Azer Sign-changing radial solutions for a semilinear problem on exterior domains with nonlinear boundary conditions

existence of sign radial solution

Abstract

In this paper we are interested to the existence and multiplicity of sign changing radial solutions of problem of elliptic equations $\Delta U(x)+\varphi(|x|)f(U)=0$ with a nonlinear boundary conditions on exterior of the unite ball centered at the origin in $\mathbb{R}^{N}$ such that $ u(x) \rightarrow 0$ as $ |x|\to \infty $, with any given number of zeros where the nonlinearity $ f(u) $ is odd, superlinear for $ u $ lager enough and $ f<0 $ on $(0,\beta)$, $ f>0$ on $(\beta,\infty) $. The function $\varphi>0$ is $ C^{1} $ on $ [R,\infty) $ where $ 0<\varphi(|x|)\leq c_0\,|x|^{-\alpha}$ with $ \alpha>2(N-1) $ and $ N>2 $ for large $ |x| $.