A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials
Description:... In this paper, we consider semilinear elliptic equations of the form −Δu−λ|x|2u+b(x)h(u)=0in Ω∖{0}, where λ is a parameter with −∞λ≤(N−2)2/4 and Ω is an open subset in RN with N≥3 such that Ω. Here, b(x) is a positive continuous function on Ω∖{0} which behaves near the origin as a regularly varying function at zero with index θ greater than −2. The nonlinearity h h is assumed continuous on R R and positive on (0,∞) with (0)=0 such that h(t)/t is bounded for small t0. We completely classify the behaviour near zero of all positive solutions of [[eqref]]one when h h is regularly varying at ∞ with index q q greater than 1 (that is, limt→∞h(ξt)/h(t)=ξq for every ξ>0). In particular, our results apply to [[eqref]]one with h(t)=tq(logt)α1 as t→∞ and b(x) = |x|θ(−log|x|)α2 as |x|→0, where α 1 α1 and α2 are any real numbers. We reveal that the solutions of [[eqref]]one generate a very complicated dynamics near the origin, depending on the interplay between q, and λ, on the one hand, and the position of λ with respect to 0 and (N−2)2/4, on the other hand. Our main results for λ=(N−2)2/4 appear here for the first time, as well as for the case λ
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