Sign changing bubble tower solutions to a slightly subcritical elliptic problem with non-power nonlinearity

Abstract

We study the following elliptic problem involving slightly subcritical non-power nonlinearity $$\{\begin{array}{ll}-\Delta u=\frac{|u{|}^{2^{\ast }-2}u}{[\ln (e+|u|){]}^{\epsilon }}& \mathrm{in}\Omega ,\\ u=0& \mathrm{on}\partial \Omega ,\end{array}$$ where $\Omega$ is a bounded smooth domain in ${\mathbb{R}}^n$, $n\ge 3$, $2^{\ast }=\frac{2n}{n-2}$ is the critical Sobolev exponent, $\epsilon >0$ is a small parameter. By the finite dimensional Lyapunov-Schmidt reduction method, we construct a sign changing bubble tower solution with the shape of a tower of bubbles as $\epsilon$ goes to zero.