Classification of radial solutions to equations related to Caffarelli-Kohn-Nirenberg inequalities (Q2297633)

Let \(N\geq 2\) be an integer, and let \(p>1\), \(a\geq -N+2\) and \(b>-N\). The author investigates the stability and the asymptotic behaviour of radially symmetric positive solutions of the equation \[ -\text{div}(|x|^a Du)=|x|^bu^p,\text{ in }\mathbb{R}^N, \tag{1} \] which is related to certain Caffarelli-Kohn-Nirenberg inequalities. More precisely, after recalling that equation \((1)\) does not admit any weak solution when \(b\leq a-2\), the author analyzes the case \(b>a-2\) and \(p\geq p_S(a,b):=\frac{N+2+2b-a}{N-2+a}\), and proves that, for each \(\alpha>0\), there exists a unique positive radially symmetric solution \(u_\alpha \in C^2(\mathbb{R}^N\setminus\{0\})\cap C(\mathbb{R}^N)\) to equation \((1)\) satisfying \(u_\alpha (0)=\alpha\). Then, he shows that any positive radial solution \(u\in C^2(\mathbb{R}^N\setminus\{0\})\cap C(\mathbb{R}^N)\) of equation \((1)\) has the following asymptotic behaviour \[ \lim_{|x|\rightarrow +\infty} |x|^{\beta}u(x)=[\beta(N-2+a-\beta)]^{\frac{1}{p-1}},\text{ if }p>p_S(a,b), \] \[ \lim_{|x|\rightarrow +\infty} |x|^{N-2+a}u(x)=\frac{1}{u(0)}[(N+b)(N-2+a-\beta)]^{\frac{N-2+a}{2+b-a}},\text{ if }p=p_S(a,b), \] where \(\beta=\frac{2+b-a}{p-1}\). In addiction, the author shows that a positive radial solution \(u\) satisfies the stability condition \[ \int_{\mathbb{R}^N}|x|^\alpha |D\varphi|^2dx\geq p\int_{\mathbb{R}^N}|x|^b|u|^{p-1}\varphi^2dx,\text{ for all }\varphi \in C_c^1(\mathbb{R}^N), \] provided that \(p\geq p_{JL}(a,b)\), where \(p_{JL}(a,b)\) is a Joseph-Lundgren-type critical exponent (explicitly computed) which is greater than \(p_S(a,b)\). As a further result, for \(\alpha_1,\alpha_2>0\) with \(\alpha_1\neq \alpha_2\), the author computes, on varying of the exponent \(p\geq p_S(a,b)\), the intersection number (that is the cardinality of the set of positive numbers \(r\) at which two radial solutions take the same value) of the couples \(u_{\alpha_1},u_{\alpha_2}\) and \(u_{\alpha_1},U_s\), where \(U_s(x):=[\beta(N-2+a-\beta)]^{\frac{1}{p-1}}|x|^{-\beta}\) is a unbounded classical solution in \(\mathbb{R}^N\setminus\{0\}\) of equation \((1)\).