Electron. J. Differential Equations, Vol. 2017 (2017), No. 214, pp. 1-13.

Existence and nonexistence of solutions for sublinear equations on exterior domains

Joseph A. Iaia

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, $B_{R}$, centered at the origin in ${\mathbb R}^{N}$ with u=0 on $\partial B_{R}$ where f is odd with f<0 on $(0, \beta) $, f>0 on $(\beta, \infty)$, $f(u)\sim u^p$ with 0<p<1 for large u and $K(r) \sim r^{-\alpha}$ for large r. We prove that if N>2 and $K(r)\sim r^{-\alpha}$ with $2< \alpha < 2(N-1)$ then there are no solutions with $\lim_{r \to \infty} u(r)=0$ for sufficiently large R>0. On the other hand, if $2< N-p(N-2) <\alpha< 2(N-1)$ and k, n are nonnegative integers with $0 \leq k \leq n$ then there exist solutions, $u_{k}$, with k zeros on $(R, \infty)$ and $\lim_{r \to \infty} u_{k}(r)=0$ if R>0 is sufficiently small.

Submitted December 29, 2016. Published September 13, 2017.
Math Subject Classifications: December 29, 2016. Published September 13, 2017.

Key Words: Exterior domains; semilinear; sublinear; 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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