A note on nonlinear biharmonic equations with negative exponents
This paper is concerned with the asymptotic behavior of positive radially symmetric solutions to the equation \(\Delta^2 u=-u^{-q}\) in \(R^2\). The following are the main results of the paper. Theorem 1. (a) For \(q>3\) and given any \(\alpha>0\) there exists a unique radially symmetric solution such that \[ \lim_{|x|\rightarrow \infty}\frac{u(x)}{|x|}=\alpha. \] (b) For \(q=3\) there exists a unique radially symmetric solution such that \[ \lim_{|x|\rightarrow \infty}\frac{u(x)}{|x|\log|x|^{1/4}}=2^{1/4}. \] (c) For \(1<q<3\) there exists a unique radially symmetric solution such that \[ \lim_{|x|\rightarrow \infty}\frac{u(x)}{|x|^{4/(q+1)}}=K^{-1/(q+1)}, \quad\text{ where }K=\tau(2-\tau)(\tau+1)(\tau-1),\quad \tau=4/(q+1). \] Theorem 2. Let \[ \alpha=\frac{1}{8\pi}\int_{R^3}u^{-q}(x)dx. \] (a) If \(q=4\) then \[ \lim_{|x|\rightarrow\infty}\frac{u(x)-\alpha|x|}{\log|x|}=-\frac{1}{2\alpha^4}. \] (b) If \(3<q<4\) then \[ \lim_{|x|\rightarrow\infty}|x|^{q-4}u(x) =\frac{1}{2\alpha^q}\Big(\frac{1}{3-q}-\frac{1}{4-q}+\frac{1}{3(5-q)}-\frac{1}{3(2-q)} \Big). \]
scientific article; zbMATH DE number 6099370