Concentration phenomenon in the critical exponent problems on hyperbolic space

ABSTRACT

This article deals with the study of the following critical exponent problem $\left(P_{\lambda ,\mu }\right)\left\{\begin{array}{l}-{\mathrm{\Delta }}_{{\mathbb{H}}^n}u+\left(\mu g\right(x)-\lambda )u=u^{p-1}\mathrm{i}\mathrm{n}{\mathbb{H}}^n,\\ u>0\mathrm{i}\mathrm{n}{\mathbb{H}}^n,u\in H^1\left({\mathbb{H}}^n\right)\end{array}\right.$ where ${\mathbb{H}}^n$ is the n-dimensional hyperbolic space, $n\ge 4$, $p=2n/(n-2)$ is the critical exponent, $\lambda ,\mu >0$ are real parameters, ${\mathrm{\Delta }}_{{\mathbb{H}}^n}$ denotes the Laplace–Beltrami operator on ${\mathbb{H}}^n$ and $g\left(x\right)$ is a real valued potential function on ${\mathbb{H}}^n$. Using variational methods, we establish the existence of positive ground state solution to $\left(P_{\lambda ,\mu }\right)$ and study the convergence of these solutions when μ approaches $+\mathrm{\infty }$.

Keywords:

• Critical exponent problems
• hyperbolic space
• Nehari manifold
• least energy solutions

2010 Mathematics Subject Classifications:

• 35R11
• 35R09
• 35A15

Acknowledgments

I am grateful to anonymous referees for carefully reading the manuscript and their suggestions which has improved the quality of this article.

Disclosure statement

No potential conflict of interest was reported by the author.