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Ground state alternative for p-Laplacian with potential term

  • Yehuda Pinchover
  • Kyril Tintarev
Article

Abstract

Let Ω be a domain in \(\mathbb{R}^d\), d  ≥  2, and 1 <  p  <  ∞. Fix \(V \in L_{\mathrm{loc}}^\infty(\Omega)\). Consider the functional Q and its Gâteaux derivative Q′ given by \( Q(u) := \mathop \int_\Omega (|\nabla u|^p+V|u|^p){\rm d}x,\,\, \frac{1}{p}Q^\prime (u) := -\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u.\) If Q  ≥  0 on\(C_0^{\infty}(\Omega)\), then either there is a positive continuous function W such that \(\int W|u|^p\,\mathrm{d}x\leq Q(u)\) for all\(u\in C_0^{\infty}(\Omega)\), or there is a sequence \(u_k\in C_0^{\infty}(\Omega)\) and a function v > 0 satisfying Q′ (v) = 0, such that Q(u k ) → 0, and \(u_k\to v\) in \(L^p_\mathrm{loc}(\Omega)\). In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every \(\psi\in C_0^\infty(\Omega)\) satisfying \(\int \psi v\,{\rm d}x \neq 0\) there exists a constant C > 0 such that \(C^{-1}\int W|u|^p\,\mathrm{d}x\le Q(u)+C\left|\int u \psi\,\mathrm{d}x\right|^p\). As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.

Keywords

Quasilinear elliptic operator p-Laplacian Ground state Positive solutions Green function Isolated singularity 

Mathematics Subject Classification (2000)

Primary 35J20 Secondary 35J60 35J70 49R50 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of MathematicsUppsala UniversityUppsalaSweden

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