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Zbl 0785.35033
Boccardo, L.; Murat, F.; Puel, J.-P.
$L\sp{\infty{}}$ estimate for some nonlinear elliptic partial differential equations and application to an existence result.
(English)
[J] SIAM J. Math. Anal. 23, No.2, 326-333 (1992). ISSN 0036-1410; ISSN 1095-7154/e

Summary: Consider the nonlinear elliptic equation $${\cal A}(u)+H(x,u,Du)=f(x)- \text{div} g(x) \tag E$$ where ${\cal A}(u)=-\text{div}(a(x,u,Du))+a\sb 0(x,u,Du)$ is a Leray-Lions operator defined on $W\sb 0\sp{1,p}(\Omega)$ with $a\sb 0(x,s,\xi)s\ge\alpha\sb 0\vert s\vert\sp p$, $\alpha\sb 0>0$, and where $H$ is a first-order term satisfying $\vert H(x,s,\xi)\vert\le C\sb 0+C\sb 1\vert\xi\vert\sp p$. The main goal of this paper is to prove an $L\sp \infty$ estimate for the bounded solutions of (E) when $f$ belongs to $L\sp q(\Omega)$ and $g$ belongs to $(L\sp r(\Omega))\sp N$ with $r=p'q$ and $\max(1,N/p)<q\le+\infty$. In view of the method and results developed in the author's previous work, this implies the existence of a solution for equation (E).
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
35B45 A priori estimates
35D05 Existence of generalized solutions of PDE
35B35 Stability of solutions of PDE

Keywords: natural growth with respect to $Du$; $L\sp \infty$ estimate; nonlinear elliptic equation; Leray-Lions operator; bounded solutions; existence

Cited in: Zbl 1097.35050

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