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Existence de solutions faibles pour des équations elliptiques quasi- linéaires à croissance quadratique. (Existence of weak solutions for quasilinear elliptic equations with quadratic growth). (French) Zbl 0588.35041
Nonlinear partial differential equations and their applications, Coll. France Semin., Vol. 4, Res. Notes Math. 84, 19-73 (1983).
[For the entire collection see Zbl 0504.00010.]
The authors prove the existence of a weak solution for the elliptic problem \[ -\sum^{n}_{i,j=1}D_{x_ i}(a_{ij}D_{x_ j}u)+a_ 0u+f(x,u,Du)=0\quad in\quad \Omega, \] u\(=0\) in \(\partial \Omega\), where \(a_ 0(x)\geq c_ 0>0\) and f has a quadratic growth in Du. The proof is obtained by a regularization method and the convergence is proved using suitable exponential test functions.

MSC:
35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35A25 Other special methods applied to PDEs