Quasilinear problems involving a perturbation with quadratic growth in the gradient and a noncoercive zeroth order term
Résumé
\begin{abstract}
In this paper we consider the problem
\begin{equation*}
\left\{\begin{array}{ll}
u\in H_{0}^{1}(\Omega),
\\ &
\\
-\textrm{div}\,(A(x)Du)=H(x,u,Du)+a_{0}(x)u+f(x)& \textrm{in } \mathcal{D}'(\Omega),
\end{array}
\right.
\end{equation*}
where $\Omega$ is an open bounded set of $\mathbb{R}^{N}$, $N \geq 3$, $A(x)$ is a coercive matrix with
coefficients in $L^\infty(\Omega)$, $f\in L^{N/2}(\Omega)$, $a_{0}\in L^{q}(\Omega )$, $ q>N/2 $, $a_{0}\geq 0$, and $H(x,s,\xi)$ is a
Carath\'eodory function which satisfies
\begin{equation*}
-c_{0}\, A(x)\, \xi\xi\leq H(x,s,\xi)\,{\rm sign}(s)\leq \gamma\,A(x)\,\xi\xi ,\;\;\;
{\rm p.p. }\; x\in \Omega,\;\;\;\forall s\in\mathbb{R},\;\;\;
\forall\xi \in \mathbb{R}^{N}.
\end{equation*}
%\rule{\linewidth}{.5pt}
\noindent For $a_{0}$ and $f$ sufficiently small, we prove the existence
of at least one solution $u$ of this problem which is moreover such that $\big(\exp(\delta_0 |u|)-1 \big)$ belongs to $H_{0}^{1}(\Omega)$ for some $\delta_0>\gamma$.
\end{abstract}
Mots clés
Quasilinear problems perturbation with quadratic growth in the gradient noncoercive zeroth order term existence Mathematics Subject Classification: 35J60 35J62
Quasilinear problems
perturbation with quadratic growth in the gradient
noncoercive zeroth order term
existence Mathematics Subject Classification: 35J60
35J62
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Origine : Fichiers produits par l'(les) auteur(s)