Electron. J. Differential Equations, Vol. 2020 (2020), No. 117, pp. 1-16.

Existence and nonexistence of radial solutions for semilinear equations with bounded nonlinearities on exterior domains

Joseph 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 $u=\delta$. With mild assumptions on f we prove that if $K(r)\sim r^{-\alpha}$ with $2< \alpha< 2(N-1)$ then there are n solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius R such that $u\to 0$ as $r \to \infty$ if R>0 is sufficiently small. We also show there are no solutions if R>0 is sufficiently large.

Submitted January 6, 2020. Published December 1, 2020.
Math Subject Classifications: 34B40, 35B05.
Key Words: Sublinear equation; radial solution; exterior domain.

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Joseph A. Iaia
Department of Mathematics
University of North Texas, P.O. Box 311430
Denton, TX 76203-5017, USA
email: iaia@unt.edu

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