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Zbl 0557.35051
Boccardo, L.; Murat, F.; Puel, J.P.
Résultats d'existence pour certains problèmes elliptiques quasilinéaires. (Existence results for certain quasilinear elliptic problems). (French)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11, 213-235 (1984). ISSN 0391-173X

The authors study the Dirichlet problem for the equation $$- \sum\sp{n}\sb{i=1}(\partial /\partial x\sb i)a\sb i(x,u,\nabla u)+f(x,u,\nabla u)=0 $$ in the Sobolev space $W\sp{1,p}(\Omega)$, $\Omega$ a bounded domain in $R\sp n$ and $p>1$. The coefficients are assumed to satisfy the usual coercivity and monotonicity conditions and, moreover, such natural growth conditions that the problem becomes meaningful in the class $W\sp{1,p}(\Omega)\cap L\sp{\infty}(\Omega)$. In particular, the lower order term f(x,$\eta$,$\xi)$ is allowed to grow as fast as $\sum a\sb i(x,\eta,\xi)\xi\sb i$ for $\vert \xi \vert \to +\infty$, namely like $\vert \xi \vert\sp p.$ \par Under mild continuity assumptions on the coefficients the authors prove the existence of a solution $u\in W\sb 0\sp{1,p}(\Omega)\cap L\sp{\infty}(\Omega)$ provided there exist a Lipschitzian subsolution $\phi$ and a Lipschitzian supersolution $\psi$ with $\phi\le \psi$ and $\phi =\psi =0$ on $\partial \Omega$. As the authors state, no such result was known before under equally mild regularity assumptions on the coefficients and at the same time allowing the natural growth of f.
[F.Tomi]

MSC 2000:: *35J65 (Nonlinear) BVP for (non)linear elliptic equations
35J25 Second order elliptic equations, boundary value problems
35B05 General behavior of solutions of PDE

Keywords: quasilinear elliptic equations; Dirichlet problem; Sobolev space; existence; Lipschitzian subsolution; Lipschitzian supersolution

Cited in: Zbl 0932.35072 Zbl 0785.35039 Zbl 0790.35039

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