Blow-up solutions of nonlinear elliptic equations in R^n with critical exponent

Details Blow-up solutions of nonlinear elliptic equations in R^n with critical exponent Blow-up solutions of nonlinear elliptic equations in R^n with critical exponent

2002-02-23

arxiv.org/abs/math/0202244v1

Mathematics

Analysis of PDEs

27 pages

Scientific paper

For an integer $n \ge 3$ and any positive number $\epsilon$ we establish the existence of smooth functions K on $R^n \setminus \{0 \}$ with $|K - 1| \le \epsilon$, such that the equation $\Delta u + n (n - 2) K u^{{n + 2}\over {n - 2}} = 0$ in $R^n \setminus \{0 \}$ has a smooth positive solution which blows up at the origin (i.e., u does not have slow decay near the origin). Furthermore, we show that in some cases K can be extended as a Lipschitz function on ${\R}^n.$ These provide counter-examples to a conjecture of C.-S. Lin when n > 4, and Taliaferro's conjecture.

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