International Journal of Mathematics ( IF 0.604 ) Pub Date : 2020-08-20 , DOI: 10.1142/s0129167x20500858
Yanjun Liu

In this paper, suppose $F:ℝN→[0,+∞)$ be a convex function of class $C2(ℝN∖{0})$ which is even and positively homogeneous of degree 1. We establish the Lions type concentration-compactness principle of singular Trudinger–Moser Inequalities involving $N$-Finsler–Laplacian operator. Let $Ω⊂ℝN(N≥2)$ be a smooth bounded domain. ${un}⊂W01,N(Ω)$ be a sequence such that anisotropic Dirichlet norm$∫ΩFN(∇un)dx=1$, $un⇀u≢0$ weakly in $W01,N(Ω)$. Denote Then we have $∫ΩeλN(1−βN)p|un|NN−1Fo(x)βdx<+∞,$ where $0≤β, $λN=NNN−1κN1N−1$ and $κN$ is the volume of a unit Wulff ball. This conclusion fails if $p≥pN(u)$. Furthermore, we also obtain the corresponding concentration-compactness principle in the entire Euclidean space $ℝN$.

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