Related papers: Existence results for mean field equations

Let us consider a semilinear boundary value problem $ - \Delta u= f(x,u),$ in $\Omega,$ with Dirichlet boundary conditions, where $ \Omega \subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions…

In this article, we firstly study the cone Moser-Trudinger inequalities and their best exponents $\alpha_2$ on both bounded and unbounded domains $\mathbb{R}^2_{+}$. Then, using the cone Moser-Trudinger inequalities, we study the existence…

We solve variationally certain equations of stellar dynamics of the form $-\sum_i\partial_{ii} u(x) =\frac{|u|^{p-2}u(x)}{{\rm dist} (x,{\mathcal A} )^s}$ in a domain $\Omega$ of $\rn$, where ${\mathcal A} $ is a proper linear subspace of…

We establish the existence of positive solutions to a general class of overdetermined semilinear elliptic boundary problems on suitable bounded open sets $\Omega\subset\mathbb{R}^n$. Specifically, for $n\leq 4$ and under mild technical…

We are devoted to the study of the following nonlinear $p$-Laplacian Schr\"odinger equation with $L^{p}$-norm constraint \begin{align*} \begin{cases} &-\Delta_{p} u=\lambda |u|^{p-2}u +|u|^{r-2}u\quad\mbox{in}\quad\Omega,\\…

We deal with existence and uniqueness of nonnegative solutions to \begin{equation*} \left\{ \begin{array}{l} -\Delta u = f(x) \text{ in }\Omega, \frac{\partial u}{\partial \nu} + \lambda(x) u = \frac{g(x)}{u^\eta} \text{ on }…

In this paper we study the following torsion problem \begin{equation*} \begin{cases} -\Delta u=1~&\mbox{in}\ \Omega,\\[1mm] u=0~&\mbox{on}\ \partial\Omega. \end{cases} \end{equation*} Let $\Omega\subset \mathbb{R}^2$ be a bounded, convex…

Let $(M,g)$ be a compact Riemann surface with unit area. We investigate the mean field equation for equilibrium turbulence: \begin{align} \begin{cases} -\Delta u = \rho_1\left(\frac{h_1e^{u}}{\int_Mh_1e^udv_g}-1\right) -…

We follow the approach employed by Y. Choquet-Bruhat, J. Isenberg and D. Pollack in the case of closed manifolds and establish existence and non-existence results for the Einstein-scalar field constraint equations on asymptotically…

We analyze the Minimal Area solution to the Loop Equations in turbulence \cite{M93}. As it follows from the new derivation in the recent paper \cite{M19}, the vorticity is represented as a normal vector to the minimal surface not just at…

We initiate the study of the finiteness condition $\int_{\Omega}u(x)^{-\beta}\,dx\leq C(\Omega,\beta)<+\infty$ where $\Omega\subseteq{\mathbb{R}}^n$ is an open set and $u$ is the solution of the Saint Venant problem $\Delta u=-1$ in…

In this paper, we prove the boundary pointwise $C^{0}$-regularity of weak solutions for Dirichlet problem of elliptic equations in divergence form with distributional coefficients, where the boundary value equals to zero. This is a…

This paper is concerned with the uniqueness of solutions to the following nonlocal semi-linear elliptic equation \begin{equation}\label{ellip}\tag{$\ast$} \Delta u-\beta u+\lambda\frac{e^u}{\int_{\Omega}e^u}=0~\mathrm{in}~\Omega,…

We extend the symmetry result of Serrin and Weinberger from the Laplacian operator to the highly degenerate game-theoretic $p$-Laplacian operator and show that viscosity solutions of $-\Delta_p^Nu=1$ in $\Omega$, $u=0$ and $\tfrac{\partial…

We study the metastable states in Ising spin models with orthogonal interaction matrices. We focus on three realizations of this model, the random case and two non-random cases, i.e.\ the fully-frustrated model on an infinite dimensional…

We consider the Brezis-Nirenberg problem: $$-\Delta u =\lambda u + |u|^{p-1}u\qquad \mbox{in}\,\, \Omega,\quad u=0\,\, \mbox{on}\,\,\ \partial\Omega,$$ where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$,…

In this paper, we establish symmetry results for solutions of the mean field equation \[ \frac{\alpha}{2} \Delta u + e^u - 1 = 0 \] on \( \mathbb{S}^2 \) for $\frac{1}{3}\leq \alpha < \frac{1}{2}$.The proofs utilize the Sphere Covering…

We show that the classical Brezis-Nirenberg problem $$\Delta u + |u|^{4 \over N-2} u + \varepsilon u = 0 ,\quad {\mbox {in}} \quad \Omega, \quad u= 0 , \quad {\mbox {on}} \quad \partial \Omega$$ admits nodal solutions clustering around a…

In this paper we prove the nonexistence of nontrivial solution to \begin{equation*} \begin{cases} -\Delta u =f(u) &\text{in }\Omega, \\ u=0 &\text{on } \partial \Omega, \end{cases} \end{equation*} being $\Omega \subset \mathbb{R}^N$ ($N\in…

We consider here a nonlinear elliptic equation in an unbounded sectorial domain of the plane. We prove the existence of a minimal solution to this equation and study its properties. We infer from this analysis some asymptotics for the…