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

• Main Authors: Janak Joshi, Joseph A. Iaia
• Format: Article
• Language: English
• Published: Texas State University 2018-05-01
• Series: Electronic Journal of Differential Equations
• Subjects: Exterior domain; superlinear; radial solution
• Online Access: http://ejde.math.txstate.edu/Volumes/2018/108/abstr.html

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 $0< \alpha < 2(N-1)$.