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Résultats d’existence pour certains problèmes elliptiques quasilinéaires. (Existence results for certain quasilinear elliptic problems). (French) Zbl 0557.35051
The authors study the Dirichlet problem for the equation \[ - \sum^{n}_{i=1}(\partial /\partial x_ i)a_ i(x,u,\nabla u)+f(x,u,\nabla u)=0 \] in the Sobolev space \(W^{1,p}(\Omega)\), \(\Omega\) a bounded domain in \(R^ n\) 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 \(W^{1,p}(\Omega)\cap L^{\infty}(\Omega)\). In particular, the lower order term f(x,\(\eta\),\(\xi)\) is allowed to grow as fast as \(\sum a_ i(x,\eta,\xi)\xi_ i\) for \(| \xi | \to +\infty\), namely like \(| \xi |^ p.\)
Under mild continuity assumptions on the coefficients the authors prove the existence of a solution \(u\in W_ 0^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) provided there exist a Lipschitzian subsolution \(\phi\) and a Lipschitzian supersolution \(\psi\) with \(\phi\leq \psi\) 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.
Reviewer: F.Tomi

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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