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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].

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