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The Singular Set for a Semilinear Unstable Problem

Abstract

We study the regularity of solutions of the following semilinear problem

$${\Delta}u = -\lambda_{+}(x) (u^{+})^{q}+\lambda_{-} (x) (u^{-})^{q} \qquad \text{in} \;\; B_{1}, $$

where B 1 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 \(\mathcal {S}_{\kappa }\), is the set of points where all the derivatives of order less than κ exist and vanish. We prove that \(\mathcal {S}_{\kappa }=\varnothing \) when κ is not an integer. Moreover, with an example we show that \(\mathcal {S}_{\kappa }\) can be nonempty if κ ∈ ℕ. Finally, a regularity investigation in the plane shows that the singular points in \(\mathcal {S}_{\kappa }\) are isolated.

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Acknowledgements

The author sincerely thanks Henrik Shahgholian for introducing the problem, and John Andersson for providing helpful feedback on a preliminary version of the paper.

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Correspondence to Morteza Fotouhi.

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Fotouhi, M. The Singular Set for a Semilinear Unstable Problem. Potential Anal 49, 411–422 (2018). https://doi.org/10.1007/s11118-017-9662-6

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  • DOI: https://doi.org/10.1007/s11118-017-9662-6

Keywords

  • Semilinear elliptic
  • Regularity
  • Unstable problem

Mathematics Subject Classification (2010)

  • 35R35
  • 35J61
  • 35B65