Existence of positive nonradial solutions for nonlinear elliptic equations in annular domains

Abstract:

We study the existence of positive nonradial solutions of equation $\Delta u + f(u) = 0$ in ${\Omega _a}$, $u = 0$ on $\partial {\Omega _a}$, where ${\Omega _a} = \{ x \in {\mathbb {R}^n}:a < |x| < 1\}$ is an annulus in ${\mathbb {R}^n}$, $n \geq 2$, and $f$ is positive and superlinear at both $0$ and $\infty$. We use a bifurcation method to show that there is a nonradial bifurcation with mode $k$ at ${a_k} \in (0,1)$ for any positive integer $k$ if $f$ is subcritical and for large $k$ if $f$ is supercritical. When $f$ is subcritical, then a Nehari-type variational method can be used to prove that there exists ${a^{\ast } } \in (0,1)$ such that for any $a \in ({a^{\ast } },1)$, the equation has a nonradial solution on ${\Omega _a}$.