Sign changing solutions of some integral equaitons with critical sobolev exponents

Abstract

In this note we prove that: \begin{theorem} for $2\leq s<\frac{n}{2}$ or $1\leq s<\frac{2n}{n+1}$ or $1\leq s<\frac{n}{2}$ but n is even, $(-Δ)^{s}(u)=|u|^{q-2}u,q=\frac{2n}{n-2s}$ has infinitely many sign changing solutions or equivalently we can say that there exist solutions $u_{k}$ such that $\int u_{k}(-Δ)^{s}(u_{k})dx \to \infty$ as $k\to \infty$ \end{theorem}