Electron. J. Differential Equations, Vol. 2018 (2018), No. 181, pp. 1-14.

Existence of solutions for sublinear equations on exterior domains

Joseph A. Iaia

In this article we consider the 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 $\partial B_{R}$ and $\lim_{r \to \infty} u(r)=0$ where N>2, 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 $K(r) \sim r^{-\alpha}$ with $\frac{(N+2)-p(N-2)}{2} \leq \alpha < N-p(N-2)$ for large r. We prove existence of n solutions - one with exactly n zeros on $[R, \infty)$ - if R>0 is sufficiently small. If Rgt;0 is sufficiently large then there are no solutions with $\lim_{r \to \infty} u(r)=0$.

Submitted May 10, 2018. Published November 6, 2018.
Math Subject Classifications: 34B40, 35B05.
Key Words: Exterior domains; semilinear; sublinear; radial solution.

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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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