NDLI: Eigenvalue problems with weights in Lorentz spaces

Content Provider	Springer Nature Link
Author	Ramaswamy, Mythily Lucia, Marcello Aop, T. V.
Copyright Year	2009
Abstract	Given V, w locally integrable functions on a general domain Ω with V ≥ 0 but w allowed to change sign, we study the existence of ground states for the nonlinear eigenvalue problem: $$-\Delta u + V u = \lambda w \|u\|^{p-2} u, \quad u\|_{\partial \Omega} =0,$$ with p subcritical. These are minimizers of the associated Rayleigh quotient whose existence is ensured under suitable assumptions on the weight w. In the present paper we show that an admissible space of weight functions is provided by the closure of smooth functions with compact support in the Lorentz space $${L(\tilde p,\infty)}$$ with $${\frac{1}{{\widetilde p}} + \frac{p}{2^{\star}} =1}$$ . This generalizes previous results and gives new sufficient conditions ensuring existence of extremals for generalized Hardy–Sobolev inequalities. The existence in such a generality of a principal eigenfunction in the linear case p = 2 is applied to study the bifurcation for semilinear problems of the type $$-\Delta u= \lambda (a(x)u + b(x) r(u)),$$ where a, b are indefinite weights belonging to some Lorentz spaces, and the function r has subcritical growth at infinity.
Ending Page	376
Page Count	22
Starting Page	355
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	3
Volume Number	36
Language	English
Publisher	Springer-Verlag
Publisher Date	2009-03-18
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Spaces of measurable functions ( $L^p$ -spaces, Orlicz spaces, KÃ¶the function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Calculus of Variations and Optimal Control; Optimization Systems Theory, Control Theoretical, Mathematical and Computational Physics Analysis Variational methods for second-order elliptic equations Bifurcation General topics in linear spectral theory
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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