Extremal functions for the Moser--Trudinger inequality of Adimurthi--Druet type in $W^{1,N}({\mathbb{R}}^N)$

Abstract

We study the existence and nonexistence of maximizers for variational problem concerning to the Moser--Trudinger inequality of Adimurthi--Druet type in $W^{1,N}({\mathbb{R}}^N)$ \[ MT(N,β, α) =\sup_{u\in W^{1,N}(\mathbb R^N), \|\nabla u\|_N^N + \|u\|_N^N\leq 1} \int_{\mathbb R^N} Φ_N(β(1+α\|u\|_N^N)^{\frac1{N-1}} |u|^{\frac N{N-1}}) dx, \] where ${\Phi }_N(t)=e^t-{\sum }_{k=0}^{N-2}\frac{t^k}{k!}$, $0\le \alpha <1$ both in the subcritical case $\beta <{\beta }_N$ and critical case $\beta ={\beta }_N$ with ${\beta }_N=N{\omega }_{N-1}^{\frac{1}{N-1}}$ and ${\omega }_{N-1}$ denotes the surface area of the unit sphere in ${\mathbb{R}}^N$. We will show that $MT(N,\beta ,\alpha )$ is attained in the subcritical case if $N\ge 3$ or $N=2$ and $\beta \in (\frac{2(1+2\alpha )}{(1+\alpha {)}^2{B}_2},{\beta }_2)$ with $B_2$ is the best constant in a Gagliardo--Nirenberg inequality in $W^{1,2}({\mathbb{R}}^2)$. We also show that $MT(2,\beta ,\alpha )$ is not attained for $\beta$ small which is different from the context of bounded domains. In the critical case, we prove that $MT(N,{\beta }_N,\alpha )$ is attained for $\alpha \ge 0$ small enough. To prove our results, we first establish a lower bound for $MT(N,\beta ,\alpha )$ which excludes the concentrating or vanishing behaviors of their maximizer sequences. This implies the attainability of $MT(N,\beta ,\alpha )$ in the subcritical case. The proof in the critical case is based on the blow-up analysis method. Finally, by using the Moser sequence together the scaling argument, we show that $MT(N,{\beta }_N,1)=\infty$. Our results settle the questions left open in \cite{doO2015,doO2016}.

37 pages, comment are welcome, fix typos, to appear in Communication in Contemporary Mathematics

References in corpus (3)

• Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space
• A sharp Trudinger-Moser type inequality involving $L^n$ norm in the entire space ${\mathbb{R}}^n$
• An improvement for the sharp Adams inequalities in bounded domains and whole space ${\mathbb{R}}^n$

Cited by in corpus (2)

• Extremal functions for the sharp Moser--Trudinger type inequalities in whole space ${\mathbb{R}}^N$
• Maximizers for the variational problems associated with Sobolev type inequalities under constraints