Electron. J. Differential Equations, Vol. 2018 (2018), No. 108, pp. 1-10.

Infinitely many solutions for a semilinear problem on exterior domains with nonlinear boundary condition

Janak Joshi, Joseph A. Iaia

Abstract:
In this article we prove the existence of an infinite number of radial solutions to $\Delta u+K(r)f(u)=0$ with a nonlinear boundary condition on the exterior of the ball of radius R centered at the origin in $\mathbb{R}^{N}$ such that $\lim_{r \to \infty} u(r)=0$ with any given number of zeros where $f:{\mathbb R} \to {\mathbb R}$ is odd and there exists a $\beta>0$ with f<0 on $(0,\beta)$, f>0 on $(\beta,\infty)$ with f superlinear for large u, and $K(r) \sim r^{-\alpha}$ with $0< \alpha < 2(N-1)$.

Submitted July 8, 2017. Published May 8, 2018.
Math Subject Classifications: 34B40, 35B05.
Key Words: Exterior domain; superlinear; radial solution.

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Janak Joshi
Department of Mathematics
University of North Texas, P.O. Box 311430
Denton, TX 76203-1430, USA
email: JanakrajJoshi@my.unt.edu
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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