Abstract
In this paper, we deal with a class of fractional critical problems. Under some suitable assumptions, we derive the existence of a positive solution concentrating at the critical point of the Robin function by using the Lyapunov–Schmidt reduction method. Comparing with previous work, we encounter some new challenges because of a nonlocal term. By making some delicate estimates for the nonlocal term we overcome the difficulty and find a bubbling solution.
Highlights
This paper is concerned with the solution for the following elliptic equation involving fractional spectral Laplacian and critical exponent:⎧ ⎨(– )su = |u|2∗s –2u + λu, ⎩u = 0, u > 0, x ∈ Ω, x ∈ ∂Ω, (1.1) where < s 2∗s =Ω is a smooth bounded domain of RN (–)s denotes the fractional Laplace operator, and λ1(Ω) is the first eigenvalue of (– )s in Ω under zero Dirichlet boundary data.The fractional power of the Laplacian (– )s appears in diverse areas including physic, biological modeling and mathematical finances; see [6, 7, 12]
[4], Brezis and Nirenberg considered the existence of positive solutions for problem (1.1) with s = 1
Using variational methods and Lyapunov–Schmidt reduction, we prove that Eq (1.1) admits a positive solution concentrating at the critical point of the Robin function
Summary
1 Introduction This paper is concerned with the solution for the following elliptic equation involving fractional spectral Laplacian and critical exponent: [4], Brezis and Nirenberg considered the existence of positive solutions for problem (1.1) with s = 1. Using variational methods and Lyapunov–Schmidt reduction, we prove that Eq (1.1) admits a positive solution concentrating at the critical point of the Robin function. We point out that we adopt in the paper the spectral definition of the fractional Laplacian in a bounded case with a Caffarelli–Silverstre type extension [9], and not the integral definition.
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