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Solutions to sublinear elliptic equations with finite generalized energy

  • Adisak Seesanea
  • Igor E. Verbitsky
Article

Abstract

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation
$$\begin{aligned} \mathcal {L}u = \sigma u^{q} + \mu \quad \text {in} \;\; \Omega , \end{aligned}$$
in the sublinear case \(0<q<1\), with finite generalized energy: \(\mathbb {E}_{\gamma }[u]:=\int _{\Omega } |\nabla u|^{2} u^{\gamma -1}dx<\infty \), for \(\gamma >0\). In this case \(u \in L^{\gamma +q}(\Omega , \sigma )\cap L^{\gamma }(\Omega , \mu )\), where \(\gamma =1\) corresponds to finite energy solutions. Here \(\mathcal {L}u:= -\,\text {div}(\mathcal {A}\nabla u)\) is a linear uniformly elliptic operator with bounded measurable coefficients, and \(\sigma \), \(\mu \) are nonnegative functions (or Radon measures), on an arbitrary domain \(\Omega \subseteq \mathbb {R}^n\) which possesses a positive Green function associated with \(\mathcal {L}\). When \(0<\gamma \le 1\), this result yields sufficient conditions for the existence of a positive solution to the above problem which belongs to the Dirichlet space \({\dot{W}}_{0}^{1,p}(\Omega )\) for \(1<p\le 2\).

Mathematics Subject Classification

Primary 35J61 42B37 Secondary 31B10 31B15 

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan
  2. 2.Department of MathematicsUniversity of MissouriColumbiaUSA

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