On Kirchhoff type problems involving critical and singular nonlinearities

Chun-Yu Lei; Chang-Mu Chu; Hong-Min Suo; Chun-Lei Tang

Displaying similar documents to “On Kirchhoff type problems involving critical and singular nonlinearities”

Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent

Lan Zeng, Chun Lei Tang (2016)

Annales Polonici Mathematici

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We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ $- [a + b (\int_{Ω} {| \nabla u | ² d x)}^{m} {] Δ u = f (x, u) + | u |}^{2 * - 2} u$ in Ω, ⎨ ⎩ u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.

Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponent

Jia-Feng Liao, Jiu Liu, Peng Zhang, Chun-Lei Tang (2016)

Annales Polonici Mathematici

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We study the following singular elliptic equation with critical exponent ⎧ $- Δ u = Q (x) u^{2 * - 1} + λ u^{- γ}$ in Ω, ⎨u > 0 in Ω, ⎩u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N≥3) is a smooth bounded domain, and λ > 0, γ ∈ (0,1) are real parameters. Under appropriate assumptions on Q, by the constrained minimizer and perturbation methods, we obtain two positive solutions for all λ > 0 small enough.

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 Lincei. Matematica e Applicazioni

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In this paper we consider a nonlinear elliptic equation with critical growth for the operator $\Delta^{2}$ in a bounded domain $\Omega\subset\mathbb{R}^{n}$ . We state some existence results when $n\geq 8$ . Moreover, we consider $5\leq n\leq 7$ , expecially when $\Omega$ is a ball in $\mathbb{R}^{n}$ .

On the nonlinear Neumann problem at resonance with critical Sobolev nonlinearity

J. Chabrowski, Shusen Yan (2002)

Colloquium Mathematicae

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We consider the Neumann problem for the equation $- Δ u - λ u = {Q (x) | u |}^{2 * - 2} u$ , u ∈ H¹(Ω), where Q is a positive and continuous coefficient on Ω̅ and λ is a parameter between two consecutive eigenvalues $λ_{k - 1}$ and $λ_{k}$ . Applying a min-max principle based on topological linking we prove the existence of a solution.

Singular $φ$ -Laplacian third-order BVPs with derivative dependance

Smaïl Djebali, Ouiza Saifi (2016)

Archivum Mathematicum

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This work is devoted to the existence of solutions for a class of singular third-order boundary value problem associated with a $φ$ -Laplacian operator and posed on the positive half-line; the nonlinearity also depends on the first derivative. The upper and lower solution method combined with the fixed point theory guarantee the existence of positive solutions when the nonlinearity is monotonic with respect to its arguments and may have a space singularity; however no Nagumo type condition...

Existence of three solutions to a double eigenvalue problem for the p-biharmonic equation

Lin Li, Shapour Heidarkhani (2012)

Annales Polonici Mathematici

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Using a three critical points theorem and variational methods, we study the existence of at least three weak solutions of the Navier problem ⎧ ${Δ (| Δ u |}^{p - 2} {Δ u) - d i v (| \nabla u |}^{p - 2} \nabla u) = λ f (x, u) + μ g (x, u)$ in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N ≥ 1) is a non-empty bounded open set with a sufficiently smooth boundary ∂Ω, λ > 0, μ > 0 and f,g: Ω × ℝ → ℝ are two L¹-Carathéodory functions.

Linking and the Morse complex

Michael Usher (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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For a Morse function $f$ on a compact oriented manifold $M$ , we show that $f$ has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in $M$ whose components have nontrivial linking number, such that the minimal value of $f$ on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of $f$ in terms of the Betti numbers of $M$ and the behavior of $f$ with respect...

