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Nehari manifold
About: Nehari manifold is a research topic. Over the lifetime, 721 publications have been published within this topic receiving 17514 citations.
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TL;DR: In this paper, general existence theorems for critical points of a continuously differentiable functional I on a real Banach space are given for the case in which I is even.
Abstract: This paper contains some general existence theorems for critical points of a continuously differentiable functional I on a real Banach space. The strongest results are for the case in which I is even. Applications are given to partial differential and integral equations.
4,081 citations
TL;DR: In this article, the existence of a fonction u satisfaisant l'equation elliptique non lineaire is investigated, i.e., a domaine borne in R n avec n ≥ 3.
Abstract: Soit Ω un domaine borne dans R n avec n≥3 On etudie l'existence d'une fonction u satisfaisant l'equation elliptique non lineaire -Δu=u P +f(x,u) sur Ω, u>0 sur Ω, u=0 sur ∂Ω, ou p=(n+2)/(n−2), f(x,0)=0 et f(x,u) est une perturbation de u P de bas ordre au sens ou lim u→+α f(x,u)/u P =0
2,676 citations
TL;DR: The variational principle states that if a differentiable function F has a finite lower bound (although it need not attain it), then, for every E > 0, there exists some point u( where 11 F'(uJj* < l, i.e., its derivative can be made arbitrarily small as discussed by the authors.
Abstract: The variational principle states that if a differentiable functional F attains its minimum at some point zi, then F’(C) = 0; it has proved a valuable tool for studying partial differential equations. This paper shows that if a differentiable function F has a finite lower bound (although it need not attain it), then, for every E > 0, there exists some point u( where 11 F’(uJj* < l , i.e., its derivative can be made arbitrarily small. Applications are given to Plateau’s problem, to partial differential equations, to nonlinear eigenvalues, to geodesics on infinite-dimensional manifolds, and to control theory.
2,105 citations
TL;DR: In this paper, a variational approach is proposed to solve a class of Schrodinger equations involving the fractional Laplacian, which is variational in nature and based on minimization on the Nehari manifold.
Abstract: We construct solutions to a class of Schrodinger equations involving the fractional Laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.
419 citations
TL;DR: In this article, it was shown that for f ∈ H − 1 satisfying a suitable condition and f ≠ 0, the Dirichlet problem admits two solutions u 0 and u 1 in H 0 1 ( Ω ).
Abstract: Let p = 2 N N − 2 , N ≧ 3 be the limiting Sobolev exponent and Ω ⊂ ℝ N open bounded set. We show that for f ∈ H −1 satisfying a suitable condition and f ≠ 0, the Dirichlet problem: { − Δ u = | u | p − 2 u + f on Ω u = 0 on ∂ Ω admits two solutions u 0 and u 1 in H 0 1 ( Ω ) . Also u 0 ≧ 0 and u 1 ≧ 0 for f ≧ 0. Notice that, in general, this is not the case if f = 0 ( see [P]).
417 citations