The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms. (September 2019) - Fiscella, Alessio; Mishra, Pawan Kumar
Abstract: In the present paper, we study the following singular Kirchhoff problem M ∬ R 2 N | u ( x ) − u ( y ) | 2 | x − y | N + 2 s d x d y ( − Δ ) s u = λ f ( x ) u − γ + g ( x ) u 2 s ∗ − 1 in Ω, u > 0 in Ω, u = 0 in R N ∖ Ω, where Ω ⊂ R N is an open bounded domain, dimension N > 2 s with s ∈ ( 0, 1 ), 2 s ∗ = 2 N ∕ ( N − 2 s ) is the fractional critical Sobolev exponent, parameter λ > 0, exponent γ ∈ ( 0, 1 ), M models a Kirchhoff coefficient, f ∈ L 2 s ∗ 2 s ∗ + γ − 1 ( Ω ) is a positive weight, while g ∈ L ∞ ( Ω ) is a sign-changing function. Using the idea of Nehari manifold technique, we prove the existence of at least two positive solutions for a sufficiently small choice of λ . This approach allows us to avoid any restriction on the boundary of Ω .