Asymptotic properties of critical points for subcritical Trudinger-Moser functional

Abstract

On a smooth bounded domain we study the Trudinger-Moser functional Eα(u)≔∫Ω(eαu2−1)dx,u∈H1(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∈(0,2π)\alpha \in \left(0,2\pi ) and its restriction Eα∣Σλ{E}_{\alpha }{| }_{{\Sigma }_{\lambda }}, where Σλ≔u∈H1(Ω)∣∫Ω(∣∇u∣2+λu2)dx=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 Eα∣Σλ{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 supu∈ΣλEα(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 .

Keywords

• asymptotic behavior
• neumann problem
• subcritical
• trudinger-moser inequality
• two dimension
• primary 35b38
• secondary 35a09
• 35b40
• 35j15
• 35j61