Existence and properties of bubbling solutions for a critical nonlinear elliptic equation

作者：Wang, Chunhua^*; Wang, Qingfang; Yang, Jing

来源：Journal of Fixed Point Theory and Applications, 2023, 25(2): 56.

DOI：10.1007/s11784-023-01059-7

• Multi-bubbling solutions
• critical exponents
• prescribed scalar curvature
• local uniqueness
• periodicity

We study the following nonlinear critical elliptic equation @@@ -Delta u + epsilon Q(y)u = u (N+2/N-2), u > 0 in R-N, @@@ where epsilon > 0 is small and N >= 5. Assuming that Q(y) is periodic in y(1) with period 1 and has a local minimum at 0 satisfying Q(0) > 0, we prove the existence and local uniqueness of infinitely many bubbling solutions of it. This local uniqueness result implies that some bubbling solutions preserve the symmetry of the potential function Q(y), i.e., the bubbling solution whose blow-up set is {(jL, 0, ... , 0) : j = 0, +/- 1, +/- 2, ... , +/- m} must be periodic in y(1) provided that epsilon goes to zero and L is any positive integer, where m is the number of the bubbles which is large enough but independent of epsilon.

• 单位
  江苏科技大学