Large number of bubble solutions for a fractional elliptic equation with almost critical exponents

作者：Wang, Chunhua; Wei, Suting^*

来源：PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, 2021, 151(5): 1642-1681.

DOI：10.1017/prm.2020.76

• The fractional Laplacian operator
• bubble solutions
• finite-dimensional reduction method
• almost critical exponents
• local Pohozaev identities

This paper deals with the following non-linear equation with a fractional Laplacian operator and almost critical exponents: @@@ (-Delta)(s)u = K(vertical bar y'vertical bar, y '')u((N+2s)/(N-2s)+/-epsilon), u > 0, u is an element of D-1,D-s(R-N), @@@ where N >= 4, 0 < s < 1, (y', y '') is an element of R-2 x RN-2, epsilon > 0 is a small parameter and K(y) is non-negative and bounded. Under some suitable assumptions of the potential function K(r, y ''), we will use the finite-dimensional reduction method and some local Pohozaev identities to prove that the above problem has a large number of bubble solutions. The concentration points of the bubble solutions include a saddle point of K(y). Moreover, the functional energies of these solutions are in the order epsilon(-((N-2s-2)/((N-2s)2)).

• 单位
  华南农业大学