Bound state solutions of Choquard equations with a nonlocal operator

作者：Guo, Lun; Li, Qi^*

来源：MATHEMATICAL METHODS IN THE APPLIED SCIENCES, 2021, 44(5): 3548-3567.

DOI：10.1002/mma.6962

• bound state solution
• Choquard equation
• Kirchhoff operator
• uniqueness

In this paper, we study the following Choquard equation with Kirchhoff operator @@@ - (a + b integral(RN)vertical bar del u vertical bar(2)) Delta u + V(x)u = (I-alpha * vertical bar u vertical bar(2 alpha)*)vertical bar u vertical bar(2 alpha)*(-2)u, x is an element of R-N, (0.1) @@@ where a >= 0, b > 0, alpha is an element of (0, N), 2(alpha)* = N+alpha/N-2 is the critical exponent respect to Hardy-Littlewood-Sobolev inequality, and V(x) is an element of L-N/2 (R-N) is a given non-negative function. By using the classical linking theorem and global compactness theorem, we prove that equation (0.1) has at least one bound state solution if parallel to V parallel to(LN2) is small. More intriguingly, our result covers a novel feature of Kirchhoff problems, which is the fact that the parameter a can be zero.

• 单位
  复旦大学