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A priori bounds and renormalized Morse indices of solutions of an elliptic system. (English) Zbl 0964.35037
Let \(\Omega\) be a smooth bounded domain in \({\mathbb R}^N\), \(N\geq 3\). Assume \(u\) and \(v\) are arbitrary solutions of the system \(-\Delta u=b(x)v^q\), \(-\Delta v=a(x)u^p\), where \(p,q>1\) and \(1/(p+1)+1/(q+1)>1-2/N\). After defining a “renormalized” Morse index, the authors prove the following Bahri-Lions type result: Set \[ f(u,v)=\int_\Omega\{\nabla u\cdot\nabla v-a(x)|u|^{p+1}/(p+1)- b(x)|v|^{q+1}/(q+1)\} dx. \] Then, for any integer \(m\geq 1\), there exists a constant \(C_m\) depending on \(a\), \(b\) and \(\Omega\) such that any critical point of \(f\) with lower Morse index \(\mu_-(z_0)\leq m\) satisfies \(\sup_\Omega\{ |u|,|v|\}\leq C_m\).

MSC:
35J60 Nonlinear elliptic equations
49J35 Existence of solutions for minimax problems
49K20 Optimality conditions for problems involving partial differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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