1. A sharp Moser-Trudinger type inequality involving Lp norm in [formula omitted] with degenerate potential.
- Author
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Sun, Jingxuan, Song, Zhen, and Zou, Wenming
- Subjects
- *
SOBOLEV spaces , *EIGENVALUES - Abstract
In this paper, we establish a Moser-Trudinger type inequality with a special degenerate potential V. Specifically, we demonstrate that for any potentials V (x) satisfying the conditions (V 1) and (V 2) , the following inequality MT (V , α , p) ≔ sup u ∈ E , ‖ u ‖ ≤ 1 ∫ R n Φ (α n (1 + α ‖ u ‖ p n) 1 n − 1 u n n − 1 ) d x < ∞ holds if α < λ 1 (V) ; where E is the usual weighted Sobolev space associated to the potential V , the norm ‖ u ‖ n : = ∫ R n | ∇ u | n + V (x) | u | n d x , and λ 1 (V) is the first Dirichlet eigenvalue of the corresponding operator − Δ + V. The potential conditions are as follows. (V 1) 0 ≤ m = inf x ∈ R n V (x) < sup x ∈ R n V (x) = lim | x | → ∞ V (x) = M ≤ ∞. (V 2) V is radially symmetric, non-decreasing. Besides, we also prove that for any potentials V (x) satisfying the condition (V 1) merely and for p = n , the previous inequality is still valid. In addition, this inequality is sharp in the sense that if p > n and α ≥ λ 1 (V) , there holds MT (V , α , p) = ∞. Meanwhile, if α > λ 1 (V) , then MT (V , α , n) = ∞. Furthermore, via a subtle blow-up analysis, we also prove the existence of an extremal function for the inequality above when α is sufficiently small. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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