A bifurcation and symmetry result for critical fractional Laplacian equations involving a perturbation

Abstract

Abstract In the present paper, by using the variational and topological methods, we obtain a multiplicity and bifurcation result for the following fractional problems involving critical nonlinearities and a lower order perturbation: −LKv=μv+|v|2s∗−2v+g(x,v)in Ω,v=0in RN∖Ω, where Ω is an open and bounded domain with Lipschitz boundary, N>2s $N>2s$, with s∈(0,1) $s\in (0,1)$, g is a lower order perturbation of the critical power |v|2s∗−2v $|v|^{2_{s}^{*}-2}v$ and μ is a positive real parameter, 2s∗=2NN−2s $2_{s}^{*}=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, while LK $\mathcal{L}_{K}$ is an integro-differential operator. Precisely, we show that the number of nontrivial solutions for this equation under suitable assumptions is at least twice the multiplicity of the eigenvalue. Our conclusions improve the related results in some respects.

Keywords

• Integro-differential operator
• Critical nonlinearities
• Fractional problems
• Variational methods