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Article Dans Une Revue Rendiconti Lincei. Matematica e Applicazioni Année : 2016

Quasilinear problems involving a perturbation with quadratic growth in the gradient and a noncoercive zeroth order term

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\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}
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hal-04521485 , version 1 (29-03-2024)

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Boussad Hamour, François Murat. Quasilinear problems involving a perturbation with quadratic growth in the gradient and a noncoercive zeroth order term. Rendiconti Lincei. Matematica e Applicazioni, 2016, 27 (2), pp.195 - 233. ⟨10.4171/rlm/731⟩. ⟨hal-04521485⟩
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