相关论文: Sharp Boundary Growth Rate Estimate of the Singula…

In this paper, we mainly investigate the critical points associated to solutions $u$ of a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a connected domain $\Omega$ in $\mathbb{R}^2$. Based on the fine…

Consider the following equation $$\partial_t u_t(x)=\frac{1}{2}\partial _{xx}u_t(x)+\lambda \sigma(u_t(x))\dot{W}(t,\,x)$$ on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution…

In this paper we consider semilinear equations $-\Delta u=f(u)$ with Dirichlet boundary conditions on certain convex domains of the two dimensional model spaces of constant curvature. We prove that a positive, semi-stable solution $u$ has…

We are concerned with the semilinear biharmonic problem under Dirichlet boundary conditions that \begin{equation*} \begin{cases} \Delta^2 u=(u^+)^{p} &{\text{in}~\Omega},\\[0.5mm] u \not\equiv 0 &{\text{in}~\Omega},\\[0.5mm] u=\partial u /…

We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we establish…

In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^{\alpha}, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n…

In this article we consider the following boundary value problem \begin{equation*}\label{abs} \left\{ \begin{aligned} F(x,u,Du,D^{2}u)+c(x)u+ p(x)u^{-\alpha}&=0~\text{in}~\Omega\\ u&=0~~\text{on}~~\partial\Omega, \end{aligned} \right.…

We study the Lane-Emden problem \[\begin{cases} -\Delta u_p =|u_p|^{p-1}u_p&\text{in}\quad \Omega, u_p=0 &\text{on}\quad\partial\Omega, \end{cases}\] where $\Omega\subset\mathbb R^2$ is a smooth bounded domain and $p>1$ is sufficiently…

We study positive solutions of the Dirichlet problem $-\Delta u = u^p$ in a uniformly convex domain $\Omega \subset \mathbb S^2$, $u= 0$ on $\partial\Omega.$ For $p=1$, we assume that the right-hand side is replaced by $\lambda_1 u$, where…

In this paper we consider the following Dirichlet problem for the $p$-Laplacian in the positive parameters $\lambda$ and $\beta$: [{{array} [c]{rcll}% -\Delta_{p}u & = & \lambda h(x,u)+\beta f(x,u,\nabla u) & \text{in}\Omega u & = & 0 &…

We introduce a new method for the analysis of singularities in the unstable problem $$\Delta u = -\chi_{\{u>0\}},$$ which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of…

We investigate quantitative properties of nonnegative solutions $u(x)\ge 0$ to the semilinear diffusion equation $\mathcal{L} u= f(u)$, posed in a bounded domain $\Omega\subset {\mathbb R}^N$ with appropriate homogeneous Dirichlet or outer…

We study the nodal solutions of the Lane Emden Dirichlet problem $-\Delta u = |u|^{p-1}u with DBC on a smooth bounded domain $\Omega$ in $\IR^2$ and where $p>1$. We consider solutions $u_p$ satisfying $p \int_{\Omega}\abs{\nabla u_p}^2\to…

In this paper we deal with uniqueness of solutions to the following problem \[ \begin{cases} \begin{split} & u_t-\Delta_p u=H(t,x,\nabla u) &\quad \text{in}\quad Q_T,\\ & u (t,x) =0 &\quad \text{on}\quad(0,T)\times \partial \Omega,\\ &…

We characterize lower growth estimates for subsolutions in halfspaces of fully nonlinear partial differential equations on the form $$ F(x,u,Du,D^2u) = 0 $$ in terms of solutions to ordinary differential equations built solely upon a growth…

This article studies a dirichlet boundary value problem for singularly perturbed time delay convection diffusion equation with degenerate coefficient. A priori explicit bounds are established on the solution and its derivatives. For…

We establish the existence and uniqueness of the solution to the Dirichlet problem for the variable exponent $p$-Laplacian on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$, where the boundary datum belongs to $W^{1,p}(\Omega)$.…

We study the admissible growth at infinity of initial data of positive solutions of $\prt\_t u-\Gd u+f(u)=0$ in $\BBR\_+\ti\BBR^N$ when $f(u)$ is a continuous function, {\it mildly} superlinear at infinity, the model case being…

In this paper we prove the existence of positive solutions of the following singular quasilinear Schr\"{o}dinger equations at critical growth \begin{eqnarray*} -\Delta u-\lambda c(x)u-\kappa\alpha(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u =…

Let us consider the singularly perturbed model problem $Lu:=-\varepsilon\Delta u-bu_x+c u =f$ with homogeneous Dirichlet boundary conditions on $\Gamma=\partial\Omega$ $u|_\Gamma =0$ on the unit-square $\Omega=(0,1)^2$. Assuming that $b>0$…