Existence of nontrivial solutions to a fourth-order Kirchhoff type elliptic equation with critical exponent

In this paper, a critical fourth-order Kirchhoff type elliptic equation with a subcritical perturbation is studied. The main feature of this problem is that it involves both a nonlocal coefficient and a critical term, which bring essential difficulty for the proof of the existence of weak solutions. When the dimension of the space is smaller than or equals to $7$, the existence of weak solution is obtained by combining the Mountain Pass Lemma with some delicate estimate on the Talenti's functions. When the dimension of the space is larger than or equals to $8$, the above argument no longer works. By introducing an appropriate truncation on the nonlocal coefficient, it is shown that the problem admits a nontrivial solution under appropriate conditions on the parameter

Index Theory, Gerbes, and Hamiltonian Quantization

We give an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and prove that this leads in a simple manner to the known Schwinger terms (Faddeev-Mickelsson cocycle) for the gauge group action. We relate the APS construction to the bundle gerbe approach discussed recently by Carey and Murray, including an explicit computation of the Dixmier-Douady class. An advantage of our method is that it can be applied whenever one has a form of the APS theorem at hand, as in the case of fermions in an external gravitational field.Comment: 16 pages, Plain TeX inputting AMSTe

Existence of a positive bound state solution for the nonlinear Schrödinger-Bopp-Podolsky system

In this paper, we study a class of Schrödinger–Bopp–Podolsky system. Under some suitable assumptions for the potentials, by developing some calculations of sharp energy estimates and using a topological argument involving the barycenter function, we establish the existence of positive bound state solution

Normalized solutions for Sobolev critical Schr\"odinger-Bopp-Podolsky systems

We study the Sobolev critical Schr\"odinger-Bopp-Podolsky system \begin{gather*} -\Delta u+\phi u=\lambda u+\mu|u|^{p-2}u+|u|^4u\quad \text{in }\mathbb{R}^3, -\Delta\phi+\Delta^2\phi=4\pi u^2\quad \text{in } \mathbb{R}^3, \end{gather*} under the mass constraint $$\int_{\mathbb{R}^3}u^2\,dx=c$$ for some prescribed $c>0$, where $2<p<8/3$, $\mu>0$ is a parameter, and $\lambda\in\mathbb{R}$ is a Lagrange multiplier. By developing a constraint minimizing approach, we show that the above system admits a local minimizer. Furthermore, we establish the existence of normalized ground state solutions.Comment: 19 page

5d Partition Functions with A Twist

We derive the partition function of 5d ${\cal N}=1$ gauge theories on the manifold $S^3_b \times \Sigma_{\frak g}$ with a partial topological twist along the Riemann surface, $\Sigma_{\frak g}$. This setup is a higher dimensional uplift of the two-dimensional A-twist, and the result can be expressed as a sum over solutions of Bethe-Ansatz-type equations, with the computation receiving nontrivial non-perturbative contributions. We study this partition function in the large $N$ limit, where it is related to holographic RG flows between asymptotically locally AdS$_6$ and AdS$_4$ spacetimes, reproducing known holographic relations between the corresponding free energies on $S^{5}$ and $S^{3}$ and predicting new ones. We also consider cases where the 5d theory admits a UV completion as a 6d SCFT, such as the maximally supersymmetric ${\cal N}=2$ Yang-Mills theory, in which case the partition function computes the 4d index of general class ${\cal S}$ theories, which we verify in certain simplifying limits. Finally, we comment on the generalization to ${\cal M}_3 \times \Sigma_{\frak g}$ with more general three-manifolds ${\cal M}_3$ and focus in particular on ${\cal M}_3=\Sigma_{\frak g'}\times S^{1}$, in which case the partition function relates to the entropy of black holes in AdS$_6$.Comment: Corrected typos, updated references, and added clarifying comments in Section

Normalized solutions for the Schr\"{o}dinger equation with combined Hartree type and power nonlinearities

We investigate normalized solutions for the Schr\"{o}dinger equation with combined Hartree type and power nonlinearities, namely \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+\lambda u=\gamma (I_{\alpha }\ast \left\vert u\right\vert ^{p})|u|^{p-2}u+\mu |u|^{q-2}u & \quad \text{in}\quad \mathbb{R}^{N}, \\ \int_{\mathbb{R}^{N}}|u|^{2}dx=c, & \end{array}% \right. \end{equation*} where $N\geq 2$ and $c>0$ is a given real number. Under different assumptions on $\gamma ,\mu ,p$ and $q$, we prove several nonexistence, existence and multiplicity results. In particular, we are more interested in the cases when the competing effect of Hartree type and power nonlinearities happens, i.e. $\gamma \mu <0,$ including the cases $\gamma 0$ and Due to the different "strength" of two types of nonlinearities, we find some differences in results and in the geometry of the corresponding functionals between these two cases

Existence of positive bound state solution for the nonlinear Schrödinger–Bopp–Podolsky system

In this paper, we study a class of Schrödinger–Bopp–Podolsky system. Under some suitable assumptions for the potentials, by developing some calculations of sharp energy estimates and using a topological argument involving the barycenter function, we establish the existence of positive bound state solution