Existence and multiplicity results for equations with nearly critical growth

Abstract

We consider the problem \begin{equation}\nonumber \left\{ \begin{array}{ll} -\Delta u=K(x)u^{p_{\epsilon}} & \hbox{ in }\mathbb R^n\\ u>0 & \hbox{ in }\mathbb R^n \end{array}\right. \end{equation} where $p=\frac{n+2}{n-2}$, $p_{\epsilon}=p-\epsilon$, $n\geq 3$, $\epsilon>0$ and $K(x)>0$ in $\mathbb R^n$. We prove an existence and multiplicity result for single peaked solutions of our problem concentrating at a fixed critical point of $K(x)$ and some other related results.