Weighted Sobolev embedding with unbounded and decaying radial potentials

Su, Jiabao; Wang, Zhiqiang; Willem, Michel

doi:10.1016/j.jde.2007.03.018

Cited by 150 publications

(157 citation statements)

References 31 publications

Supporting

Mentioning

143

Contrasting

Unclassified

Order By: Relevance

“…More recently, the zero mass case of equations (1.1) with noncritical nonlinearities behaving as a single power has been widely studied in both the autonomous and nonautonomous cases (see e.g. [10,12,22,25,26,37] and [1,3,21,31,37] respectively, and the references therein), showing essentially that the existence of solutions relies on suitable compatibility conditions between the power of u and the growth and decaying rates of V (x) (and possibly of the nonlinearity) at zero and infinity.…”

Section: Introductionmentioning

confidence: 99%

Sum of weighted Lebesgue spaces and nonlinear elliptic equations

Badiale

Pisani

Rolando

2011

Nonlinear Differ. Equ. Appl.

View full text Add to dashboard Cite

Abstract.We study the sum of weighted Lebesgue spaces, by considering an abstract measure space (Ω, A, µ) and investigating the main properties of both the Banach spaceand the Nemytskiȋ operator defined on it. Then we apply our general results to prove existence and multiplicity of solutions to a class of nonlinear p-Laplacian equations of the formwhere V is a nonnegative measurable potential, possibly singular and vanishing at infinity, and f is a Carathéodory function satisfying a double-power growth condition in u. Mathematics Subject Classification (2010). Primary 35J62; Secondary 46E30.

show abstract

Section: Introductionmentioning

confidence: 99%

Sum of weighted Lebesgue spaces and nonlinear elliptic equations

Badiale

Pisani

Rolando

2011

Nonlinear Differ. Equ. Appl.

View full text Add to dashboard Cite

show abstract

“…When radial potentials are involved, the compactness of the embeddings may be valid for a wider range of q. In [13,14], the authors developed techniques and ideas in establishing weighted Sobolev type embeddings from W 1,p r (R N ; V ) into L q (R N ; Q) with singular radial potentials V and Q for p q. The embedding results in [13,14] include some cases that the embedding …”

Section: Introductionmentioning

confidence: 99%

“…In [12], the authors further explored the effects of the potentials and extend the techniques and ideas developed in [13,14] to establish the embedding W 1,p r (R N ; V ) → L q (R N ; Q) with 1 < q < p and then to study (P) with sub-p-linear nonlinearity. In this paper, we study (P) with bounded nonlinearity f by applying a compact embedding from W 1,p r (R N ; V ) into L 1 (R N ; Q), which will be stated in Sect.…”

Section: Introductionmentioning

confidence: 99%

Quasilinear elliptic equations on $${\mathbb{R}^{N}}$$ with singular potentials and bounded nonlinearity

2011

Z. Angew. Math. Phys.

Self Cite

View full text Add to dashboard Cite

We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equationwith singular radial potentials V, Q and bounded nonlinearity f . The approaches used here are based on a compact embedding from W 1,p r (R N ; V ) into L 1 (R N ; Q) and minimax methods. A uniqueness result is given for f ≡ 1. (2000). 35J05 · 35J20 · 35J60 · 58C20. Mathematics Subject Classification

show abstract

“…, second, a weighted Sobolev embedding for radial function proved by Su, Wang and Wilem [10] gives the inclusionḢ…”

mentioning

confidence: 94%

Sharp lower bounds for Coulomb energy

Bellazzini

Ghimenti

Ozawa

2016

Mathematical Research Letters

View full text Add to dashboard Cite

Abstract. We prove L p lower bounds for Coulomb energy for radially symmetric functions inḢ s (R 3 ) with 1 2 < s < 3 2 . In case 1 2 < s ≤ 1 we show that the lower bounds are sharp.In this paper we prove lower bounds for the Coulomb energy In the general case, without restricting to radial functions, the upper bound for the Coulomb energy is given by the well known Hardy-Littlewood-Sobolev inequality while lower bounds have been proved only very recently. In particular if one can control suitable homogeneous Sobolev spaceḢ s (R 3 ) the L p lower bound for the Coulomb energy is given by the following inequalitieswith θ = 6−5p 3−2ps−2p. Here the parameters s > 0 and 1 < p ≤ ∞ satisfy .We shall underline that in many physical applications involving Sobolev norms and Coulomb energy the radially symmetric assumption of ϕ is natural due to the rotational invariance of energy functionals (see e.g [8] in the context of stability of matter). Our purpose is to see if it is possible to control lower L p norms if one assumes radial symmetry of ϕ. In the sequel we use two theorems that are crucial for our improvement in case of radial symmetry. The first is the following pointwise decay for radial functions inḢis the weighted Lebesgue space with the normwhere θ = where Jd−2 2 is the Bessel function of order. The argument is similar to the one developed in [5] for the pointwise decay of radial function inḢ s (R d ), i.e to split ϕ into low and high frequency parts. The pointwise decay of high frequency part of ϕ will be controlled by the boundess of Sobolev norm while the decay of low frequency part by the boundness of the weighted Lebesgue norm.The second theorem is the following lower bound for the Coulomb energy by Ruiz, see [9]. , there exists c = c(α) > 0 such that for any measurable ϕ : R d → R we have SHARP LOWER BOUNDS FOR COULOMB ENERGY 3Let us defineFollowing the argument of Ruiz [9] it is easy to show that || · || E s is a norm andRuiz proved that for E 1 the following continuous embeddingThe result by Ruiz follows from two steps: first, Theorem 0.2 proves that , second, a weighted Sobolev embedding for radial function proved by Su, Wang and Wilem [10] gives the inclusionḢThe aim of our paper is to find continuous embeddings and hence better lower bounds for the Coulomb energy assuming radial symmetry when 1 2

show abstract

Weighted Sobolev embedding with unbounded and decaying radial potentials

Abstract: We prove embedding results of weighted W 1,p (R N ) spaces of radially symmetric functions. The results then are used to obtain ground and bound state solutions of quasilinear equations with unbounded or decaying radial potentials.

Cited by 150 publications

References 31 publications

Sum of weighted Lebesgue spaces and nonlinear elliptic equations

Sum of weighted Lebesgue spaces and nonlinear elliptic equations

Quasilinear elliptic equations on $${\mathbb{R}^{N}}$$ with singular potentials and bounded nonlinearity

Sharp lower bounds for Coulomb energy

Contact Info

Product

Resources

About