Singular solutions of conformal Hessian equation

Journal de Mathématiques Pures et Appliquées, 2013

Advances in Mathematics, 2011

Calculus of Variations and Partial Differential Equations, 2006

We give conditions on a positive Hölder continuous function K(x) such that every C 2 positive solution u(x) of the conformal scalar curvature equation

Differential and Integral Equations, 2008

Nonlinear Analysis

We study a class of elliptic problems, involving a k-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. An important ingredient of this study is analyzing the number of intersection points between the singular and regular solutions for rescaled problems. In the particular case of the exponential nonlinearity, we obtain the convergence of regular solutions to the singular.

2021

We study the asymptotic behavior for singular solutions to a critical fourth order system generalizing the constant Q-curvature equation. Our main result extends to the case of strongly coupled systems, the celebrated asymptotic classification due to [L. A. Caffarelli, B. Gidas and J. Spruck, Comm. Pure Appl. Math. (1989)] and [N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Invent. Math., (1999)]. On the technical level, we use an involved spectral analysis to study the Jacobi fields' growth properties in the kernel of the linearization of our system around a blow-up limit solution. Besides, we obtain sharp a priori estimates for the decay rate of singular solutions near the origin. Consequently, we prove that sufficiently close to the isolated singularity solutions behave like the so-called Emden–Fowler solution. Our main theorem positively answers a question posed by [R. L. Frank and T. König, Anal. PDE (2019)] concerning the local behavior close to the isolated singularity ...

Archive for Rational Mechanics and Analysis, 1980

Journal of Mathematical Analysis and Applications, 2020

Let (M, g) be a smooth and compact Riemannian manifold with dimensions n ≥ 3. This paper deals with the existence of nontrivial solutions for a nonlinear secondorder elliptic equation with a singular term and critical Sobolev growth. We provide a geometric application of this singular equation.

Mathematical Research Letters, 1997

Nonlinear Analysis: Theory, Methods & Applications, 1981

We give a classification of isolated singularities of any solution of the equation-Au + u julq-' = 0. If 0 is the singular point, we prove that there exist two types of singularities when 1 < 4 < N/(N-2) and that we have essentially the following in 0 (i) either u(x) h k [q,NI~j-2'(q-1), with Iq, N = ((2/(q-1))(2q/(q-1)-N))"(y-l', (ii) or u(x) N c&4, where c is any constant and cp the fundamental solution of the Laplacian in W'.