Fourth Order Equations of Critical Sobolev Growth. Energy Function and Solutions of Bounded Energy in the Conformally Flat Case
نویسنده
چکیده
In 1983, Paneitz [23] introduced a conformally fourth order operator defined on 4-dimensional Riemannian manifolds. Branson [1] generalized the definition to n-dimensional Riemannian manifolds, n ≥ 5. Such operators have a geometrical meaning. While the conformal Laplacian is associated to the scalar curvature, the Paneitz-Branson operator is associated to a notion of Q-curvature. Possible references are Chang [2] and Chang-Yang [3]. When the manifold (M, g) is Einstein, the Paneitz-Branson operator PBg has constant coefficients. It expresses as PBg(u) = ∆ 2 gu+ ᾱ∆gu+ āu , (0.1)
منابع مشابه
The Solvability of Concave-Convex Quasilinear Elliptic Systems Involving $p$-Laplacian and Critical Sobolev Exponent
In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.
متن کاملCompactness of conformal metrics with constant Q-curvature. I
We establish compactness for nonnegative solutions of the fourth order constant Qcurvature equations on smooth compact Riemannian manifolds of dimension ≥ 5. If the Q-curvature equals −1, we prove that all solutions are universally bounded. If the Qcurvature is 1, assuming that Paneitz operator’s kernel is trivial and its Green function is positive, we establish universal energy bounds on manif...
متن کاملEinstein equation with quantum corrections reduced to second order.
We consider the Einstein equation with first-order (semiclassical) quantum corrections. Although the quantum corrections contain up to fourth-order derivatives of the metric, the solutions which are physically relevant satisfy reduced equations which contain derivatives no higher than second order. We obtain the reduced equations for a range of stress-energy tensors. These reduced equations are...
متن کاملRenormalized Solutions for Strongly Nonlinear Elliptic Problems with Lower Order Terms and Measure Data in Orlicz-Sobolev Spaces
The purpose of this paper is to prove the existence of a renormalized solution of perturbed elliptic problems$ -operatorname{div}Big(a(x,u,nabla u)+Phi(u) Big)+ g(x,u,nabla u) = mumbox{ in }Omega, $ in the framework of Orlicz-Sobolev spaces without any restriction on the $M$ N-function of the Orlicz spaces, where $-operatorname{div}Big(a(x,u,nabla u)Big)$ is a Leray-Lions operator defined f...
متن کاملA Modified Energy Balance Method to Obtain Higher-order Approximations to the Oscillators with Cubic and Harmonic Restoring Force
This article analyzes a strongly nonlinear oscillator with cubic and harmonic restoring force and proposes an efficient analytical technique based on the modified energy balance method (MEBM). The proposed method incorporates higher-order approximations. After applying the proposed MEBM, a set of complicated higher-order nonlinear algebraic equations are obtained. Higher-order nonlinear algebra...
متن کامل