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New a priori estimates for \(q\)-nonlinear elliptic systems with strong nonlinearities in the gradient, \(1< q< 2\). (Russian, English) Zbl 1102.35037
J. Math. Sci., New York 132, No. 3, 255-273 (2006); translation from Zap. Nauchn. Semin. POMI 310, 19-48, 226 (2004).
In this paper, the author considers the Dirichlet problem for the following nonlinear elliptic system: \[ \begin{aligned} - \text{div} \, a(x,u,u_x) + b(x,u,u_x) =0, & x \in \Omega, \\ u| _{\partial \Omega} = 0, &\end{aligned} \] where \(\Omega \subset \mathbb{R}^n\) is a bounded domain with \(C^1\)-boundary \(\partial \Omega\) and \(n \geq 2\). Moreover,
\(u : \overline{\Omega} \rightarrow \mathbb{R}^N , \, N\geq 1, \, u= (u^1, \dots, u^N), \, u_x = \{ u^k_{x_{\alpha}} \}^{k \leq N}_{\alpha \leq n}, \, a = \{ a^k_{\alpha} \}^{k \leq N}_{\alpha \leq n}\) and \(b = \{ b^k \}^{k \leq N}.\)
Under a smallness condition on the gradient of a solution in the Morrey space \(L^{q, n-q}, \, (1 < q < 2)\) and by using a theorem on quasireverse Hölder inequalities proven by herself in [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Rend. Lancei (9) Mat. Appl. 14, No. 2, 91–108 (2003; Zbl 1225.35082)], the author produces estimates of the \(L^p\)-norm of the gradient (\(p> q\)) and of the Hölder norm of the solution in the case of \(n=2\). In the last theorem, the additional assumption on the regularity of the boundary: \(\partial \Omega \in C^2\), is made.
MSC:
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B45 A priori estimates in context of PDEs
35J60 Nonlinear elliptic equations
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