Sharp Affine and Improved Moser–Trudinger–Adams Type Inequalities on Unbounded Domains in the Spirit of Lions

Content Provider	Semantic Scholar
Author	Lâm, Nguyễn Xuân Lu, Guozhen Tang, Hanli
Copyright Year	2017
Abstract	The purpose of this paper is threefold. First, we prove sharp singular affine Moser–Trudinger inequalities on both bounded and unbounded domains in $${\mathbb {R}}^{n}$$Rn. In particular, we will prove the following much sharper affine Moser–Trudinger inequality in the spirit of Lions (Rev Mat Iberoamericana 1(2):45–121, 1985) (see our Theorem 1.4): Let $$\alpha _{n}=n\left( \frac{n\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2}+1)}\right) ^{\frac{1}{n-1}}$$αn=nnπn2Γ(n2+1)1n-1, $$0\le \beta 0$$τ>0. Then there exists a constant $$C=C\left( n,\beta \right) >0$$C=Cn,β>0 such that for all $$0\le \alpha \le \left( 1-\frac{\beta }{n}\right) \alpha _{n}$$0≤α≤1-βnαn and $$u\in C_{0}^{\infty }\left( {\mathbb {R}}^{n}\right) \setminus \left\{ 0\right\} $$u∈C0∞Rn\0 with the affine energy $$~{\mathcal {E}}_{n}\left( u\right) <1$$Enu<1, we have $$\begin{aligned} {\displaystyle \int \nolimits _{{\mathbb {R}}^{n}}} \frac{\phi _{n,1}\left( \frac{2^{\frac{1}{n-1}}\alpha }{\left( 1+{\mathcal {E}}_{n}\left( u\right) ^{n}\right) ^{\frac{1}{n-1}}}\left| u\right| ^{\frac{n}{n-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( n,\beta \right) \frac{\left\| u\right\| _{n}^{n-\beta }}{\left| 1-{\mathcal {E}}_{n}\left( u\right) ^{n}\right| ^{1-\frac{\beta }{n}}}. \end{aligned}$$∫Rnϕn,121n-1α1+Enun1n-1unn-1xβdx≤Cn,βunn-β1-Enun1-βn.Moreover, the constant $$\left( 1-\frac{\beta }{n}\right) \alpha _{n}$$1-βnαn is the best possible in the sense that there is no uniform constant $$C(n, \beta )$$C(n,β) independent of u in the above inequality when $$\alpha >\left( 1-\frac{\beta }{n}\right) \alpha _{n}$$α>1-βnαn. Second, we establish the following improved Adams type inequality in the spirit of Lions (Theorem 1.8): Let $$0\le \beta <2m$$0≤β<2m and $$\tau >0$$τ>0. Then there exists a constant $$C=C\left( m,\beta ,\tau \right) >0$$C=Cm,β,τ>0 such that $$\begin{aligned} \underset{u\in W^{2,m}\left( {\mathbb {R}}^{2m}\right) , \int _{ {\mathbb {R}}^{2m}}\left| \Delta u\right| ^{m}+\tau \left| u\right| ^{m} \le 1}{\sup } {\displaystyle \int \nolimits _{{\mathbb {R}}^{2m}}} \frac{\phi _{2m,2}\left( \frac{2^{\frac{1}{m-1}}\alpha }{\left( 1+\left\| \Delta u\right\| _{m}^{m}\right) ^{\frac{1}{m-1}}}\left| u\right| ^{\frac{m}{m-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( m,\beta ,\tau \right) , \end{aligned}$$supu∈W2,mR2m,∫R2mΔum+τum≤1∫R2mϕ2m,221m-1α1+Δumm1m-1umm-1xβdx≤Cm,β,τ,for all $$0\le \alpha \le \left( 1-\frac{\beta }{2m}\right) \beta (2m,2)$$0≤α≤1-β2mβ(2m,2). When $$\alpha >\left( 1-\frac{\beta }{2m}\right) \beta (2m,2)$$α>1-β2mβ(2m,2), the supremum is infinite. In the above, we use $$\begin{aligned} \phi _{p,q}(t)=e^{t}- {\displaystyle \sum \limits _{j=0}^{j_{\frac{p}{q}}-2}} \frac{t^{j}}{j!},\,\,\,j_{\frac{p}{q}}=\min \left\{ j\in {\mathbb {N}} :j\ge \frac{p}{q}\right\} \ge \frac{p}{q}. \end{aligned}$$ϕp,q(t)=et-∑j=0jpq-2tjj!,jpq=minj∈N:j≥pq≥pq.The main difficulties of proving the above results are that the symmetrization method does not work. Therefore, our main ideas are to develop a rearrangement-free argument in the spirit of Lam and Lu (J Differ Equ 255(3):298–325, 2013; Adv Math 231(6): 3259–3287, 2012), Lam et al. (Nonlinear Anal 95: 77–92, 2014) to establish such theorems. Third, as an application, we will study the existence of weak solutions to the biharmonic equation $$\begin{aligned} \left\{ \begin{array}{l} \Delta ^{2}u+V(x)u=f(x,u)\text { in }{\mathbb {R}}^{4}\\ u\in H^{2}\left( {\mathbb {R}}^{4}\right) ,~u\ge 0 \end{array} \right. , \end{aligned}$$Δ2u+V(x)u=f(x,u)inR4u∈H2R4,u≥0,where the nonlinearity f has the critical exponential growth.
Starting Page	300
Ending Page	334
Page Count	35
File Format	PDF HTM / HTML
DOI	10.1007/s12220-016-9682-2
Volume Number	27
Alternate Webpage(s)	https://www2.math.uconn.edu/~guozhenlu/papers/LamLuTang-JGEA2017.pdf
Alternate Webpage(s)	https://doi.org/10.1007/s12220-016-9682-2
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article