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Liouville-type theorems and bounds of solutions for Hardy-Hénon elliptic systems. (English) Zbl 1262.35108

Summary: We consider the Hardy-Hénon system \(-\Delta u =|x|^a v^p, -\Delta v =|x|^b u^q\) with \(p,q>0\) and \(a,b\in {\mathbb R}\) and we are concerned in particular with the Liouville property, i.e., the nonexistence of positive solutions in the whole space \({\mathbb R}^N\). In view of known results, it is a natural conjecture that this property should be true if and only if \((N+a)/(p+1)+(N+b)/(q+1)>(N-2)\). In this paper, we prove the conjecture for dimension \(N=3\) in the case of bounded solutions and in dimensions \(N\leq 4\) when \(a,b\leq 0\), among other partial nonexistence results. As far as we know, this is the first optimal Liouville-type result for the Hardy-Hénon system. Next, as applications, we give results on singularity and decay estimates as well as a priori bounds of positive solutions.

MSC:

35J60 Nonlinear elliptic equations
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35B33 Critical exponents in context of PDEs
35B45 A priori estimates in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J47 Second-order elliptic systems
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Full Text: arXiv Euclid