Concentration compactness of Moser functionals on manifolds

Li, Yuxiang

doi:10.1007/s10455-007-9062-z

Cited by 3 publications

(1 citation statement)

References 12 publications

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“…This type of inequality was first proved in [22] on compact Riemannian manifolds of any dimension n. When the dimension is two, this inequality was proved in [24] on manifolds with and without boundary, and the existence of extremal functions was established. We refer also to [25,26,34,35] for related results and generalizations.…”

Section: Introductionmentioning

confidence: 99%

Bubbling solutions for Moser–Trudinger type equations on compact Riemann surfaces

Figueroa

Musso

2018

Journal of Functional Analysis

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We study an elliptic equation related to the Moser-Trudinger inequality on a compact Riemann surface (S, g),where λ > 0 is a small parameter, |S| is the area of S, ∆g is the Laplace-Beltrami operator and dvg is the area element. Given any integer k ≥ 1, under general conditions on S we find a bubbling solution u λ which blows up at exactly k points in S, as λ → 0. When S is a flat two-torus in rectangular form, we find that either seven or nine families of such solutions do exist for k = 2. In particular, in any square flat two-torus actually nine families of bubbling solutions with two bubbling points do exist. If S is a Riemann surface with non-constant Robin's function then at least two bubbling solutions with k = 1 exists.

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Section: Introductionmentioning

confidence: 99%

Bubbling solutions for Moser–Trudinger type equations on compact Riemann surfaces

Figueroa

Musso

2018

Journal of Functional Analysis

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Concentration-Compactness principle for Trudinger–Moser inequalities with logarithmic weights and their applications

Zhang

2020

Nonlinear Analysis

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Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ⁴

Chen

et al. 2018

Advanced Nonlinear Studies

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In this paper, we establish a sharp concentration-compactness principle associated with the singular Adams inequality on the second-order Sobolev spaces in {\mathbb{R}^{4}} . We also give a new Sobolev compact embedding which states {W^{2,2}(\mathbb{R}^{4})} is compactly embedded into {L^{p}(\mathbb{R}^{4},|x|^{-\beta}\,dx)} for {p\geq 2} and {0<\beta<4} . As applications, we establish the existence of ground state solutions to the following bi-Laplacian equation with critical nonlinearity: \displaystyle\Delta^{2}u+V(x)u=\frac{f(x,u)}{|x|^{\beta}}\quad\mbox{in }% \mathbb{R}^{4}, where {V(x)} has a positive lower bound and {f(x,t)} behaves like {\exp(\alpha|t|^{2})} as {t\to+\infty} . In the case {\beta=0} , because of the loss of Sobolev compact embedding, we use the principle of symmetric criticality to obtain the existence of ground state solutions by assuming {f(x,t)} and {V(x)} are radial with respect to x and {f(x,t)=o(t)} as {t\rightarrow 0} .

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Concentration compactness of Moser functionals on manifolds

Abstract: We will prove a concentration compactness property of the Moser functional on a compact Riemannian manifold.

Cited by 3 publications

References 12 publications

Bubbling solutions for Moser–Trudinger type equations on compact Riemann surfaces

Bubbling solutions for Moser–Trudinger type equations on compact Riemann surfaces

Concentration-Compactness principle for Trudinger–Moser inequalities with logarithmic weights and their applications

Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ⁴

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Concentration compactness of Moser functionals on manifolds

Abstract: We will prove a concentration compactness property of the Moser functional on a compact Riemannian manifold.

Cited by 3 publications

References 12 publications

Bubbling solutions for Moser–Trudinger type equations on compact Riemann surfaces

Bubbling solutions for Moser–Trudinger type equations on compact Riemann surfaces

Concentration-Compactness principle for Trudinger–Moser inequalities with logarithmic weights and their applications

Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4

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Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ⁴