Linear and semilikear eigenvalue problems in R<sup>n</sup>

Edelson, Allan L.; Rumbos, Adolfo J.

doi:10.1080/03605309308820928

Cited by 33 publications

(10 citation statements)

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“…In the next theorem, we exhibit another class of admissible weights. The analogous result for the first order Hardy-Sobolev inequalities is obtained in [14](see lemma 1.1).…”

Section: Introduction and Main Resultssupporting

confidence: 62%

On the generalized Hardy-Rellich inequalities

Anoop

Das²,

Sarkar

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this article, we look for the weight functions (say g) that admits the following generalized Hardy-Rellich type inequality:for some constant C > 0, where Ω is an open set in R N with N ≥ 1. We find various classes of such weight functions, depending on the dimension N and the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of D 2,2 0 (Ω) into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger. (2010): 35A23, 46E30, 46E35. Mathematics Subject ClassificationSports. f * (t) := ess sup f, t = 0 inf{s > 0 : α f (s) < t}, t > 0.

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“…In the next theorem, we exhibit another class of admissible weights. The analogous result for the first order Hardy-Sobolev inequalities is obtained in [14](see lemma 1.1).…”

Section: Introduction and Main Resultssupporting

confidence: 62%

On the generalized Hardy-Rellich inequalities

Anoop

Das²,

Sarkar

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…where f and h are continuous functions that satisfy some technical conditions. Here we point out that many estimates in our paper are totally different from those used in [10,11], because in the present work the problem is non-local, while in the previous papers the problem is local.…”

Section: Introduction and Main Resultsmentioning

confidence: 58%

“…However, we must be careful because, in the above paper, Ω is a smooth bounded domain, and thus it is possible to use compact embeddings for Sobolev spaces and Schauder spaces. Since we are working in whole R N , we need to show new estimates, and to this end, our inspiration comes from some papers by Edelson and Rumbos [10,11], where bifurcation theory has been used to study the existence of a solution for a problem of the type…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Existence of a Solution for a Non-Local Problem in ℝ^Nvia Bifurcation Theory

Alves¹,

Lima²,

Souto³

2018

Proceedings of the Edinburgh Mathematical Society

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In this paper, we study the existence of a solution for the following class of non-local problems: P$$\eqalign{\big\{&-\Delta u=\left(\lambda f(x)-\int_{{open R}^N}K(x,y)\vert u(y)\vert ^{\gamma}\hbox{d}y\right)u\quad \mbox{in } \R^{N}, \cr &\lim_{\vert x\vert \to +\infty}u(x)=0,\quad u \gt 0 \quad \text{in } {open R}^{N},}$$ where N ≥ 3, λ > 0, γ ∈ [1, 2), f : ℝ → ℝ is a positive continuous function and K : ℝN × ℝN → ℝ is a non-negative function. The functions f and K satisfy some conditions that permit us to use bifurcation theory to prove the existence of a solution for (P).

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“…The reader can find in the literature some recent papers involving bifurcation theory, see for example Alves, Delgado, Souto and Suárez , Alves, de Lima and Souto , Ambrosetti and Arcoya , Bartsch and Mandel , Brown and Stavrakakis , de Lima and Souto , Edelson and Rumbos , Kryszewski and Szulkin , Sun, Shi and Wang , Yan and Ma and their references.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Bifurcation properties for a class of fractional Laplacian equations in

Alves

Lima

Nóbrega

2018

Mathematische Nachrichten

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This paper concerns with the study of some bifurcation properties for the following class of nonlocal problems trueright90.0pt{(−Δ)su=λffalse(xfalse)false(u+h(u)false),indouble-struckRN,ufalse(xfalse)>0,forallx∈double-struckRN,falseprefixlim|x|→∞ufalse(xfalse)=0,where N>2s, s∈(0,1), λ>0, f:double-struckRN→R is a positive continuous function, h:R→R is a bounded continuous function and (−Δ)su is the fractional Laplacian. The main tools used are the Leray–Shauder degree theory and the global bifurcation result due to Rabinowitz.

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Linear and semilikear eigenvalue problems in Rⁿ

Cited by 33 publications

References 3 publications

On the generalized Hardy-Rellich inequalities

On the generalized Hardy-Rellich inequalities

Existence of a Solution for a Non-Local Problem in ℝ^Nvia Bifurcation Theory

Bifurcation properties for a class of fractional Laplacian equations in

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Linear and semilikear eigenvalue problems in Rn

Cited by 33 publications

References 3 publications

On the generalized Hardy-Rellich inequalities

On the generalized Hardy-Rellich inequalities

Existence of a Solution for a Non-Local Problem in ℝNvia Bifurcation Theory

Bifurcation properties for a class of fractional Laplacian equations in

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Linear and semilikear eigenvalue problems in Rⁿ

Existence of a Solution for a Non-Local Problem in ℝ^Nvia Bifurcation Theory