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Pohozaev type obstructions and solutions of bounded energy for quasilinear elliptic equations with critical Sobolev growth. The conformally flat case. (English) Zbl 1066.35032
Let $$(M,g)$$ be a smooth, compact, Riemannian manifold of dimension $$n\geq 3$$, and consider the equation $\Delta_g u + \alpha u = u^{2^*-1}, \qquad u>0 \quad\text{in}\quad M, (*)_\alpha$ where $$\Delta_g u=-\text{ div}_g(\nabla u)$$ and $$2^*=2n/(n-2)$$ is the critical Sobolev exponent. If $$\Omega$$ is a smooth, bounded, star-shaped domain in $${\mathbb R}^n$$ with the Euclidean metric, and $$u$$ is required to vanish on $$\partial\Omega$$, the well known Pohozaev identity implies that there is no positive solution of $$(*)_\alpha$$ with $$M=\Omega$$ if $$\alpha\geq 0$$.
Here the authors are interested in proving some generalization of this result in the Riemannian setting. Observing first that there is no solution of $$(*)_\alpha$$ if $$\alpha=0$$, and that $$u_\alpha=\alpha^{(n-2)/4}$$ is a solution for $$\alpha>0$$, one possible generalization would be that there exists an $$\alpha_0$$ such that $$(*)_\alpha$$ has no nonconstant solution if $$\alpha\geq \alpha_0$$. However, this turns out to be false: the authors exhibit examples of conformally flat manifolds $$(M,g)$$ for which there are nonconstant solutions of $$(*)_\alpha$$ with $$\alpha\geq\alpha_0$$ for arbitrarily large $$\alpha_0$$.
The main result of the paper is the following. Define ${\mathcal S}_\Lambda = \left\{ \alpha>0: \text{there exists a solution $$u$$ of $$(*)_\alpha$$ with $$\left(\int_M u^{2^*}\right)^{1/{2^*}} \leq \Lambda$$ } \right\}.$ Then, if $$(M,g)$$ is conformally flat, $${\mathcal S}_\Lambda$$ is bounded for any $$\Lambda>0$$. Moreover, $${\mathcal S}_\Lambda$$ is closed in the set of positive real numbers if $$\Lambda$$ is sufficiently large. The proof is by a blow up analysis. The main difficult is to deal with blowing up sequences that have multiple concentration points.
This result was proved by E. Hebey and M. Vaugon [Duke Math. J. 79, 235–279 (1995; Zbl 0839.53030)] in the special case that $$\Lambda=K(n,2)^{-2/(2^*-2)}$$ where $$K(n,2)$$ is the best constant in the Euclidean Sobolev inequality on $${\mathbb R}^n$$.

MSC:
 35J60 Nonlinear elliptic equations 35B33 Critical exponents in context of PDEs 58J05 Elliptic equations on manifolds, general theory
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References:
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