Entire large solutions for semilinear elliptic equations

Abstract

We analyze the semilinear elliptic equation $\Delta u=\rho (x)f(u)$, $u>0$ in ${\mathbf{R}}^D$ $(D\ge 3)$, with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions $u$ such that ${\lim }_{|x|\to +\infty }u(x)=+\infty$. Assuming that $f$ satisfies the Keller-Osserman growth assumption and that $\rho$ decays at infinity in a suitable sense, we prove the existence of entire large solutions. We then discuss the more delicate questions of asymptotic behavior at infinity, uniqueness and symmetry of solutions.

Journal of Differential Equations 2012, 28 pages

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