Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity

In this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: a + b [ u ] s , p p ( - Δ ) p s u = λ | u | q - 2 u ln | u | 2 + | u | p s ∗ - 2 u in Ω , u = 0 in R N \ Ω , where N > s p with s ∈ ( 0 , 1 ) , p > 1 , and [ u ] s , p p = ∬ R 2 N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y , p s ∗ = N p / ( N - p s ) is the fractional critical Sobolev exponent, Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with Lipschitz boundary and λ is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution u b . Moreover, for any λ > 0 , we show that the energy of u b is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as b → 0 .