On the maximizing problem associated with Sobolev-type embeddings under inhomogeneous constraints

ABSTRACT

We consider the attainability of a maximizing problem associated with the Sobolev embedding $W^{1,p}\left({\mathbb{R}}^N\right)\hookrightarrow L^q\left({\mathbb{R}}^N\right)$ of the form $D:=\underset{u\in W^{1,p}\left({\mathbb{R}}^N\right),}{sup}\parallel \mathrm{\nabla }u{\parallel }_p^a+\parallel u{\parallel }_p^b=1\left(\parallel u{\parallel }_p^p+\alpha \parallel u{\parallel }_q^q\right)$, where $N\ge 2$, $1<p\le N$, $p<q<\left\{\begin{array}{c}\left(Np/\right(N-p\left)\right)\mathrm{if}p<N,\\ =+\mathrm{\infty }\mathrm{if}p=N\end{array}\right.$ and $\alpha ,a,b>0$. The existence and non-existence of a maximizer for D are related to the exponents $\alpha ,N,p,q,a$ and b. Indeed, for some range of α, we have a non-existence result in spite of the sub-critical setting and the radial nature of the problem. We show that the value $\alpha ={\alpha }_{\ast }:=\underset{u\in W^{1,p}\left({\mathbb{R}}^N\right),\parallel \mathrm{\nabla }u{\parallel }_p^a+\parallel u{\parallel }_p^b=1}{inf}\left(\left(1-\parallel u{\parallel }_p^p\right)/\parallel u{\parallel }_q^q\right)$ is the threshold in terms of the attainability of D. We also consider the attainability of D with $\alpha ={\alpha }_{\ast }$ and compute the exact value ${\alpha }_{\ast }$ for some range of the exponents.

KEYWORDS:

• Attainability of maximizing problem
• Sobolev's embedding theorem
• normalized vanishing sequence

AMS Subject Classifications:

• 47J30
• 46E35
• 26D10

Acknowledgments

The authors would like to express their hearty thanks to the referees for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the authors.