Search

In this paper, we focus on some class of Stefan type problems. We prove the existence and uniqueness of renormalized solution in anisotropic Sobolev spaces with data belonging to L1 -data, based on the properties of the renormalized solutions and the generalized monotonicity method in the functional spaces.

This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by \alignat2&-{\rm div}( b(|u|)|\nabla u|^{p-2}\nabla u) + d(|u|)|\nabla u|^p = f - {\rm div}(c(x)|u|^{\alpha}) &\quad&in \Omega, & u = 0 &\quad&on \partial\Omega, where $\Omega$ is a bounded open set of $\mathbb{R}^N$ ($N\geq2$) with $1

In this article, we study the existence of renormalized solution for the nonlinear $p(x)$-parabolic problem of the form:\\ $\begin{cases} \frac{\partial b(x,u)}{\partial t} - div (a(x,t,u,\nabla u)) + H(x,t,u,\nabla u) = f - divF $ \; in $Q= \Omega\times(0,T)\\$ $b(x,u)\mid_{t=0} = b(x,u_{0})$ \; in $\Omega\\$ $ u = 0 $ \quad on $\partial\Omega\times(0,T)\\$ $\end{cases}$ with $ f $ $ \in L^{1} (Q),$\; $b(x,u_{0}) \in L^{1} (\Omega)$ and $ F \in (L^{P'(.)}(Q))^{N}. $\\ The main contribution of our work is to prove the existence of renormalized solution of the variable exponent Soblev spaces, and we suppose that\;$ H(x,t,u,\nabla u)$\; is the non linear term satisfying some growth condition but no sig

In this article, we study the problem $$displaylines{ frac{partial}{partial t} b(x, u)-hbox{div}(a(x,t,u,D u)) +H(x,t,u,Du) = fquad hbox{in } Omegaimes ]0,T[,cr b(x,u)(t=0)=b(x,u_0)quadhbox{in } Omega,cr u=0quadhbox{in } partialOmegaimes ]0,T[ }$$ in the framework of weighted Sobolev spaces, with $b(x,u)$ unbounded function on u. The main contribution of our work is to prove the existence of a renormalized solution without the sign condition and the coercivity condition on $H(x,t,u,Du)$. The critical growth condition on $H$ is with respect to Du and no growth condition with respect to u. The second term f belongs to $L^1(Q)$, and $b(x,u_0)in L^1(Omega)$.

In this paper, we study the existence of solutions for the nonlinear degenerated elliptic problem $$ -{mathop{ m div}}(a(x,u, abla u)) = Fquad mbox{in } Omega, $$ where $Omega$ is a bounded domain of $mathbb{R}^N$, $N geq 2$, $a:Omegaimesmathbb{R}imesmathbb{R}^Nomathbb{R}^N $ is a Caratheodory function satisfying the coercivity condition, but they verify the general growth condition and only the large monotonicity. The second term $F$ belongs to $W^{-1, p'}(Omega, w^*)$.

We prove the existence of solutions for nonlinear degenerate elliptic boundary-value problems of higher order. Solutions are obtained using pseudo-monotonicity theory in a suitable weighted Sobolev space.

We prove an existence theorem for a strongly nonlinear degenerated elliptic inequalities involving nonlinear operators of the form $Au+g(x,u,abla u)$. Here $A$ is a Leray-Lions operator, $g(x,s,xi)$ is a lower order term satisfying some natural growth with respect to $|abla u|$. There is no growth restrictions with respect to $|u|$, only a sign condition. Under the assumption that the second term belongs to $W^{-1,p'}(Omega,w^*)$, we obtain the main result via strong convergence of truncations.

Extending the theory of pseudo-monotone mappings in weighted Sobolev spaces, we prove some existence results for degenerate or singular elliptic equations generated by the second-order differential operator $$ Au(x)=-mathop{m div}a(x,u,abla u))+a_0(x,u,abla u), $$ (in particular, when only large monotonicity is satisfied)