4 SHARP ESTIMATES AND EXISTENCE FOR ANISOTROPIC ELLIPTIC PROBLEMS WITH GENERAL GROWTH IN THE GRADIENT
Francesco Della Pietra, Pietra, Nunzia Gavitone · 2016
In this paper, we prove sharp estimates and existence results for anisotropic nonlinear elliptic problems with lower order terms depending on the gradient. Our prototype is: \left\{ \begin{alignedat}{2} -\mathcal Q_{p}u &=[H(Du)]^{q}+f(x)&\quad &\text{in }\Omega,\\ u&=0&&\text{on }\partial \Omega. \end{alignedat} \right. Here \Omega is a bounded open set of \mathbb R^{N} , N\ge 2 , 0 < p-1 < q \le p < N , and \mathcal Q_{p} is the anisotropic operator \mathcal Q_{p} u =\mathrm {div} \left( [H(Du)]^{p-1}H_{\xi}(Du) \right), where H is a suitable norm of \mathbb R^{N} . Moreover, f belongs to a suitable Marcinkiewicz space.