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Displaying similar documents to “Perturbed nonlinear degenerate problems in N

Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity

Olivier Rey, Juncheng Wei (2005)

Journal of the European Mathematical Society

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We show that the critical nonlinear elliptic Neumann problem Δ u μ u + u 7 / 3 = 0 in Ω , u > 0 in Ω , u ν = 0 on Ω , where Ω is a bounded and smooth domain in 5 , has arbitrarily many solutions, provided that μ > 0 is small enough. More precisely, for any positive integer K , there exists μ K > 0 such that for 0 < μ < μ K , the above problem has a nontrivial solution which blows up at K interior points in Ω , as μ 0 . The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella (2005)

Journal of the European Mathematical Society

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We consider the Yamabe type family of problems ( P ε ) : Δ u ε = u ε ( n + 2 ) / ( n 2 ) , u ε > 0 in A ε , u ε = 0 on A ε , where A ε is an annulus-shaped domain of n , n 3 , which becomes thinner as ε 0 . We show that for every solution u ε , the energy A ε | u | 2 as well as the Morse index tend to infinity as ε 0 . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on n , a half-space or an infinite strip. Our argument also involves a Liouville...

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In this paper we consider a nonlinear elliptic equation with critical growth for the operator Δ 2 in a bounded domain Ω n . We state some existence results when n 8 . Moreover, we consider 5 n 7 , expecially when Ω is a ball in n .

Recent results on stationary critical Kirchhoff systems in closed manifolds

Emmanuel Hebey, Pierre-Damien Thizy (2013-2014)

Séminaire Laurent Schwartz — EDP et applications

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We report on results we recently obtained in Hebey and Thizy [11, 12] for critical stationary Kirchhoff systems in closed manifolds. Let ( M n , g ) be a closed n -manifold, n 3 . The critical Kirchhoff systems we consider are written as a + b j = 1 p M | u j | 2 d v g Δ g u i + j = 1 p A i j u j = U 2 - 2 u i for all i = 1 , , p , where Δ g is the Laplace-Beltrami operator, A is a C 1 -map from M into the space M s p ( ) of symmetric p × p matrices with real entries, the A i j ’s are the components of A , U = ( u 1 , , u p ) , | U | : M is the Euclidean norm of U , 2 = 2 n n - 2 is the critical Sobolev exponent, and...

Positive solution for a quasilinear equation with critical growth in N

Lin Chen, Caisheng Chen, Zonghu Xiu (2016)

Annales Polonici Mathematici

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We study the existence of positive solutions of the quasilinear problem ⎧ - Δ N u + V ( x ) | u | N - 2 u = f ( u , | u | N - 2 u ) , x N , ⎨ ⎩ u(x) > 0, x N , where Δ N u = d i v ( | u | N - 2 u ) is the N-Laplacian operator, V : N is a continuous potential, f : × N is a continuous function. The main result follows from an iterative method based on Mountain Pass techniques.

A bifurcation theory for some nonlinear elliptic equations

Biagio Ricceri (2003)

Colloquium Mathematicae

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We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ( P λ ) ⎩ u Ω = 0 where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ( P λ ) admits a non-zero, non-negative strong solution u λ p 2 W 2 , p ( Ω ) such that l i m λ 0 | | u λ | | W 2 , p ( Ω ) = 0 for all p ≥ 2. Moreover, the function λ I λ ( u λ ) is negative and decreasing in ]0,λ*[, where I λ is the energy functional related to ( P λ ). ...

A priori bounds for positive radial solutions of quasilinear equations of Lane–Emden type

Soohyun Bae (2023)

Archivum Mathematicum

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We consider the quasilinear equation Δ p u + K ( | x | ) u q = 0 , and present the proof of the local existence of positive radial solutions near 0 under suitable conditions on K . Moreover, we provide a priori estimates of positive radial solutions near when r - K ( r ) for - p is bounded near .

Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight functions

Honghui Yin, Zuodong Yang (2011)

Annales Polonici Mathematici

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Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ - Δ p u + | u | p - 2 u = f 1 λ ( x ) | u | q - 2 u + 2 α / ( α + β ) g μ | u | α - 2 u | v | β , x ∈ Ω, ⎨ - Δ p v + | v | p - 2 v = f 2 λ ( x ) | v | q - 2 v + 2 β / ( α + β ) g μ | u | α | v | β - 2 v , x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and Ω N is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and f i λ i ( x ) = λ i f i + ( x ) + f i - ( x ) (i = 1,2) are sign-changing functions, where f i ± ( x ) = m a x ± f i ( x ) , 0 , g μ ( x ) = a ( x ) + μ b ( x ) , and Δ p u = d i v ( | u | p - 2 u ) denotes the p-Laplace operator. We use variational methods.

