Zhongwei Tang

We study a class of Dirac equations on $ \mathbb{R}^n $ with critical nonlinearity and lower-order perturbations. More precisely, we consider $ \not{D}_{g_{\mathbb{R}^n}}\psi = \varepsilon h(x)|\psi|^{q-1}\psi+(1+\varepsilon K(x))|\psi|^{\frac{2}{n-1}}\psi, \qquad 1\le q<\frac{n+1}{n-1}, $ where $ \varepsilon>0 $ is a small parameter and $ h, K $ are scalar functions. The equation can be viewed as a perturbed Euclidean spinorial Yamabe equation. Using a finite-dimensional reduction method, we reduce the problem to the study of a reduced functional defined on the critical manifold of the unperturbed equation. For the case $ K\equiv0 $, we establish the existence of solutions when $ h\in L^1( \mathbb{R}^n)\cap L^\infty( \mathbb{R}^n) $ and further extend the result to the weaker assumption $ h\in L^r( \mathbb{R}^n) $ for suitable $ r $. Moreover, if $ h $ admits a decomposition into finitely many components satisfying suitable assumptions, we prove the existence of multiple solutions. We also consider the case $ q = 1 $, under appropriate assumptions on $ h $ and $ K $, we obtain existence result by analyzing the behavior of the reduced functional.

This paper is devoted to establishing some results on the density and multiplicity of solutions to the fractional Nirenberg problem which is equivalent to studying the conformally invariant equations $ P_\sigma(v) = K v^{\frac{n+2\sigma}{n-2\sigma}} $ on the standard unit sphere $ (\mathbb{S}^n,g_0) $ with $ \sigma\in (0,1) $ and $ n\geq 2 $, where $ P_\sigma $ is the intertwining operator of order $ 2 \sigma $ and $ K $ is the prescribed curvature function. More specifically, by using the variational gluing method, refined analysis of bubbling behavior, extension formula, as well as the blow up analysis arguments, we obtain the existence of infinitely many multi-bump solutions. In particular, we show that the smooth curvature functions of metrics conformal to $ g_0 $ are dense in the $ C^{0} $ topology. Moreover, the related fractional Laplacian equations $ (-\Delta)^{ \sigma} u = K(x) u^{\frac{n+2\sigma}{n-2\sigma}} $ in $ \mathbb{R}^n $, with $ K(x) $ being asymptotically periodic in one of the variables, are also studied and infinitely many solutions are obtained under natural flatness assumptions.

In the present paper, we consider the coupled Schrödinger systems with critical exponent: − Δ u i + λ V i ( x ) + a i u i = ∑ j = 1 d β i j u j 3 u i u i in R 3 , u i ∈ H 1 ( R N ) , i = 1,2 , … , d , $$\begin{cases}-{\Delta}{u}_{i}+\left(\lambda {V}_{i}\left(x\right)+{a}_{i}\right){u}_{i}=\sum _{j=1}^{d}{\beta }_{ij}{\left\vert {u}_{j}\right\vert }^{3}\left\vert {u}_{i}\right\vert {u}_{i}\quad \,\text{in}\,{\mathbb{R}}^{3},\quad \hfill \\ {u}_{i}\in {H}^{1}\left({\mathbb{R}}^{N}\right),\quad i=1,2,\dots ,d,\quad \hfill \end{cases}$$ where d ≥ 2, β _ii > 0 for every i , β _ij = β _ji when i ≠ j , λ > 0 is a parameter and 0 ≤ V i ∈ L loc ∞ R N $0\le {V}_{i}\in {L}_{\text{loc\,}}^{\infty }\left({\mathbb{R}}^{N}\right)$ have a common bottom int V i − 1 ( 0 ) ${V}_{i}^{-1}\left(0\right)$ composed of ℓ 0 ℓ 0 ≥ 1 ${\ell }_{0}\left({\ell }_{0}\ge 1\right)$ connected components Ω k k = 1 ℓ 0 ${\left\{{{\Omega}}_{k}\right\}}_{k=1}^{{\ell }_{0}}$ , where int V i − 1 ( 0 ) ${V}_{i}^{-1}\left(0\right)$ is the interior of the zero set V i − 1 ( 0 ) = x ∈ R N ∣ V i ( x ) = 0 ${V}_{i}^{-1}\left(0\right)=\left\{x\in {\mathbb{R}}^{N}\mid {V}_{i}\left(x\right)=0\right\}$ of V _i . We study the existence of least energy positive solutions to this system which are trapped near the zero sets int V i − 1 ( 0 ) ${V}_{i}^{-1}\left(0\right)$ for λ > 0 large for weakly cooperative case β i j > 0 s m a l l $\left({\beta }_{ij}{ >}0 \mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\right)$ and for purely competitive case β i j ≤ 0 $\left({\beta }_{ij}\le 0\right)$ . Besides, when d = 2, we construct a one-bump fully nontrivial solution which is localised at one prescribed components Ω k k = 1 ℓ ${\left\{{{\Omega}}_{k}\right\}}_{k=1}^{\ell }$ for large λ .

We prove some results on the density and multiplicity of positive solutions to the prescribed Webster scalar curvature problem on the $(2n+1)$-dimensional standard unit CR sphere $(\mathbb{S} ^{2n+1},θ_0)$. Specifically, we construct arbitrarily many multi-bump solutions via the variational gluing method. In particular, we show the Webster scalar curvature functions of contact forms conformal to $θ_0$ are $C^{0}$-dense among bounded functions which are positive somewhere. Existence results of infinitely many positive solutions to the related equation $-Δ_{\mathbb{H}} u=R(ξ) u^{(n+2) /n}$ on the Heisenberg group $\Hn $ with $R(ξ)$ being asymptotically periodic with respect to left translation are also obtained. Our proofs make use of a refined analysis of bubbling behavior, gradient flow, Pohozaev identity, as well as blow up arguments.