# Large number of bubble solutions for a perturbed fractional Laplacian equation

## TL;DR

This paper proves the existence of many bubble solutions for a perturbed fractional Laplacian equation with a small parameter, using local Pohozaev identities and finite reduction methods, especially near stable critical points of the coefficient function.

## Contribution

It introduces a novel approach using local Pohozaev identities to locate concentration points, expanding the understanding of solution multiplicity for fractional Laplacian equations.

## Key findings

- Existence of large numbers of bubble solutions for small perturbations.
- Solutions concentrate near stable critical points of the coefficient function.
- The energy of some solutions scales as a negative power of epsilon.

## Abstract

This paper deals with the following nonlinear perturbed fractional Laplacian equation $$(-\Delta)^s u = K(|y'|,y'')u^{\frac{N+2s}{N-2s}\pm\epsilon},\,\,u>0,\,\,u\in D^{1,s}(\mathbb{R}^N),$$ where $0<s<1, N\geq 4,$ $(y',y'')\in \mathbb{R}^2\times \mathbb{R}^{N-2},$ $\epsilon>0$ is a small parameter and $K(y)$ is nonnegative and bounded. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\geq 4,\max\{\frac{N+1-\sqrt{N^{2}-2N+9}}{4},\frac{3-\sqrt{N^{2}-6N+13}}{2}\}<s<1$ and $K(r,y'')$ has a stable critical point $(r_0, y_0'')$ with $r_0>0$ and $K(r_0, y_0'')>0,$ then the above problem has large number of bubble solutions if $\epsilon>0$ is small enough. Also there exist solutions whose functional energy is in the order $\epsilon^{-\frac{N-2s-2}{(N-2s)^{2}}}$. Here, instead of estimating directly the derivatives of the reduced functional, we apply some local Pohozaev identities to locate the concentration points of the bubble solutions. Moreover, the concentration points of the bubble solutions include a saddle point of $K(y)$.