JournalsaihpcVol. 30, No. 2pp. 315–336

Fading absorption in non-linear elliptic equations

  • Moshe Marcus

    Department of Mathematics, Technion Haifa, Israel
  • Andrey Shishkov

    Institute of Appl. Math. and Mech., NAS of Ukraine, Donetsk, Ukraine
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Abstract

We study the equation Δu+h(x)uq1u=0−\mathrm{\Delta }u + h(x)|u|^{q−1}u = 0, q>1q > 1, in R+N=RN1×R+\mathbb{R}_{ + }^{N} = \mathbb{R}^{N−1} \times \mathbb{R}_{ + } where h \in C(\mathbb{R}_{ + }^{N}\limits^{¯}), h0h⩾0. Let (x1,,xN)(x_{1},…,x_{N}) be a coordinate system such that R+N=[xN>0]\mathbb{R}_{ + }^{N} = [x_{N} > 0] and denote a point xRNx \in \mathbb{R}^{N} by (x,xN)(x^{′},x_{N}). Assume that h(x,xN)>0h(x^{′},x_{N}) > 0 when x0x^{′} \neq 0 but h(x,xN)0h(x^{′},x_{N})\rightarrow 0 as x0|x^{′}|\rightarrow 0. For this class of equations we obtain sharp necessary and sufficient conditions in order that singularities on the boundary do not propagate in the interior.

Cite this article

Moshe Marcus, Andrey Shishkov, Fading absorption in non-linear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 2, pp. 315–336

DOI 10.1016/J.ANIHPC.2012.08.002