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Quasilinear problems having quadratic growth in the gradient: An existence result when the source term is small. (English) Zbl 0917.35039
Équations aux dérivées partielles et applications. Articles dédiés à Jacques-Louis Lions. Gauthier-Villars: Paris. 497-515 (1998).
Let \(\Omega\) be a bounded open set in \(\mathbb{R}^N\), \(N\geq 3\), \(A(x)\) a coercive matrix with bounded measurable elements, and \(H(x,u, Du)\) a nonlinear function which satisfies \(-c_0\zeta A(x)\zeta\leq H(x,s,\zeta)\text{sign }s\leq \gamma\zeta A(x)\zeta\) for a given \(\gamma>0\) and \(c_0>0\). The authors prove that if \(\| f\|_{N/2}\) is sufficiently small, then there exists at least one solution \(u\) to the elliptic problem \[ -\text{div}(A(x) Du)= H(x,u,Du)+ f\quad\text{in }D'(\Omega),\quad u\in H^1_0(\Omega) \] such that \(w= \delta^{-1}(e^{\delta| u|}- 1)\text{sign }u\in H^1_0(\Omega)\) for every \(\delta> \gamma\) not too large. The bound on \(f\), namely, \(\| f\|_{N/2}< \alpha(\gamma C_N)^{-1}\), where \(\alpha\) is the coercivity constant for \(A(x)\) and \(C_N\) is the best Sobolev constant in \(\|\varphi\|^2_{2^*}\leq C_N\| D\varphi\|^2_2\) for all \(\varphi\in H^1_0(\Omega)\) and \(2^*= 2N/(N- 2)\), is shown to be sharp. The proof of the existence consists of obtaining a priori estimates on the solution of an approximate equation and then passing to the limit by means of these estimates.
For the entire collection see [Zbl 0899.00020].

35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs