NDLI: Ground state alternative for p-Laplacian with potential term

Content Provider	SpringerLink
Author	Pinchover, Yehuda Tintarev, Kyril
Copyright Year	2006
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.
Ending Page	201
Page Count	23
Starting Page	179
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	2
Volume Number	28
Language	English
Publisher	Springer-Verlag
Publisher Date	2006-07-15
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	p-Laplacian Systems Theory, Control Mathematical and Computational Physics Variational methods for second-order elliptic equations Ground state Nonlinear elliptic equations Isolated singularity Green function Degenerate elliptic equations Quasilinear elliptic operator Positive solutions Calculus of Variations and Optimal Control; Optimization Analysis
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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