Vanishing-concentration-compactness alternative for critical Sobolev embedding with a general integrand in $\mathbb R^2$

Abstract

A maximizing problem associated with the Moser–Pohozaev–Trudinger–Yudovich type inequality proved in Ruf (J Funct Anal 219:340–367, 2005) states:

$$\begin{aligned} d(\alpha ):=\sup _{u\in H^1(R^2), \,\Vert u\Vert _{H^1}=1} \int _{R^2}(e^{\alpha |u|^2}-1)<\infty \end{aligned}$$

for $0<\alpha \le 4\pi$. Do Ó–Sani–Tarsi (Commun Contemp Math 20:1650036, 2018) established the vanishing-concentration-compactness (VCC) alternative with respect to any maximizing sequence for $d(\alpha )$. In this paper, we consider a maximizing problem with a general integrand:

$$\begin{aligned} d_\varPhi (\alpha ):=\sup _{u\in H^1(R^2), \,\Vert u\Vert _{H^1}=1}\int _{R^2}\varPhi _\alpha (|u|), \end{aligned}$$

where $\varPhi \in C([0,\infty ))$ with $\alpha >0$ and $\varPhi _\alpha (s):=\varPhi (\sqrt{\alpha }s)$ for $s\ge 0$. Our aim is to extract sufficient conditions for $\varPhi$ under which the (VCC) alternative still holds for $d_\varPhi (\alpha )$. As a result, assuming $\lim _{s\rightarrow \infty }\frac{\varPhi (s)}{\varPhi ((1+\varepsilon )s)}=0$ with any small $\varepsilon >0$ as an essential condition, we prove the (VCC) alternative for $d_\varPhi (\alpha )$ which extends not only $d(\alpha )$ but other various examples untreated so far.