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A priori bounds for positive solutions of a non-variational elliptic system. (English) Zbl 0997.35015
From the introduction: A general class of systems, namely \[ \begin{aligned} & -\Delta u_1=f(x, u_1, u_2,\nabla u_1,\nabla u_2)\\& -\Delta u_2=g(x, u_1, u_2,\nabla u_1,\nabla u_2)\end{aligned}\quad \text{in }\Omega,\tag{1} \] subject to zero Dirichlet boundary conditions on \(\partial\Omega\), where \(\Omega\) is some smooth bounded domain in \(\mathbb{R}^N\). Separating the leading part in (1), it becomes a system of the form \[ \begin{aligned} & -\Delta u_1=a(x)u^{\alpha_{11}}_1+b(x) u_2^{\alpha_{12}}+h_1(x, u_1, u_2,\nabla u_1,\nabla u_2)\\ & -\Delta u_2=c(x) u_1^{\alpha_{21}}+d(x) u^{\alpha_{22}}_2+h_2(x, u_1, u_2,\nabla u_1,\nabla u_2).\end{aligned}\tag{2} \] Since (2) is not variational, topological methods are used to prove the existence of positive solutions, i.e., a priori bounds for the solutions of (2) are obtained via the so-called blow-up method.

MSC:
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B45 A priori estimates in context of PDEs
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