Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝn and n-Laplace Equations
By Caifeng Zhang, Lu Chen. 2017
58 citations recorded by the source corpus.
Abstract In this paper, we use the rearrangement-free argument, in the spirit of the work by Li, Lu and Zhu [25], on the concentration-compactness principle on the Heisenberg group to establish a sharpened version of the singular Lions concentration-compactness principle for the Trudinger–Moser inequality in ℝn{\mathbb{R}^{n}}. Then we prove a compact embedding theorem, which states that W1,n(ℝn){W^{1,n}(\mathbb{R}^{n})} is compactly embedded into Lp(ℝn,|x|-βdx){L^{p}(\mathbb{R}^{n},|x|^{-\beta}\,dx)} for p≥n{p\geq n} and 0<β<n{0<\beta<n}. As an application of the above results, we establish sufficient conditions for the existence of ground state solutions to the following n-Laplace equation with critical nonlinearity: ($*$) { - div ( | ∇ u | n - 2 ∇ u ) + V ( x ) | u | n - 2 u = f ( x , u ) | x | β , u ∈ W 1 , n ( ℝ n ) , u ≥ 0 , {}\left\{\begin{aligned} &\displaystyle{-}\operatorname{div}(|\nabla u|^{n-2}% \nabla u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^{\beta}},\\ &\displaystyle u\in W^{1,n}(\mathbb{R}^{n}),\quad u\geq 0,\end{aligned}\right. where V(x)≥c0{V(x)\geq c_{0}} for some positive constant c0{c_{0}} and f(x,t){f(x,t)} behaves like exp(α|t|nn-1){\exp(\alpha|t|^{\frac{n}{n-1}})} as t→+∞{t\rightarrow+\infty}. This work improves substantially related results found in the literature.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/c1fb45c3c30c25b7875091faf2d613f5e352f6d5
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In Applied Mathematics
In Computational Theory and Mathematics
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Citations in the corpus
- Trudinger–Moser Inequalities in Fractional Sobolev–Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation (2018, background): “For this case, the authors of [18, 52, 64] gave the existence of ground state solutions.”