Local behavior of solutions of some elliptic equations

Elliptic Equations Involving Measures

“…Finally, although this was noticed before by Lions [66], Theorem 4.14 implies that the Dirac mass δ 0 is q-admissible. The classification of isolated singularities of positive solutions of (4.77) has been performed by Lions [66] in the case 1 < q < n/(n − 2), Aviles [6] in the case q = n/(n − 2), Gidas and Spruck [46] when n/(n − 2) < q < (n + 2)/(n − 2) and Caffarelli, Gidas and Spruck [24] in the case q = (n + 2)/(n − 2). The case q > (n + 2)/(n − 2) remains essentially open, except if the solutions are supposed to be radial.…”

“…Thus the problem may not be solved for all the measures, but only for specific ones. A natural condition is to assume that the measure λ satisfies 6) where G Ω L (|λ|), defined by…”

A priori estimates for solutions of anisotropic elliptic equations

“…Whereas, we note that these estimates are no more sharp in the case of the equation −∆u = u q−1 when 2 < q ≤ 2 * . In this case, the local behavior near an isolated singularity was established by Lions [46] for q in (2, 2 * ) (see Bidaut-Véron [47] for an extension to the p-Laplace operator) and by Aviles [48] for q = 2 * .…”

Separable solutions of some quasilinear equations with source reaction

“…Because H B−1 (e B ) = 0, H B−1 (x) − H B−1 (e B ) = H B−1 (ξ )(x − e B ) and x ranges onto (H B,1 (d − 1), H B,2 (d − 1)) := (x 1,B , x 2,B ), when B → ln d, (x 1,B , x 2,B ) → (x 1,ln d , x 2,ln d ), it follows |H B−1 (ξ )| 1/μ > 0 independent on B. Moreover H B (x) − H B (e B ) = (1/2)H B (ξ )(x − e B ) 2 = −(1/2ξ )(x − e B )2 . Thus there exists m > 0 such thatH B (x) − H B e B m 2 x − e B 2 m 2 μ 2 H 2 B−1 (x).Therefore, near ln d,taking x = L B,i (d − √ 1 − λ 2 u * 2 ), one derive ψ i,u * (λ) 2 mμ d − √ 1 − λ 2 u * 2 − e B = 2 mμ √ 1 − u * 2 − √ 1 − λ 2 u * 2 4 mμ √ 1 − λ 2 .…”