Pohozaev type obstructions and solutions of bounded energy for quasilinear elliptic equations with critical Sobolev growth. The conformally flat case.

*(English)*Zbl 1066.35032Let \((M,g)\) 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 \({\mathbb R}^n\).

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 \({\mathbb R}^n\).

Reviewer: John Urbas (Canberra)

##### MSC:

35J60 | Nonlinear elliptic equations |

35B33 | Critical exponents in context of PDEs |

58J05 | Elliptic equations on manifolds, general theory |

##### Keywords:

quasilinear elliptic equations; Riemannian manifolds; Pokhozhaev obstruction; critical Sobolev exponent
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\textit{O. Druet} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 51, No. 1, 79--94 (2002; Zbl 1066.35032)

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##### References:

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