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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