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A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponent
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-03-17 , DOI: 10.1016/j.jfa.2021.108991
Thomas Bartsch , Tian Xu

Let (M,g,σ) be a compact Riemannian spin manifold of dimension m2, let S(M) denote the spinor bundle on M, and let D be the Atiyah-Singer Dirac operator acting on spinors ψ:MS(M). We study the existence of solutions of the nonlinear Dirac equation with critical exponent(NLD)Dψ=λψ+f(|ψ|)ψ+|ψ|2m1ψ where λR and f(|ψ|)ψ is a subcritical nonlinearity in the sense that f(s)=o(s2m1) as s. A model nonlinearity is f(s)=αsp2 with 2<p<2mm1, αR. In particular we study the nonlinear Dirac equation(BND)Dψ=λψ+|ψ|2m1ψ,λR. This equation is a spinorial analogue of the Brezis-Nirenberg problem. As corollary of our main results we obtain the existence of least energy solutions (λ,ψ) of (BND) for every λ>0, even if λ is an eigenvalue of D. For some classes of nonlinearities f we also obtain solutions of (NLD) for every λR, except for non-positive eigenvalues. If m3 (mod 4) we obtain solutions of (BND) for every λR, except for a finite number of non-positive eigenvalues. In certain parameter ranges we obtain multiple solutions of (NLD) and (BND), some near the trivial branch, others away from it.

The proofs of our results are based on variational methods using the strongly indefinite energy functional associated to (NLD).



中文翻译:

涉及临界Sobolev指数的Brezis-Nirenberg定理的一个脊髓类似物

中号Gσ 成为一个尺寸紧凑的黎曼自旋流形 2个, 让 小号中号表示M上的旋转子束,并令D为作用于旋转子上的Atiyah-Singer Dirac算子ψ中号小号中号。我们研究具有临界指数(NLD)的非线性Dirac方程的解的存在性dψ=λψ+F|ψ|ψ+|ψ|2个-1个ψ 在哪里 λ[RF|ψ|ψ 在某种意义上是亚临界非线性 Fs=Øs2个-1个 作为 s。模型非线性为Fs=αsp-2个2个<p<2个-1个α[R。特别地,我们研究非线性狄拉克方程(BND)dψ=λψ+|ψ|2个-1个ψλ[R该方程是布雷齐斯-尼伦贝格问题的自旋模拟。作为我们主要结果的推论,我们获得了最小能量解的存在λψ 每(BND)的 λ>0,即使λD的特征值。对于某些类别的非线性f,我们还获得了(NLD)的解λ[R,但非正特征值除外。如果3 (mod 4)我们获得每个(BND)的解 λ[R,但有限数量的非正特征值除外。在某些参数范围内,我们获得(NLD)和(BND)的多个解,其中一些靠近琐碎的分支,而另一些则远离它。

我们的结果证明基于使用与(NLD)相关的强不确定能量函数的变分方法。

更新日期:2021-03-19
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