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Remarks on perturbated systems with critical growth. (English) Zbl 0786.35061
The author considers the question on convergence almost everywhere for gradients \(Du_ j\) of a bounded sequence \(u_ j\) in the Sobolev space \(W_ 0^{1,p} (\Omega;\mathbb{R}^ N)\) which appears in the study of the Dirichlet problem for the elliptic system with critical growth perturbation: \[ A(u)+G(u)=f,\quad u |_{\partial \Omega} =0 \] where \(\Omega \subset \mathbb{R}^ n\) is a bounded domain, \(A(u)\) is a quasilinear operator and the perturbation \(G(u)\) is dependent on the (vector-valued) function \(u\) and its gradient \(Du\). The growth of the perturbation is called critical if it can be controlled by \(| Du |^ p\) only, where \(p\) is the same exponent for which \(A(u)\) defines a coercive mapping from \(W_ 0^{1,p}(\Omega;\mathbb{R}^ N)\) to its dual space \((W_ 0^{1,p}(\Omega;\mathbb{R}^ N))^*\).

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
Full Text: DOI
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