A Moser-Trudinger inequality on the boundary of a compact Riemann surface

Moser–Trudinger trace inequalities on a compact Riemannian surface with boundary

“…Theorem D [Li and Liu 2005]. Let (M, g) be a compact Riemannian surface with boundary ∂ M. Then (1-2) sup…”

“…Our method to prove Theorem 1.1 is similar to that of [Li and Liu 2005]. Precisely speaking, we divide the proof into two steps.…”

“…The weak compactness of L p (M) ( p > 1) gives the existence of the extremal function, i.e., (1-3) holds. It should be mentioned that x ε lies on ∂ M naturally in [Li and Liu 2005] because u ε is a harmonic function there. But in our case, passing to any subsequence, we cannot assume x ε ∈ ∂ M and u ε (x ε ) → +∞ simultaneously.…”

“…But in our case, passing to any subsequence, we cannot assume x ε ∈ ∂ M and u ε (x ε ) → +∞ simultaneously. Also, in the second step, the blowing up sequence we constructed (see Section 5) is different from that of [Li and Liu 2005].…”

Elliptic Equations and Systems with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition

“…Li et al [25][26][27], Ruf [38], Lu and Yang [30,31], etc. using the classical Critical Point Theory as first developed by Ambrosetti-Rabinowitz in their celebrated work [5].…”