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The singular set for a semilinear unstable problem

Fotouhi, M ; Sharif University of Technology | 2018

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  1. Type of Document: Article
  2. DOI: 10.1007/s11118-017-9662-6
  3. Publisher: Springer Netherlands , 2018
  4. Abstract:
  5. We study the regularity of solutions of the following semilinear problemΔu=−λ+(x)(u+)q+λ−(x)(u−)qinB1,where B1 is the unit ball in ℝn, 0 < q < 1 and λ± satisfy a Hölder continuity condition. Our main results concern local regularity analysis of solutions and their nodal set {u = 0}. The desired regularity is C[κ],κ−[κ] for κ = 2/(1 − q) and we divide the singular points in two classes. The first class contains the points where at least one of the derivatives of order less than κ is nonzero, the second class which is named Sκ, is the set of points where all the derivatives of order less than κ exist and vanish. We prove that Sκ= ∅ when κ is not an integer. Moreover, with an example we show that Sκ can be nonempty if κ ∈ ℕ. Finally, a regularity investigation in the plane shows that the singular points in Sκ are isolated. © 2017, Springer Science+Business Media B.V
  6. Keywords:
  7. Regularity ; Semilinear elliptic ; Unstable problem
  8. Source: Potential Analysis ; Volume 49, Issue 3 , 2018 , Pages 411-422 ; 09262601 (ISSN)
  9. URL: https://link.springer.com/article/10.1007/s11118-017-9662-6