Isolated singularities of polyharmonic operator in even dimension
R. Dhanya,Abhishek Sarkar +1 more
TLDR
In this paper, the authors considered the problem of finding the barrier function in higher dimensions (N >= 5) with a specific weight function a(x) = |x|(sigma), where a is a nonnegative measurable function in some Lebesgue space.Abstract:
We consider the equation Delta(2)u = g(x, u) >= 0 in the sense of distribution in Omega' = Omega\textbackslash {0} where u and -Delta u >= 0. Then it is known that u solves Delta(2)u = g(x, u) + alpha delta(0) - beta Delta delta(0), for some nonnegative constants alpha and beta. In this paper, we study the existence of singular solutions to Delta(2)u = a(x) f (u) + alpha delta(0) - beta Delta delta(0) in a domain Omega subset of R-4, a is a nonnegative measurable function in some Lebesgue space. If Delta(2)u = a(x) f (u) in Omega', then we find the growth of the nonlinearity f that determines alpha and beta to be 0. In case when alpha = beta = 0, we will establish regularity results when f (t) 0. This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions (N >= 5) with a specific weight function a(x) = |x|(sigma). Later, we discuss its analogous generalization for the polyharmonic operator.read more
References
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Journal ArticleDOI
Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)eu in two dimensions
Haim Brezis,Frank Merle +1 more
TL;DR: In this article, uniform estimates and blow-up behavior for solutions of −δ(u) = v(x)eu in two dimensions are presented, with a focus on partial differential equations.
Journal ArticleDOI
A classification of solutions of a conformally invariant fourth order equation in Rn
TL;DR: In this article, a necessary and sufficient condition for solutions obtained from the smooth conformal metrics on S 4>>\s was established for the stereograph projection of the biharmonic operator.
BookDOI
Polyharmonic Boundary Value Problems
TL;DR: The preprint version of this paper has different page and line numbers from the final version which appeared at Springer-Verlag as mentioned in this paper, which can be found on their personal web pages.
Journal ArticleDOI
Concentration–compactness phenomena in the higher order Liouville's equation☆
TL;DR: In this paper, the authors investigated different concentration-compactness and blow-up phenomena related to the Q-curvature in arbitrary even dimension and showed that on a locally conformally flat manifold of non-positive Euler characteristic, one always has compactness.
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