Convergence of minimax structures and continuation of critical points for singularly perturbed systems

Benedetta Noris, Hugo Tavares, Susanna Terracini, Gianmaria Verzini (2012)

Journal of the European Mathematical Society

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In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system $- Δ u + u^{3} + β u v^{2} = λ u, - Δ v + v^{3} + β u^{2} v = μ v, u, v \in H_{0}^{1} (Ω), u, v > 0$ , as the interspecies scattering length $β$ goes to $+ \infty$ . For this system we consider the associated energy functionals $J_{β}, β \in (0, + \infty)$ , with $L^{2}$ -mass constraints, which limit $J_{\infty}$ (as $β \to + \infty$ ) is strongly irregular. For such functionals, we construct multiple critical points...

On cusps and flat tops

Neil Dobbs (2014)

Annales de l’institut Fourier

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Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to $C^{1 + ϵ}$ . The critical points are not required to verify a non-flatness condition, so the results are applicable to $C^{1 + ϵ}$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.

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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Existence of renormalized solutions for parabolic equations without the sign condition and with three unbounded nonlinearities

Y. Akdim, J. Bennouna, M. Mekkour, H. Redwane (2012)

Applicationes Mathematicae

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We study the problem ∂b(x,u)/∂t - div(a(x,t,u,Du)) + H(x,t,u,Du) = μ in Q = Ω×(0,T), ${b (x, u) |}_{t = 0} = b (x, u ₀)$ in Ω, u = 0 in ∂Ω × (0,T). The main contribution of our work is to prove the existence of a renormalized solution without the sign condition or the coercivity condition on H(x,t,u,Du). The critical growth condition on H is only with respect to Du and not with respect to u. The datum μ is assumed to be in $L ¹ (Q) + L^{p^{'}} (0, T; W^{- 1, p^{'}} (Ω))$ and b(x,u₀) ∈ L¹(Ω).

Sharp $L^{1}$ estimates for singular transport equations

Sergiu Klainerman, Igor Rodnianski (2008)

Journal of the European Mathematical Society

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We provide $L^{1}$ estimates for a transport equation which contains singular integral operators. The form of the equation was motivated by the study of Kirchhoff–Sobolev parametrices in a Lorentzian space-time satisfying the Einstein equations. While our main application is for a specific problem in General Relativity we believe that the phenomenon which our result illustrates is of a more general interest.

Asymptotic properties of ground states of scalar field equations with a vanishing parameter

Vitaly Moroz, Cyrill B. Muratov (2014)

Journal of the European Mathematical Society

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We study the leading order behaviour of positive solutions of the equation $- Δ u + ϵ u - {| u |}^{p - 2} u + {| u |}^{q - 2} u = 0, x \in ℝ^{N}$ , where $N \geq 3$ , $q > p > 2$ and when $ϵ > 0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$ , $q$ and $N$ . The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $2^{*} = \frac{2 N}{N - 2}$ . For $p < 2^{*}$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p > 2^{*}$ the solution asymptotically...

Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential

Jaeyoung Byeon, Kazunaga Tanaka (2013)

Journal of the European Mathematical Society

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We consider a singularly perturbed elliptic equation $ϵ^{2} Δ u - V (x) u + f (u) = 0, u (x) > 0$ on $ℝ^{N}$ , ${𝚕𝚒𝚖}_{|x| \to \infty} u (x) = 0$ , where $V (x) > 0$ for any $x \in ℝ^{N}$ . The singularly perturbed problem has corresponding limiting problems $Δ U - c U + f (U) = 0, U (x) > 0$ on $ℝ^{N}$ , ${𝚕𝚒𝚖}_{|x| \to \infty} U (x) = 0, c > 0$ . Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity $f$ for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential $V$ under possibly general conditions...

On asymptotic critical values and the Rabier Theorem

Zbigniew Jelonek (2004)

Banach Center Publications

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Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let $f = (f ₁, . . ., f_{m}) : X \to k^{m}$ be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with...

Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in $ℝ^{N}$

Caisheng Chen, Hongxue Song (2016)

Applications of Mathematics

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In this work, we study the existence of nonnegative and nontrivial solutions for the quasilinear Schrödinger equation $- Δ_{N} u + b {| u |}^{N - 2} u - Δ_{N} (u^{2}) u = h (u), x \in ℝ^{N},$ where $Δ_{N}$ is the $N$ -Laplacian operator, $h (u)$ is continuous and behaves as $exp (α | u |^{N / (N - 1)})$ when $| u | \to \infty$ . Using the Nehari manifold method and the Schwarz symmetrization with some special techniques, the existence of a nonnegative and nontrivial solution $u (x) \in W^{1, N} (ℝ^{N})$ with $u (x) \to 0$ as $| x | \to \infty$ is established.

Lieb–Thirring inequalities on the half-line with critical exponent

Tomas Ekholm, Rupert Frank (2008)

Journal of the European Mathematical Society

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We consider the operator $- d^{2} / d r^{2} - V$ in $L_{2} (ℝ_{+})$ with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound $tr {(- d^{2} / d r^{2} - V)}_{-}^{γ} \leq C_{γ, α} \int_{ℝ_{+}} {(V (r) - 1 / (4 r^{2}))}_{+}^{γ + (1 + α) / 2} r^{α} d r$ for any $α \in [0, 1)$ and $γ \geq (1 - α) / 2$ . This includes a Lieb-Thirring inequality in the critical endpoint case.

Bubbling along boundary geodesics near the second critical exponent

Manuel del Pino, Monica Musso, Frank Pacard (2010)

Journal of the European Mathematical Society

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The role of the second critical exponent $p = (n + 1) / (n - 3)$ , the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem $Δ u + u^{p} = 0$ , $u > 0$ under zero Dirichlet boundary conditions, in a domain $Ω$ in $ℝ^{n}$ with bounded, smooth boundary. Given $Γ$ , a geodesic of the boundary with negative inner normal curvature we find that for $p = (n + 1) / (n - 3 - ε)$ , there exists a solution $u_{ε}$ such that $| \nabla u_{ε} |^{2}$ converges weakly to a Dirac measure on $Γ$ as $ε \to 0^{+}$ , provided that $Γ$ is nondegenerate in the sense of second...

A singular initial value problem for the equation $u^{(n)} (x) = g (u (x))$

Wojciech Mydlarczyk (1998)

Annales Polonici Mathematici

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We consider the problem of the existence of positive solutions u to the problem $u^{(n)} (x) = g (u (x))$ , $u (0) = u^{'} (0) = . . . = u^{(n - 1)} (0) = 0$ (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition $\int ₀^{δ} 1 / s {[s / g (s)]}^{1 / n} d s < \infty$ is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.

Behaviour of $L^{r}$ -Dini singular integrals in weighted $L^{1}$ spaces

Osvaldo Capri, Carlos Segovia (1989)

Studia Mathematica

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Real $C^{k}$ Koebe principle

Weixiao Shen, Michael Todd (2005)

Fundamenta Mathematicae

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We prove a $C^{k}$ version of the real Koebe principle for interval (or circle) maps with non-flat critical points.

Commutators on ${(\sum ℓ_{q})}_{p}$

Dongyang Chen, William B. Johnson, Bentuo Zheng (2011)

Studia Mathematica

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Let T be a bounded linear operator on $X = {(\sum ℓ_{q})}_{p}$ with 1 ≤ q < ∞ and 1 < p < ∞. Then T is a commutator if and only if for all non-zero λ ∈ ℂ, the operator T - λI is not X-strictly singular.

