Asymptotic properties of critical points for subcritical Trudinger-Moser functional

Abstract

Abstract On a smooth bounded domain we study the Trudinger-Moser functional E α ( u ) ≔ ∫ Ω ( e α u 2 − 1 ) d x , u ∈ H 1 ( Ω ) {E}_{\alpha }\left(u):= \mathop{\int }\limits_{\Omega }({e}^{\alpha {u}^{2}}-1){\rm{d}}x,\hspace{1.0em}u\in {H}^{1}\left(\Omega ) for α ∈ ( 0 , 2 π ) \alpha \in \left(0,2\pi ) and its restriction E α ∣ Σ λ {E}_{\alpha }{| }_{{\Sigma }_{\lambda }} , where Σ λ ≔ u ∈ H 1 ( Ω ) ∣ ∫ Ω ( ∣ ∇ u ∣ 2 + λ u 2 ) d x = 1 {\Sigma }_{\lambda }:= \left\{u\in {H}^{1}\left(\Omega )| {\int }_{\Omega }(| \nabla u{| }^{2}+\lambda {u}^{2}){\rm{d}}x=1\right\} for λ > 0 \lambda \gt 0 . By applying the asymptotic analysis and the variational method, we obtain asymptotic behavior of critical points of E α ∣ Σ λ {E}_{\alpha }{| }_{{\Sigma }_{\lambda }} both as λ → 0 \lambda \to 0 and as λ → + ∞ \lambda \to +\infty . In particular, we prove that when α \alpha is sufficiently small, maximizers for sup u ∈ Σ λ E α ( u ) {\sup }_{u\in {\Sigma }_{\lambda }}{E}_{\alpha }\left(u) tend to 0 in C ( Ω ¯ ) C\left(\overline{\Omega }) as λ → + ∞ \lambda \to +\infty .