Isolated singularities of polyharmonic operator in even dimension
TL;DR: 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.
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TL;DR: In this article, a positive answer to a question of Lucio Boccardo concerning existence of solutions of an elliptic system with absorption was provided, and the authors provided a positive solution to the same question of BOCcardo.
Abstract: we study the equation -Delta u + g(x, u) = mu, where g(., s) is finite outside sets of zero H-1-capacity, for all s is an element of R, and mu is a diffuse measure. As an application, we provide a positive answer to a question of Lucio Boccardo concerning existence of solutions of an elliptic system with absorption.
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TL;DR: In this paper, the local behavior of singular solutions of the Laplace equation was studied and an existence result for the singularity together with an a priori estimate of the radially symmetric solutions was established.
Abstract: This paper is mainly concerned with the local behavior of singular solutions of the biharmonic equation
$$\Delta ^2 u = |x|^\sigma u^p $$
with u ≤ 0 ,
$$- \Delta u $$
≤ 0 in
$$\Omega \backslash \{ 0\} \subset \mathbb{R}^N ,N$$
≥ 4, and Ω = B(0, R) is a ball centered at the origin of radius R > 0 The complete description of the singularity together with an existence result will be given when
$$$$
≤ 0, for N > 4, or 1 < p < +∞, for N = 4 Moreover, an a priori estimate of the radially symmetric solutions will be established when p ≥
$$\frac{{N+ \sigma}}{{N-4}}, -4 < \sigma$$
≤ 0, N > 4 This paper generalizes the results in Brezis and Lions (1981) and Lions (1980) for the corresponding Laplace equation
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