Common zero sets of equivalent singular inner functions

Keiji Izuchi (2004)

Studia Mathematica

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Let μ and λ be bounded positive singular measures on the unit circle such that μ ⊥ λ. It is proved that there exist positive measures μ₀ and λ₀ such that μ₀ ∼ μ, λ₀ ∼ λ, and $| ψ_{μ ₀} | < 1 \cap | ψ_{λ ₀} | < 1 = \emptyset$ , where $ψ_{μ}$ is the associated singular inner function of μ. Let $(μ) = ⋂_{ν; ν \sim μ} Z (ψ_{ν})$ be the common zeros of equivalent singular inner functions of $ψ_{μ}$ . Then (μ) ≠ ∅ and (μ) ∩ (λ) = ∅. It follows that μ ≪ λ if and only if (μ) ⊂ (λ). Hence (μ) is the set in the maximal ideal space of $H^{\infty}$ which relates naturally to the set of measures equivalent...

Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity

Djairo Guedes de Figueiredo, Jean-Pierre Gossez, Pedro Ubilla (2006)

Journal of the European Mathematical Society

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We study the existence, nonexistence and multiplicity of positive solutions for the family of problems $- Δ u = f_{λ} (x, u)$ , $u \in H_{0}^{1} (Ω)$ , where $Ω$ is a bounded domain in $ℝ^{N}$ , $N \geq 3$ and $λ > 0$ is a parameter. The results include the well-known nonlinearities of the Ambrosetti–Brezis–Cerami type in a more general form, namely $λ a (x) u^{q} + b (x) u^{p}$ , where $0 \leq q < 1 < p \leq 2^{*} - 1$ . The coefficient $a (x)$ is assumed to be nonnegative but $b (x)$ is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential...

The Cauchy problem for the liquid crystals system in the critical Besov space with negative index

Sen Ming, Han Yang, Zili Chen, Ls Yong (2017)

Czechoslovak Mathematical Journal

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The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space ${\dot{B}}_{p, 1}^{n / p - 1} (ℝ^{n}) \times {\dot{B}}_{p, 1}^{n / p} (ℝ^{n})$ with $n < p < 2 n$ is established by using the heat semigroup theory and the Littlewood-Paley theory. The global well-posedness for the system is obtained with small initial datum by using the fixed point theorem. The blow-up results for strong solutions to the system are also analysed.

Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition

Zonghu Xiu, Caisheng Chen (2013)

Annales Polonici Mathematici

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We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ $- {d i v (| x |}^{- a p} {| \nabla u |}^{p - 2} {\nabla u) + h (x) | u |}^{p - 2} u = g (x) {| u |}^{r - 2} u$ , x ∈ Ω, ⎨ ⎩ ${| x |}^{- a p} {| \nabla u |}^{p - 2} \partial u / \partial ν = λ f (x) {| u |}^{q - 2} u$ , x ∈ ∂Ω, where Ω is an exterior domain in $ℝ^{N}$ , that is, $Ω = ℝ^{N} ∖ D$ , where D is a bounded domain in $ℝ^{N}$ with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function...

Optimal embeddings of critical Sobolev-Lorentz-Zygmund spaces

Hidemitsu Wadade (2014)

Studia Mathematica

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We establish the embedding of the critical Sobolev-Lorentz-Zygmund space $H_{p, q, λ ₁, . . ., λ ₘ}^{n / p} (ℝ ⁿ)$ into the generalized Morrey space $ℳ_{Φ, r} (ℝ ⁿ)$ with an optimal Young function Φ. As an application, we obtain the almost Lipschitz continuity for functions in $H_{p, q, λ ₁, . . ., λ ₘ}^{n / p + 1} (ℝ ⁿ)$ . O’Neil’s inequality and its reverse play an essential role in the proofs of the main theorems.

Rigidity of critical circle mappings I

Edson de Faria, Welington de Melo (1999)

Journal of the European Mathematical Society

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We prove that two $C^{3}$ critical circle maps with the same rotation number in a special set $𝔸$ are $C^{1 + α}$ conjugate for some $α > 0$ provided their successive renormalizations converge together at an exponential rate in the $C^{0}$ sense. The set $𝔸$ has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of $C^{\infty}$ critical circle maps with the same rotation number that are not $C^{1 + β}$ conjugate for any $β > 0$ . The class of rotation numbers for which such examples exist...