On the solutions to some elliptic equations with nonlinear Neumann boundary conditions (Q1906690)

Let \(H\) be the half space of \(\mathbb{R}^n\) \((n\geq 3)\), defined by \(x_1> 0\). The authors describe all nonnegative nontrivial solutions \(u\in C^2(H)\cap C^1(\overline H)\) of the semilinear elliptic problem \[ - \Delta u= a u^{(n+ 2)/(n- 2)}\quad\text{in } H,\quad \partial u/\partial\nu= bu^{n/(n- 2)}\quad\text{on }\partial H,\tag{1} \] where \(a\) and \(b\) are real constants, \(\nu\) is the outward normal vector on \(\partial H\). Their main result states the following: i) if \(a> 0\) or \(a\leq 0\) and \(b> B= \sqrt{- a(n- 2)/n}\), then \(u\) has the form \(u(x)= {\alpha\over (|x- x^0|^2+ \beta)^{(n- 2)/2}}\), with \(x^0\in \mathbb{R}^n\) fixed, \(\alpha, \beta\in \mathbb{R}\), \(\alpha> 0\), ii) if \(\alpha= 0\), \(b< 0\), then \(u(x)= \alpha x_1+ (-\alpha/b)^{(n- 2)/n}\), \(\alpha> 0\), iii) if \(\alpha< 0\), \(b= B\), then \(u(x)= ((2/(n- 2)) Bx_1+ \alpha)^{-(n- 2)/2}\), \(\alpha> 0\), iv) if \(\alpha= b= 0\), then \(u(x)=\text{constant}\), v) in the remaining case \(a< 0\), \(b< B\), there is no nontrivial nonnegative solution of (1). To prove this result, the authors use a variant of the moving plane method which they call the shrinking sphere method, and the maximum principle.