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Keywords:
renormalized solution; nonlinear elliptic equation; non-coercive problem
renormalized solution; nonlinear elliptic equation; non-coercive problem
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.
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