Critical points of the Moser-Trudinger functional on a disk

Andrea Malchiodi, Luca Martinazzi (2014)

Journal of the European Mathematical Society

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On the unit disk B 1 2 we study the Moser-Trudinger functional E ( u ) = B 1 e u 2 - 1 d x , u H 0 1 ( B 1 ) and its restrictions E | M Λ , where M Λ : = { u H 0 1 ( B 1 ) : u H 0 1 2 = Λ } for Λ > 0 . We prove that if a sequence u k of positive critical points of E | M Λ k (for some Λ k > 0 ) blows up as k , then Λ k 4 π , and u k 0 weakly in H 0 1 ( B 1 ) and strongly in C loc 1 ( B ¯ 1 { 0 } ) . Using this fact we also prove that when Λ is large enough, then E | M Λ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.

L p , q spaces

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the L p , q spaces................................. 114. Integral representation of bounded linear functionals on L p , q ( B ) ........ 235. Examples in L p , q theory...................................................................................

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

A localization property for B p q s and F p q s spaces

Hans Triebel (1994)

Studia Mathematica

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Let f j = k a k f ( 2 j + 1 x - 2 k ) , where the sum is taken over the lattice of all points k in n having integer-valued components, j∈ℕ and a k . Let A p q s be either B p q s or F p q s (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on n . The aim of the paper is to clarify under what conditions f j | A p q s is equivalent to 2 j ( s - n / p ) ( k | a k | p ) 1 / p f | A p q s .

Generalized versions of Ilmanen lemma: Insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove that for a normed linear space X , if f 1 : X is continuous and semiconvex with modulus ω , f 2 : X is continuous and semiconcave with modulus ω and f 1 f 2 , then there exists f C 1 , ω ( X ) such that f 1 f f 2 . Using this result we prove a generalization of Ilmanen lemma (which deals with the case ω ( t ) = t ) to the case of an arbitrary nontrivial modulus ω . This generalization (where a C l o c 1 , ω function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

Location of the critical points of certain polynomials

Somjate Chaiya, Aimo Hinkkanen (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let 𝔻 denote the unit disk { z : | z | < 1 } in the complex plane . In this paper, we study a family of polynomials P with only one zero lying outside 𝔻 ¯ .  We establish  criteria for P to satisfy implying that each of P and P '   has exactly one critical point outside 𝔻 ¯ .

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Theoretical analysis for 1 - 2 minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

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We investigate the recovery of k -sparse signals using the 1 - 2 minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume k -sparse signals 𝐱 with the prior support T which is composed of g true indices and b wrong...

On hereditary normality of ω * , Kunen points and character ω 1

Sergei Logunov (2021)

Commentationes Mathematicae Universitatis Carolinae

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We show that ω * { p } is not normal, if p is a limit point of some countable subset of ω * , consisting of points of character ω 1 . Moreover, such a point p is a Kunen point and a super Kunen point.

A compactness result for polyharmonic maps in the critical dimension

Shenzhou Zheng (2016)

Czechoslovak Mathematical Journal

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For n = 2 m 4 , let Ω n be a bounded smooth domain and 𝒩 L a compact smooth Riemannian manifold without boundary. Suppose that { u k } W m , 2 ( Ω , 𝒩 ) is a sequence of weak solutions in the critical dimension to the perturbed m -polyharmonic maps d d t | t = 0 E m ( Π ( u + t ξ ) ) = 0 with Φ k 0 in ( W m , 2 ( Ω , 𝒩 ) ) * and u k u weakly in W m , 2 ( Ω , 𝒩 ) . Then u is an m -polyharmonic map. In particular, the space of m -polyharmonic maps is sequentially compact for the weak- W m , 2 topology.

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

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Let Ω be a bounded domain of class C 2 in N and let K be a compact subset of Ω . Assume that q ( N + 1 ) / ( N 1 ) and denote by U K the maximal solution of Δ u + u q = 0 in Ω which vanishes on Ω K . We obtain sharp upper and lower estimates for U K in terms of the Bessel capacity C 2 / q , q ' and prove that U K is σ -moderate. In addition we describe the precise asymptotic behavior of U K at points σ K , which depends on the “density” of K at σ , measured in terms of the capacity C 2 / q , q ' .