In this paper, we study the following fractional Kirchhoff-type problem: [a+b(∬R2N|u(x)-u(y)|2|x-y|N+2sdxdy)θ-1](-Δ)su=|u|2s∗-2u+λf(x)|u|q-2u,inRN,where (- Δ) s is the fractional Laplacian operator with 0 < s< 1 , λ≥ 0 , a≥ 0 , b> 0 , 1 < q< 2 , N> 2 s, and 2s∗=2NN-2s is fractional critical Sobolev exponent. When λ= 0 , under suitable values of the parameters θ, a and b, we obtain a non-existence result and the existence of infinitely many nontrivial solutions for the above problem. Also, for suitable weight function f(x), using the Nehari manifold technique and the fibbing maps, we prove the existence of at least two positive solutions for a sufficiently small choice of λ.

Existence and non-existence results for fractional Kirchhoff Laplacian problems / Nyamoradi, N.; Ambrosio, V.. - In: ANALYSIS AND MATHEMATICAL PHYSICS. - ISSN 1664-2368. - 11:3(2021). [10.1007/s13324-020-00435-7]

Existence and non-existence results for fractional Kirchhoff Laplacian problems

Ambrosio V.
2021-01-01

Abstract

In this paper, we study the following fractional Kirchhoff-type problem: [a+b(∬R2N|u(x)-u(y)|2|x-y|N+2sdxdy)θ-1](-Δ)su=|u|2s∗-2u+λf(x)|u|q-2u,inRN,where (- Δ) s is the fractional Laplacian operator with 0 < s< 1 , λ≥ 0 , a≥ 0 , b> 0 , 1 < q< 2 , N> 2 s, and 2s∗=2NN-2s is fractional critical Sobolev exponent. When λ= 0 , under suitable values of the parameters θ, a and b, we obtain a non-existence result and the existence of infinitely many nontrivial solutions for the above problem. Also, for suitable weight function f(x), using the Nehari manifold technique and the fibbing maps, we prove the existence of at least two positive solutions for a sufficiently small choice of λ.
2021
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/294895
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