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I am interested the following variation of the Brezis-Nirenberg problem:

$$ -\Delta u(x) = (1+g(x)) u(x)^p \quad \mbox{ in } \quad \Omega$$
with $u=0$ on $ \partial \Omega$. Here $\Omega$ is a bounded domain in $ \mathbb{R}^N$ with smooth boundary and where $g(x) \ge 0$ is continuous and bounded and $ p:=\frac{N+2}{N-2}$.

My question is whether equation has been studied; in particular whether people have shown the existence of a positive solution under suitable conditions on $g(x)$.

Any comments would be greatly appreciated. greg

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In a paper by Wang Xu-jia your problem is analyzed in a even more general way.

(FTR: I can't comment yet..)

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  • $\begingroup$ oh impressive find... never heard of this paper (and it looks like its never been cited...which seems weird). After looking at the paper I am not sure whether it covers my situation; so if your familiar with it maybe you happen to just know the answer. Suppose $ \Omega=B_1(0)$ (unit ball at origin) and suppose $g(r)$ is increasing (with extra conditions or parameters if needed). Do this paper show the existence of a positive solution? Thanks for your comments $\endgroup$
    – Math604
    Commented Dec 6, 2017 at 22:09
  • $\begingroup$ There are some results regarding functions like $g(r)$, but I didn't study the paper in depth. $\endgroup$
    – Malte Winckler
    Commented Dec 7, 2017 at 8:22

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