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I was reading about Pohazaev type obstructions: precisely, I mean the following kind of results. Let $h\in C^1(\mathbb{R}^n)$ and consider the following Dirichlet problem $$ \begin{cases} \Delta u + hu = |u|^{\frac{4}{n-2}}u &\text{ in }\Omega,\\ u = 0 &\text{ on }\partial\Omega. \end{cases} $$ where $\Omega$ is star-shaped (with respect to the origin) smoothy-bounded domain in $\mathbb{R}^n$ with $n\geq 3$. Then, if $h$ satisfies the inequality $$ h(x) + \frac{1}{2}\langle x,\nabla h(x)\rangle \geq 0, $$ Pohozaev says that there are no non-trivial solutions for the above boundary value problem.

Question. Is there any similar kind of obstruction for higher order elliptic operators?

Final note. In the case of higher order operators, we obviously need a better regularity for $h$ but I am just curious to know if there is there some analogous result for higher order, conformally invariant operators.

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  • $\begingroup$ arxiv.org/pdf/1010.4614.pdf $\endgroup$
    – Liviu Nicolaescu
    Commented Jan 21 at 16:05
  • $\begingroup$ Thanks. But I am just curious for ex. say for bi-Laplacian $\Delta^2u$ and say the PDE is $\Delta^2u+hu=|u|^{\frac{8}{n-4}}u$ does $h$ satisfy some similar kind of obstruction like $h(x)+\frac{1}{2}<x,\Delta h(x)>\ \geq 0$ in order to have no solutions? $\endgroup$
    – Sarthak
    Commented Jan 22 at 8:25
  • $\begingroup$ Phohozaev identity os about conformal invariance. For operators of order 4 and their associated Yamabe-type equation see (1.5) here ams.org/journals/era/2004-10-15/S1079-6762-04-00138-6/… $\endgroup$
    – Liviu Nicolaescu
    Commented Jan 22 at 8:44

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