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A monotonicity formula and Liouville-type theorems for stable solutions of the weighted elliptic system. (English) Zbl 1364.35094

In this paper the author proves Liouville-type theorems for stable solutions (positive or sign-changing) of weighted elliptic systems of the form \[ -\Delta u = |x|^{\beta}v^{\vartheta} \;\;\text{ in } \Omega, \]
\[ -\Delta v=|x|^{\alpha}|u|^{p-1}u \;\;\text{ in } \Omega, \] where \(\Omega\) is a subset of \(\mathbb{R}^N, N\geq 5\), \(\alpha>-4\), \(0\leq\beta <N-4\) and \(p\vartheta>1\).
The key ingredients of this paper are an application of the Pohozaev identity to construct a monotonicity formula, and a blow-down procedure, which allow the author to examine the nonexistence of classical stable solutions to the system above.
The first main result of the paper states that if \(N\) is in a certain interval depending on \(\alpha, \beta\) and \(p\), if \(u\in W^{2,2}_{\text{loc}}(\mathbb{R}^N\setminus \{0\})\) is a homogeneous, stable solution of the system above with \(|x|^{\alpha}|u|^{p+1}\in L^1_{\text{loc}}(\mathbb{R}^N\setminus\{0\})\) and \(|x|^{-\beta}|\Delta u|^2\in L^1_{\text{loc}}(\mathbb{R}^N\setminus\{0\})\), then \(u\equiv 0\).
The second main result states that if \(u\in C^4(\mathbb{R}^N)\) is a stable solution of the system in question in \(\mathbb{R}^N\), with \(N\) in a certain interval depending on \(\alpha,\beta\), and \(p\), then \(u\equiv 0\).

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
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