A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials

作者：Cirstea Florica C^*

来源：Memoirs of the American Mathematical Society, 2014, 227(1068): 1-+.

DOI：10.1090/memo/1068

• Nonlinear elliptic equations
• isolated singularities
• regular variation theory
• inverse square potentials
• removable singularities

In this paper, we consider semilinear elliptic equations of the form
(0.1) -Delta u - lambda/vertical bar x vertical bar(2)u + b(x) h(u) = 0 in Omega\{0},
where lambda is a parameter with -infinity < lambda <= (N - 2)(2)/4 and Omega is an open subset in R-N with N >= 3 such that 0 is an element of Omega. Here, b(x) is a positive continuous function on <(Omega)over bar>\{0} which behaves near the origin as a regularly varying function at zero with index theta greater than -2. The nonlinearity h is assumed continuous on R and positive on (0, infinity) with h(0) = 0 such that h(t)/t is bounded for small t > 0. We completely classify the behaviour near zero of all positive solutions of (0.1) when h is regularly varying at infinity with index q greater than 1 (that is, lim(t ->infinity) h(xi t)/h(t) = xi(q) for every xi > 0). In particular, our results apply to (0.1) with h(t) = t(q)(log t)(alpha 1) as t -> infinity and b(x) = vertical bar x vertical bar(theta)(-log vertical bar x vertical bar)(alpha 2) as vertical bar x vertical bar -> 0, where alpha(1) and alpha(2) are any real numbers.
We reveal that the solutions of (0.1) generate a very complicated dynamics near the origin, depending on the interplay between q, N, theta and lambda, on the one hand, and the position of. with respect to 0 and (N - 2)(2)/4, on the other hand. Our main results for lambda = (N-2)(2)/4 appear here for the first time, as well as for the case lambda < 0. We establish a trichotomy of positive solutions of (0.1) under optimal conditions, hence generalizing and improving through a different approach a previous result with Chaudhuri on (0.1) with 0 < lambda < (N - 2)(2)/4 and b - 1. Moreover, recent results of the author with Du on (0.1) with lambda = 0 are here sharpened and extended to any -infinity < lambda < (N - 2)(2)/4. In addition, we unveil a new single-type behaviour of the positive solutions of (0.1) specific to 0 < lambda < (N - 2)(2)/4. We also provide necessary and sufficient conditions for the existence of positive solutions of (0.1) that are comparable with the fundamental solutions of
-Delta u - lambda/vertical bar x vertical bar(2)u = 0 in R-N\{0}.
In particular, for b - 1 and lambda - 0, we find a sharp condition on h such that the origin is a removable singularity for all non-negative solutions of (0.1), thus addressing an open question of Vazquez and Veron.

• 出版日期2014-1