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On the existence of weak solutions of perturbed systems with critical growth. (English) Zbl 0664.35027
The author proves the existence of weak solutions u: $$\Omega$$ $$\to {\mathbb{R}}^ M$$ for systems of the type $(1)\quad A(u)+G(u)=f\quad on\quad \Omega \subset {\mathbb{R}}^ n$ (with zero data on $$\partial \Omega)$$, where A(u) is of the form of the p-Laplacian $$D_ i$$ $$(| Du|^{p-2} D_ iu)$$, $$p>1$$, $$G(u)=(g^ 1(.,u,Du),...,g^ M(.,u,Du))$$ is of critical growth in Du, i.e. G(u) grows of order $$| Du|^ p$$, and satisfies the angle condition which roughly says that u and G(u) point in the same direction. Then it is possible to show that a sequence $$\{u_ k\}$$ of solutions to truncated problems $(2)\quad A(u_ k)+G_ k(u_ k)=f\quad on\quad \Omega,\quad u_ k|_{\partial \Omega}=0$ (obtained by the theory of pseudomonotone operators), is weakly compact with pointwise convergence of the gradients and the additional property that the sequence $$| Du_ k|^ p$$ is equiintegrable. This is sufficient to deduce convergence $$u_ k\to u$$ with u solving (1).
Reviewer: M.Fuchs

##### MSC:
 35J60 Nonlinear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35A35 Theoretical approximation in context of PDEs 47H05 Monotone operators and generalizations
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