On the resonant Lane–Emden problem for the p-Laplacian

Author:

Ercole Grey1

Affiliation:

1. Departamento de Matemática – ICEx, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, 30161-970, Belo Horizonte, Minas Gerais, Brazil

Abstract

We study the positive solutions of the Lane–Emden problem -Δpu = λp|u|q-2u in Ω, u = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded and smooth domain, N ≥ 2, λp is the first eigenvalue of the p-Laplacian operator Δp, p > 1, and q is close to p. We prove that any family of positive solutions of this problem converges in [Formula: see text] to the function θpep when q → p, where ep is the positive and L-normalized first eigenfunction of the p-Laplacian and [Formula: see text]. A consequence of this result is that the best constant of the immersion [Formula: see text] is differentiable at q = p. Previous results on the asymptotic behavior (as q → p) of the positive solutions of the nonresonant Lane–Emden problem (i.e. with λp replaced by a positive λ ≠ λp) are also generalized to the space [Formula: see text] and to arbitrary families of these solutions. Moreover, if uλ,q denotes a solution of the nonresonant problem for an arbitrarily fixed λ > 0, we show how to obtain the first eigenpair of the p-Laplacian as the limit in [Formula: see text], when q → p, of a suitable scaling of the pair (λ, uλ,q). For computational purposes the advantage of this approach is that λ does not need to be close to λp. Finally, an explicit estimate involving L- and L1-norms of uλ,q is also derived using set level techniques. It is applied to any ground state family {vq} in order to produce an explicit upper bound for ‖vq which is valid for q ∈ [1, p + ϵ] where [Formula: see text].

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,General Mathematics

Cited by 8 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Hardy-Lane-Emden inequalities for $$\pmb {p}$$-Laplacian on arbitrary domains;Nonlinear Differential Equations and Applications NoDEA;2022-06-22

2. The limiting behaviour of solutions to a family of eigenvalue problems;Complex Variables and Elliptic Equations;2019-02-20

3. On a singular minimizing problem;Journal d'Analyse Mathématique;2018-06

4. On the existence threshold for positive solutions of p-Laplacian equations with a concave–convex nonlinearity;Communications in Contemporary Mathematics;2015-10-29

5. Remarks on the behavior of the best Sobolev constants;Contributions to Nonlinear Elliptic Equations and Systems;2015

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