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Register a new userOn a Paneitz Type Equation in Six Dimensional Domains
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In this paper we consider a fourth order equation involving the critical Sobolev exponent on a bounded and smooth domain in $R^6$. Using theory of critical points at infinity, we give some topological conditions on a given function defined on a domain to ensure some existence results.
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We give a sup+inf inequality on $S_4$ for Paneitz operator.
We present another proof of the sharp inequality for Paneitz operator on the standard three sphere, in the spirit of subcritical approximation for the classical Yamabe problem. To solve the perturbed problem, we use a symmetrization process which only works for extremal functions. This gives a new example of symmetrization for higher order variational problems.
The purpose of the present paper is to establish the local energy decay estimates and dispersive estimates for 3-dimensional wave equation with a potential to the initial-boundary value problem on exterior domains. The geometrical assumptions on domains are rather general, for example non-trapping condition is not imposed in the local energy decay result. As a by-product, Strichartz estimates is obtained too.
We prove Strichartz estimates with a loss of derivatives for the Schrodinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a result of the second author regarding the Schrodinger equation on the Euclidean cone.
We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain $$left{ begin{array}{ll} -Delta u=lambda_1dfrac{V_1 e^{u}}{ int_{Omega_{boldsymbolepsilon}} V_1 e^{u} dx } - lambda_2tau dfrac{ V_2 e^{-tau u}}{ int_{Omega_{boldsymbolepsilon}}V_2 e^{ - tau u} dx}&text{in $Omega_{boldsymbolepsilon}=Omegasetminus displaystyle bigcup_{i=1}^m overline{B(xi_i,epsilon_i)}$} u=0 &text{on $partial Omega_{boldsymbolepsilon}$}, end{array} right. $$ where $B(xi_i,epsilon_i)$ is a ball centered at $xi_iinOmega$ with radius $epsilon_i$, $tau$ is a positive parameter and $V_1,V_2>0$ are smooth potentials. When $lambda_1>8pi m_1$ and $lambda_2 tau^2>8pi (m-m_1)$ with $m_1 in {0,1,dots,m}$, there exist radii $epsilon_1,dots,epsilon_m$ small enough such that the problem has a solution which blows-up positively and negatively at the points $xi_1,dots,xi_{m_1}$ and $xi_{m_1+1},dots,xi_{m}$, respectively, as the radii approach zero.
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