A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity

Details A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity

2008-10-28

arxiv.org/abs/0810.5140v1

Mathematics

Analysis of PDEs

26 pages

Scientific paper

We consider positive solutions of $\Delta u-\mu u+Ku^{\frac{n+2}{n-2}}=0$ on $B_1$ ($n\ge 5$) where $\mu $ and $K>0$ are smooth functions on $B_1$. If $K$ is very sub-harmonic at each critical point of $K$ in $B_{2/3}$ and the maximum of $u$ in $\bar B_{1/3}$ is comparable to its maximum over $\bar B_1$, then all positive solutions are uniformly bounded on $\bar B_{1/3}$. As an application, a priori estimate for solutions of equations defined on $\mathbb S^n$ is derived.

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