TY - JOUR T1 - Nonradial Entire Large Solutions of Semilinear Elliptic Equations AU - Alan V. Lair JO - Journal of Partial Differential Equations VL - 4 SP - 366 EP - 373 PY - 2010 DA - 2010/11 SN - 23 DO - http://doi.org/10.4208/jpde.v23.n4.4 UR - https://global-sci.org/intro/article_detail/jpde/5239.html KW - Large solution KW - elliptic equation KW - semilinear equation AB -

We consider the problem of whether the equation $Δu = p(x) f (u)$ on $R^N, N ≥ 3$, has a positive solution for which $lim_{|x|→∞} u(x)=∞$ where f is locally Lipschitz continuous, positive, and nondecreasing on (0,∞) and satisfies $∫^∞_1[F(t)]^{-1/2}dt=∞$ where $F(t)=∫^t_0f(s)ds$. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies $∫^∞_0r\min_{|x|=r}p(x)dr=∞$. Conversely, we show that a necessary condition for the solution to exist is that p satisfies $∫^∞_0r^{1+ε}\min_{|x|=r}p(x)dr=∞$ for all $ε > 0$.