Bound state solutions of Choquard equations with a nonlocal operator

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.

Automated summary

In this paper, a Choquard equation with Kirchhoff operator is studied. By using the classical linking theorem and global compactness theorem, it is proved that the equation has at least one bound state solution if the norm of V in L^(N/2) is small. Furthermore, a novel feature of Kirchhoff problems is covered, allowing the parameter a to be zero.