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Electronic Journal of Differential Equations 2002-Fez conference on Partial Differental Equations,
Electron. J. Diff. Eqns., Conf. 09, 2002, pp. 183-202.

Nonlinear equations with natural growth terms and measure data

Alessio Porretta

We consider a class of nonlinear elliptic equations containing a $p$-Laplacian type operator, lower order terms having natural growth with respect to the gradient, and bounded measures as data. The model example is the equation
$$ -\Delta_p(u) + g(u)|\nabla u|^p=\mu    $$
in a bounded set $\Omega\subset \mathbb{R}^N$, coupled with a Dirichlet boundary condition. We provide a review of the results recently obtained in the absorption case (when $g(s)s\geq0$) and prove a new existence result without any sign condition on $g$, assuming only that $g\in L^1({\bf R})$. This latter assumption is proved to be optimal for existence of solutions for any measure $\mu$.

Published December 28, 2002.
Subject classfications: 35J60, 35J65, 35R05.
Key words: Nonlinear elliptic equations, natural growth terms, measure data.

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Alessio Porretta
Dipartimento di Matematica,
Universita di Roma "Tor Vergata",
Via della Ricerca Scientifica 1,
00133, Roma, Italia.

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