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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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