Sign changing solutions of some integral equaitons with critical sobolev exponents

TL;DR

This paper proves the existence of infinitely many sign-changing solutions for certain fractional Laplacian equations with critical Sobolev exponents, expanding understanding of solution behaviors in nonlinear integral equations.

Contribution

It establishes the existence of infinitely many sign-changing solutions for fractional Laplacian equations with critical exponents, a novel result in this area.

Findings

Existence of infinitely many sign-changing solutions.

Solutions' energy tends to infinity as index increases.

Results apply under specified ranges of s and n.

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, $(-\Delta)^{s}(u)=|u|^{q-2}u,q=\frac{2n}{n-2s}$ has infinitely many sign changing solutions or equivalently we can say that there exist solutions $u_{k}$ such that $\int u_{k}(-\Delta)^{s}(u_{k})dx \to \infty$ as $k\to \infty$ \end{theorem}