Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity. (September 2020) - Xiang, Mingqi; Hu, Die; Yang, Di
Abstract: In this paper, we study the existence of least energy solutions to the following fractional Kirchhoff problem with logarithmic nonlinearity M ( [ u ] s, p p ) ( − Δ ) p s u = h ( x ) | u | θ p − 2 u ln | u | + λ | u | q − 2 u x ∈ Ω, u = 0 x ∈ R N ∖ Ω, where s ∈ (0, 1), 1 < p < N ∕ s, Ω ⊂ R N is a bounded domain with Lipschitz boundary, M ( [ u ] s, p p ) = [ u ] s, p ( θ − 1 ) p with θ ≥ 1 and [ u ] s, p is the Gagliardo seminorm of u, h ∈ C ( Ω ¯ ) may change sign, λ > 0 is a parameter, q ∈ ( 1, p s ∗ ) and ( − Δ ) p s is the fractional p − Laplacian. When θ p < q < p s ∗ and h is a positive function on Ω, the existence of least energy solutions is obtained by restricting the discussion on Nehari manifold. When 1 < q < θ p and h is a sign-changing function on Ω, two local least energy solutions are obtained by using the Nehari manifold approach.