Electron. J. Diff. Equ., Vol. 2016 (2016), No. 227, pp. 1-9.

Existence and nonexistence of solutions for semilinear equations on exterior domains

Joseph A. Iaia

Abstract:
In this article we study radial solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius R>0 centered at the origin in ${\mathbb R}^{N}$ where f is odd with f<0 on $(0, \beta) $, f>0 on $(\beta, \delta)$, $f\equiv 0$ for $u> \delta$, and where the function K(r) is assumed to be positive and $K(r)\to 0$ as $r \to \infty$. The primitive $F(u) = \int_0^u f(t) \, dt$ has a "hilltop" at $u=\delta$. We prove that if $K(r) \sim r^{-\alpha}$ with $\alpha> 2(N-1)$ and if R>0 is sufficiently small then there are a finite number of solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius R such that $u \to 0$ as $r \to \infty$. We also prove the nonexistence of solutions if R is sufficiently large.

Submitted July 20, 2016. Published August 22, 2016.
Math Subject Classifications: 34B40, 35B05.
Key Words: Exterior domains; semilinear; superlinear; radial.

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Joseph A. Iaia
Department of Mathematics
University of North Texas
P.O. Box 311430
Denton, TX 76203-1430, USA
email: iaia@unt.edu

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