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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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References:
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