Related papers: Boundary value problem for the mean field equation…

Let $(M,g)$ be a compact Riemann surface, $h$ be a positive smooth function on $M$. It is well known the functional $$J(u)=\frac{1}{2}\int_M|\nabla u|^2dv_g+8\pi\int_M udv_g-8\pi\log\int_Mhe^{u}dv_g$$ achieves its minimum under…

Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving the eigenvalue problem $Lu=\lambda u$ for an elliptic partial differential…

In this paper, by variational and topological arguments based on linking and $\nabla$-theorems, we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$ \left\{…

We consider the mean field equation: (1) \Delta u+\rho\frac{e^u}{\int_\Omega e^u}=0 & \hbox{in} \;\Omega, u=0 & \hbox{on}\;\partial\Omega, where $\Omega\subset \mathbb{R}^2$ is an open and bounded domain of class $C^1$. In his 1992 paper,…

In this paper, we prove two existence results of solutions to mean field equations $$\Delta u+e^u=\rho\delta_0$$ and $$\Delta u=\lambda e^u(e^u-1)+4 \pi \sum_{j=1}^{M}{\delta_{p_j}}$$ on an arbitrary connected finite graph, where $\rho>0$…

In this work we study the existence of solutions to the critical Brezis-Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet-Neumann boundary conditions, i.e., $$ \left\{\begin{array}{rcl}…

Let $\Omega\subset\BBR^N$ be a bounded $C^2$ domain and $\CL_\gk=-\Gd-\frac{\gk}{d^2}$ the Hardy operator where $d=\dist (.,\prt\Gw)$ and $0<\gk\leq\frac{1}{4}$. Let $\ga_{\pm}=1\pm\sqrt{1-4\gk}$ be the two Hardy exponents, $\gl_\gk$ the…

We study the existence of solutions of mixed Riemann-Hilbert or Cherepanov boundary value problem with simply connected fibers on the unit disk ${\Delta}$. Let $L$ be a closed arc on $\partial{\Delta}$ with the end points $\omega_{-1},…

We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold $\mathbb{M}$ without boundary, \begin{equation*} -\Delta_g u_i = H_i(u_1,\cdots,u_m) \ \ \text{on} \ \ \mathbb{M},…

In this paper we study the boundary value problem for the equation $\mbox{div}\left(D(\nabla u)\nabla\left(\mbox{div}\left(|\nabla u|^{p-2}\nabla u+\beta\frac{\nabla u}{|\nabla u|}\right)\right)\right)+au=f$ in the $z=(x,y)$ plane. This…

In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation $\frac{\partial u}{\partial t} = \Delta\left(\Delta u\right)^{-3}$. This problem arises in the mathematical modeling…

In this paper we discuss the existence and non--existence of weak solutions to parametric equations involving the Laplace-Beltrami operator $\Delta_g$ in a complete non-compact $d$--dimensional ($d\geq 3$) Riemannian manifold…

We obtain necessary and sufficient existence conditions for solutions of the boundary value problem $$ \Delta_p u = f \quad \mbox{on } M, \quad \left. \left| \nabla u \right|^{p - 2} \frac{\partial u}{\partial \nu} \right|_{ \partial M } =…

In this paper we discuss the solvability of the Neumann and Regularity boundary value problem of elliptic Schr\"odinger-type equation $-\DIV(A(x)\nabla u(x,t))+V(x)u(x,t)=0$ with bounded measurable uniformly elliptic coefficinets $A(x)$…

This article uses the iterative schemes and perturbation methods to completely solve the Han-Li conjecture, i.e. the general boundary Yamabe problem with prescribed constant scalar curvature and constant mean curvature on compact manifolds…

Let $(M,g)$ be a compact Riemannian surface without boundary, $W^{1,2}(M)$ be the usual Sobolev space, $J: W^{1,2}(M)\rightarrow \mathbb{R}$ be the functional defined by $$J(u)=\frac{1}{2}\int_M|\nabla u|^2dv_g+8\pi \int_M…

Let $(M,g)$ be a compact Riemannian manifold with a boundary of class $\mathscr{C}^{1}$. We are interested in the spectrum of the weighted Laplacian on $M$ with Neumann boundary conditions. More precisely, given $\rho$ and $\sigma$ two…

In this paper, we study the existence and the summability of solutions to a Robin boundary value problem whose prototype is the following: $$ \begin{cases} -\text{div}(b(|u|)\nabla u)=f &\text{in }\Omega,\\[.2cm] \displaystyle\frac{\partial…

We consider the following class of equations with exponential nonlinearities on a compact surface $M$: $$ - \Delta u = \rho_1 \left( \frac{h_1 \,e^{u}}{\int_M h_1 \,e^{u} } - \frac{1}{|M|} \right) - \rho_2 \left( \frac{h_2 \,e^{-u}}{\int_M…

We study an inhomogeneous Neumann boundary value problem for functions of least gradient on bounded domains in metric spaces that are equipped with a doubling measure and support a Poincar\'e inequality. We show that solutions exist under…