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Zeitschrift für Analysis und ihre Anwendungen


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Volume 35, Issue 1, 2016, pp. 61–80
DOI: 10.4171/ZAA/1555

Published online: 2015-12-23

Sharp Estimates and Existence for Anisotropic Elliptic Problems with General Growth in the Gradient

Francesco Della Pietra[1] and Nunzia Gavitone[2]

(1) Università degli Studi di Napoli Federico II, Italy
(2) Università degli Studi di Napoli Federico II, Italy

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: \begin{equation*} \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. \end{equation*} 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.

Keywords: Nonlinear elliptic problems with gradient dependent terms, anisotropic Laplacian, convex symmetrization, a priori estimates

Della Pietra Francesco, Gavitone Nunzia: Sharp Estimates and Existence for Anisotropic Elliptic Problems with General Growth in the Gradient. Z. Anal. Anwend. 35 (2016), 61-80. doi: 10.4171/ZAA/1555