Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity

By Mingqi Xiang, Die Hu, Di Yang. 2020

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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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

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