Abstract
For dimensions $N \geq 4$, we consider the Brezis-Nirenberg variational problem of finding $ S(\epsilon V) : = \inf\limits_{0\not\equiv u\in H^1_0(\Omega)} \frac{\int_\Omega |\nabla u|^2\, dx +\epsilon \int_\Omega V\, |u|^2\, dx}{\left(\int_\Omega |u|^q \, dx \right)^{2/q}}, $ where $q = \frac{2N}{N-2}$ is the critical Sobolev exponent, $\Omega \subset \mathbb{R}^N$ is a bounded open set and $V:\overline{\Omega}\to \mathbb{R}$ is a continuous function. We compute the asymptotics of $S(0) - S(\epsilon V)$ to leading order as $\epsilon \to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and, in particular, we characterize the concentration points as being extrema of a quotient involving the Robin function. This complements the results from our recent paper in the case $N = 3$.
Highlights
To prepare the statement of our main results, we introduce some key objects for the following analysis
It is natural to study general positive solutions of this equation, even if they do not arise as minimizers of (1.2)
In the special case where V is a negative constant, Brézis and Peletier [4] discussed the concentration behavior of such general solutions and made some conjectures, which were later proved by Han [7] and Rey [8]
Summary
The purpose of this paper is, for N ě 4, to describe the asymptotics of SN Sp V q to leading order as Ñ 0, as well as the asymptotic behavior of corresponding (almost) minimizing sequences and, in particular, their concentration behavior This is the higher-dimensional complement to our recent paper [6], where analogous results are shown in the more difficult case N “ 3. In our paper we generalize Takahashi’s results to non-constant V and to almost minimizing sequences and we give an alternative, self-contained proof which does not rely on the works of Han and Rey. The present work is a companion paper to [6] relying on the techniques developed there in the three dimensional case.
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