Study on the behaviors of rupture solutions for a class of elliptic MEMS equations in \(\mathbb{R}^2\) (Q6635966)

Let \(\lambda>0\), \(\alpha>-2\) and \(p>0\) and set \(\beta = (\alpha+2)/(p+1)>0\). The solutions \(u\) to the singular elliptic equation\N\[\N\Delta u(x) = \lambda |x|^\alpha u^{-p}(x), \qquad x\in\mathbb{R}^2\setminus\{0\},\N\]\Nsatisfying\N\[\N0 < \inf_{\mathbb{R}^2\setminus\{0\}} \big\{ |x|^{-\beta} u(x) \big\} \le \sup_{\mathbb{R}^2\setminus\{0\}} \big\{ |x|^{-\beta} u(x) \big\} < \infty\N\]\N(which implies in particular that \(u(0)=0\) and \(u>0\) in \(\mathbb{R}^2\setminus\{0\}\)) are identified according to the range of \((\alpha,p)\). The proof relies on a complete description of the connected components of the set\N\[\N\mathfrak{S} = \left\{ w\in C^2\big(\mathbb{S}^1\big)\ :\ w'' + \beta^2 w - \lambda w^{-p} = 0, \ w>0\text{ on }\mathbb{S}^1 \right\}\N\]\Nand the analysis of the behaviour of \(u(x)\) as \(|x|\to 0\) and \(|x|\to\infty\). Indeed, introducing polar coordinates \((r,\theta)\in (0,\infty)\times\mathbb{S}^1\) and setting \(u(x)=u(r,\theta)\), it is shown that there are \((w_0,w_\infty)\in\mathfrak{S}^2\) such that\N\[\N\lim_{r\to 0} \left\| r^{-\beta} u(r,\cdot) - w_0 \right\|_{C^2(\mathbb{S}^1)} = 0\text{ and }\lim_{r\to \infty} \left\| r^{-\beta} u(r,\cdot) - w_\infty \right\|_{C^2(\mathbb{S}^1)} = 0.\N\]\NIn particular, when \(\mathfrak{S}\) is reduced to the singleton \(\left\{ \big(\lambda\beta^{-2}\big)^{1/(p+1)}\right\}\), then necessarily \(u(x) = \big(\lambda\beta^{-2}\big)^{1/(p+1)} |x|^\beta\) for \(x\in\mathbb{R}^2\).