!R^{N})\>$ and $\mu$ belongs to ${\mathcal{M}}_{0}^{b}(\Omega)$