Construction of sign-changing solutions for a harmonic equation with subcritical exponent

Abstract

In this paper, we study the harmonic equation involving subcritical exponent $(P_{\varepsilon })$: $\Delta u = 0$, in $\mathbb {B}^n$ and $\displaystyle \frac{\partial u}{\partial \nu } + \displaystyle \frac{n-2}{2}u = \displaystyle \frac{n-2}{2} K u^{\frac{n}{n-2}-\varepsilon }$ on $\mathbb {S}^{n-1}$ where $\mathbb {B}^n$ is the unit ball in $\mathbb {R}^n$, $n\ge 5$ with Euclidean metric $g_0$, $\partial \mathbb {B}^n = \mathbb {S}^{n-1}$ is its boundary, K is a function on $\mathbb {S}^{n-1}$ and $\varepsilon$ is a small positive parameter. We construct solutions of the subcritical equation $(P_{\varepsilon })$ which blow up at two different critical points of K. Furthermore, we construct solutions of $(P_{\varepsilon })$ which have two bubbles and blow up at the same critical point of K.