Volume 23, Issue 4
Nonradial Entire Large Solutions of Semilinear Elliptic Equations

Alan V. Lair


J. Part. Diff. Eq., 23 (2010), pp. 366-373.

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  • Abstract

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 $ε > 0$.

  • History

Published online: 2010-11

  • AMS Subject Headings

35J61, 35J25

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