Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms
The author considers the rotationally symmetric cone \(\Omega_a = \{\, (x', x_n) \in \mathbb R^{n - 1} \times \mathbb R : x_n > a \, |x'| \,\}\), \(a \geq 0\), \(n \geq 3\), and proves that every solution \(u(x) = u(x',x_n)\) to \[ \begin{cases} -\Delta u(x) = \frac{u^{2^*(t) - 1}(x) \,}{|x|^t} &\text{in \(\Omega_a\),}\\ u \geq 0 &\text{in \(\Omega_a\),}\\ u = 0 &\text{on \(\partial\Omega_a\),} \end{cases} \eqno{(1)} \] belonging to an appropriate function space, is also rotationally symmetric in the sense that \(u(x',x_n) = u(\tilde x', x_n)\) if \(|x'| = |\tilde x'|\). Moreover \(u(x',x_n) \leq u(\tilde x', x_n)\) if \(|x'| \geq |\tilde x'|\). Here \(2^*(t) = 2 \, (n - t)/(n - 2)\). Problem~\((1)\) is known to possess at least one solution for \(t \in (0,2)\). Rotational symmetry was proved by \textit{N. Ghoussoub} and \textit{F. Robert} [Geom.\ Funct.\ Anal.\ 16, No. 6, 1201--1245 (2006; Zbl 1232.35044)] in the special case \(a = 0\) (half-space) and under an assumption on the behavior of~\(u(x)\) as \(|x| \to +\infty\). The interest of problem~\((1)\) is motivated in connection to the Hardy-Sobolev inequality \[ C \left( \int_\Omega \frac{\, |u|^{2^*(t)} \,}{|x|^t} \, dx \right) ^{\frac2{2^*(t)}} \leq \int_\Omega |\nabla u|^2 \, dx , \] which reduces to the Sobolev inequality for \(t = 0\) and to the Hardy inequality for \(t = 2\). The same symmetry result holds if the equation in~\((1)\) is replaced by \[ -\Delta u(x) = b \, \frac{u^{2^*(t) - 1}(x)}{|x|^t} + \frac{u^{2^*(t) - 1}(x)}{|x|^s}, \] where \(0 \leq s < t \leq 2\) and \(b > 0\). The last equation arises when searching for minimizers to the Caffarelli-Kohn-Nirenberg inequality. Lane-Emden systems with Hardy weights are also considered. Namely, the author proves that if \(u,v \geq 0\) are (entire) solutions to \[ \begin{cases} -\Delta u(x) = |x|^{-s} \, v^p(x), &x \in \mathbb R^n,\\ -\Delta v(x) = |x|^{-t} \, u^q(x), &x \in \mathbb R^n, \end{cases} \eqno{(2)} \] where \(0 \leq s,t < 2\), \((s,t) \neq (0,0)\), \(0 < p \leq 2^*(s) - 1\) and \(0 < q \leq 2^*(t) - 1\), then \(u(x)\) and~\(v(x)\) are radial, and nonincreasing with respect to~\(|x|\). System~\((2)\) is then investigated in the upper half-space~\(\mathbb R^n_+\). All problems are very well introduced, and detailed citations are provided. Proofs are based on the moving sphere method, which proceeds as the celebrated method of the moving plane using spheres instead of planes and exploiting properties of the Kelvin transformation. Using the Pohozaev identity, a nonexistence result for system~\((2)\) is also obtained.
scientific article; zbMATH DE number 5992249
scientific article