Asymptotic behaviour of a semilinear elliptic system with a large exponent

Consider the problem \begin{eqnarray*} -\Delta u &=& v^{\frac 2{N-2}},\quad v>0\quad {in}\quad \Omega, -\Delta v &=& u^{p},\:\:\:\quad u>0\quad {in}\quad \Omega, u&=&v\:\:=\:\:0 \quad {on}\quad \partial \Omega, \end{eqnarray*} where $\Omega$ is a bounded convex domain in $\R^N,$ $N>2,$ with smooth boundary $\partial \Omega.$ We study the asymptotic behaviour of the least energy solutions of this system as $p\to \infty.$ We show that the solution remain bounded for $p$ large and have one or two peaks away form the boundary. When one peak occurs we characterize its location.Comment: 16 pages, submmited for publicatio

Spectral properties of Schr\"{o}dinger-type operators and large-time behavior of the solutions to the corresponding wave equation

Let $L$ be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations &(1) \quad \ddot{w}+ Lw=0, \quad w(0)=0,\quad \dot{w}(0)=f, \quad \dot{w}=\frac{dw}{dt}, \quad f \in H. &(2) \quad \ddot{u}+Lu=f e^{-ikt}, \quad u(0)=0, \quad \dot{u}(0)=0, where $k>0$ is a constant. Necessary and sufficient conditions are given for the operator $L$ not to have eigenvalues in the half-plane Re$z<0$ and not to have a positive eigenvalue at a given point $k_d^2 >0$. These conditions are given in terms of the large-time behavior of the solutions to problem (1) for generic $f$. Sufficient conditions are given for the validity of a version of the limiting amplitude principle for the operator $L$. A relation between the limiting amplitude principle and the limiting absorption principle is established

Heat flow method to Lichnerowicz type equation on closed manifolds

In this paper, we establish existence results for positive solutions to the Lichnerowicz equation of the following type in closed manifolds -\Delta u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where $p>1, q>0$, and $A(x)>0$, $B(x)\geq0$ are given smooth functions. Our analysis is based on the global existence of positive solutions to the following heat equation {ll} u_t-\Delta u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M\times\mathbb{R}^{+}, u(x,0)=u_0,\quad in\quad M with the positive smooth initial data $u_0$.Comment: 10 page