Related papers: Existence theorems for a generalized Chern-Simons …

This paper deals with existence of solutions to the following fractional $p$-Laplacian system of equations \begin{equation*} %\tag{$\mathcal P$}\label{MAT1} \begin{cases} (-\Delta_p)^s u =|u|^{p^*_s-2}u+…

Via a new inequality \`a la Gagliardo--Nirenberg, we prove the existence and nonexistence of solutions to \begin{equation*} \begin{cases} (-\Delta)^s u + \frac{\mu}{|y|^{2s}} u + \lambda u = f(u), \quad \mathbb{R}^N \ni x = (y,z) \in…

We investigate the Chern-Simons Higgs models for p-Laplacian on a connected finite graph, employing topological degree theory as our main tool. Notably, we overcome the difficulties arising from the nonlinearity of p-Laplacian operator and…

We compute the exact effective action for \cN=3 U(N)_k and \cN=4,6 U(N)_k\times U(N')_{-k} Chern-Simons theories with minimal matter content in the 't Hooft vector model limit under which N and k go to infinity holding N/k, N' fixed. We…

In this paper, the following critical biharmonic elliptic problem \begin{eqnarray*} \begin{cases} \Delta^2u= \lambda u+\mu u\ln u^2+|u|^{2^{**}-2}u, &x\in\Omega,\\ u=\dfrac{\partial u}{\partial \nu}=0, &x\in\partial\Omega \end{cases}…

We study the following Lane-Emden system \[ -\Delta u=|v|^{q-1}v \quad \text{ in } \Omega, \qquad -\Delta v=|u|^{p-1}u \quad \text{ in } \Omega, \qquad u_\nu=v_\nu=0 \quad \text{ on } \partial \Omega, \] with $\Omega$ a bounded regular…

We prove existence of a positive radial solution to the Choquard equation $$-\Delta u +V u=(I_\alpha\ast |u|^p)|u|^{p-2}u\qquad\text{in}\,\,\,\Omega$$ with Neumann or Dirichlet boundary conditions, when $\Omega$ is an annulus, or an…

We study four-point functions in Chern-Simons vector models in the large $N$ limit. We compute the four-point function of the scalar primary to all orders in the `t Hooft coupling $\lambda=N/k$ in $U(N)_k$ Chern-Simons theory coupled to a…

We establish the existence of finitely many sign-changing solutions to the Lane-Emden system $$-\Delta u=|v|^{q-2}v,\quad -\Delta v=|u|^{p-2}u \quad \text{ in }\mathbb{R}^N, \ \ N\geq 4,$$ where the exponents $p$ and $q$ lie on the critical…

In the present paper, we consider the Chern-Simons-Schr\"{o}dinger system \begin{equation} \left\{ \begin{aligned} &-\varepsilon^{2}\Delta u+V(x)u+(A_{0}+A_{1}^{2}+A_{2}^{2})u=|u|^{p-2}u,\,\,\,\,x\in \mathbb{R}^2,\\ &\partial_1 A_0 = A_2…

The aim of this paper is to study the critical elliptic equations with Stein-Weiss type convolution parts $$ \displaystyle-\Delta u =\frac{1}{|x|^{\alpha}}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{\alpha,…

Via a constrained minimization, we find a solution $(\lambda,u)$ to the problem \begin{equation*} \begin{cases} (-\Delta)^m u+\frac{\mu}{|x|^{2m}}u + \lambda u = \eta u^3 + g(u)\\ \int_{\mathbb{R}^{2m}} u^2 \, dx = \rho \end{cases}…

We proceed with the investigation of the problem $(P_\lambda): $ $-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } \Omega, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial \Omega$, where $\Omega$ is a…

We discuss 3d $\mathcal{N}=1$ supersymmetric SU(N) and U(N) Chern-Simons-matter theories, with $N_f$ matter superfields in the fundamental representation of SU(N) or U(N). In the large N 't Hooft limit with fixed 't Hooft coupling $\lambda$…

In this paper, we study the existence of solution for the following class of nonlocal problem, $$ \left\{ \begin{array}{lcl} -\Delta u=\left(\lambda f(x)-\int_{\R^N}K(x,y)|u(y)|^{\gamma}dy\right)u,\quad \mbox{in} \quad \R^{N}, \\…

We introduce a modified non-linear heat equation $\partial_t u = \Delta u + \Gamma u$ as a substitute of $\log P_t f$ where $P_t$ is the heat semigroup. We prove an exponential decay of $\Gamma u$ under the Bakry Emery curvature condition…

This paper is devoted to studies of non-negative, non-trivial (classical, punctured, or distributional) solutions to the higher order Hardy-H\'enon equations \[ (-\Delta)^m u = |x|^\sigma u^p \] in $\mathbf R^n$ with $p > 1$. We show that…

We consider the following elliptic system with Neumann boundary: \begin{equation} \begin{cases} -\Delta u + \mu u=v^p, &\hbox{in } \Omega, \\-\Delta v + \mu v=u^q, &\hbox{in } \Omega, \\\frac{\partial u}{\partial n} = \frac{\partial…

In this paper, we investigate the existence and nonexistence of positive solutions to the Lane-Emden equations $$ -\Delta u = Q |u|^{p-2}u $$ on the $d$-dimensional integer lattice graph $\mathbb{Z}^d$, as well as in the half-space and…

In this paper, we study the existence of solutions for the following superlinear elliptic equation with nonlinear boundary value condition $$ \left\{ \begin{array}{ll} -\Delta u+u=|u|^{r-2}u &\text{in} \; \Omega,\\ \\ \frac{\partial…