The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms

By A. Fiscella, P. Mishra. Nonlinear Analysis, 2019

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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 .

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c0145c37ac3b2bd224976230331d98b7009fb9d2

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Citations in the corpus

On double phase Kirchhoff problems with singular nonlinearity (2021, methodology): “This method has become a very powerful tool and has been used, for example, in the works of Chen-Kuo-Wu [9] (for degenerate Kirchhoff Laplacian problems with sign-changing weight), Fiscella-Mishra [22] (for fractional Kirchhoff problems), Kumar-Rădulescu-Sreenadh [32] (for critical (p, q)-equations)”