Quasilinear problems involving a perturbation with quadratic growth in the gradient and a noncoercive zeroth order term
Boussad Hamour, François Murat · Rendiconti Lincei Matematica e Applicazioni · 2016
In this paper we consider the problem \begin{array}{ll} u \in H_{0}^{1}(\Omega), \\ -\textrm{div}\,(A(x)Du)=H(x,u,Du)+f(x)+a_{0}(x)\, u& \textrm{in} \;\;\;\mathcal{D}'(\Omega), \end{array} 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) , H(x,s,\xi) is a Carathéodory function which satisfies for some \gamma >0 -c_{0}\, A(x)\, \xi\xi\leq H(x,s,\xi)\,{\rm sign}(s)\leq \gamma\,A(x)\,\xi\xi \;\; {\rm a.e. }\; x \in \Omega,\;\;\forall s \in\mathbb{R},\;\; \forall \xi \in \mathbb{R}^{N}\!\!, f belongs to L^{N/2}(\Omega) and a_{0} \geq 0 to L^{q}(\Omega ) , q>N/2 . For f and a_{0} sufficiently small, we prove the existence of at least one solution u of this problem which is such that e^{\delta_0 |u|} -1 belongs to H_{0}^{1}(\Omega) for some \delta_0 \geq \gamma . This solution satisfies some a priori estimate.