Finite Morse index solutions of an elliptic equation with supercritical exponent

By Edward Norman Dancer, Yihong Du, Zongming Guo. 2011

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Abstract We study the behavior of finite Morse index solutions of the equation − Δ u = | x | α | u | p − 1 u in Ω ⊂ R N , where p > 1 , α > − 2 , and Ω is a bounded or unbounded domain. We show that there is a critical power p = p ¯ ( α ) larger than the usual critical exponent N + 2 N − 2 such that this equation with Ω = R N has no nontrivial stable solution for 1 p p ¯ ( α ) but it admits a family of stable positive solutions when p ⩾ p ¯ ( α ) . For a positive solution u with finite Morse index, we classify the singularity of u at the origin when Ω is a punctured ball B R ( 0 ) \ { 0 } , and we classify its behavior at infinity when Ω = R N ∖ B R ( 0 ) . We show that the behavior depends crucially on whether p is below or above the critical power p ¯ ( α ) . We also demonstrate how a duality method can be used to obtain sharper results.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0bcca74e1e8b5699fbe658557046f687b146c170

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In Applied Mathematics

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In Computational Theory and Mathematics

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  2. Chao Wang
  3. Y. Yang

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