We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, for which we establish the classical gluing method of Mazzeo and Pacard ([63]) for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional order ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.
We consider the following nonlinear Schrodinger equa-where V is a potential satisfying some decay condition and f (u) is a superlinear nonlinearity satisfying some nondegeneracy condition. Using localized energy method, we prove that there exists some δ 0 such that for 0 < δ < δ 0 , the above problem has infinitely many positive solutions. This generalizes and gives a new proof of the results by Cerami-Passaseo-Solimini [11]. The new techniques allow us to establish the existence of infinitely many positive bound states for elliptic systems.
For the generalized surface quasi-geostrophic equation { a m p ; ∂ t θ + u ⋅ ∇ θ = 0 , in R 2 × ( 0 , T ) , a m p ; u = ∇ ⊥ ψ , ψ = ( − Δ ) − s θ in R 2 × ( 0 , T ) , \begin{equation*} \left \{ \begin {aligned} & \partial _t \theta +u\cdot \nabla \theta =0, \quad \text {in } \mathbb {R}^2 \times (0,T), \\ & u=\nabla ^\perp \psi , \quad \psi = (-\Delta )^{-s}\theta \quad \text {in } \mathbb {R}^2 \times (0,T) , \end{aligned} \right . \end{equation*} 0 > s > 1 0>s>1 , we consider for k ≥ 1 k\ge 1 the problem of finding a family of k k -vortex solutions θ ε ( x , t ) \theta _\varepsilon (x,t) such that as ε → 0 \varepsilon \to 0 θ ε ( x , t ) ⇀ ∑ j = 1 k m j δ ( x − ξ j ( t ) ) \begin{equation*} \theta _\varepsilon (x,t) \rightharpoonup \sum _{j=1}^k m_j\delta (x-\xi _j(t)) \end{equation*} for suitable trajectories for the vortices x = ξ j ( t ) x=\xi _j(t) . We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem ( − Δ ) s W = ( W − 1 ) + γ , in R 2 , 1 > γ > 1 + s 1 − s \begin{equation*} (-\Delta )^sW = (W-1)^\gamma _+ , \quad \text {in } \mathbb {R}^2, \quad 1>\gamma > \frac {1+s}{1-s} \end{equation*} whose existence and uniqueness have recently been proven in Chan, del Mar González, Huang, Mainini, and Volzone [Calc. Var. Partial Differential Equations 59 (2020), p. 42].
For any rank 2 of simple Lie algebra, the relativistic Chern-Simons system has the following form: * MSC: 35J60 (Primary); 35B10, 58J37 (Secondary)
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