A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

By A. Fiscella. 2017

93 citations recorded by the source corpus.

Abstract In this paper, we consider the following critical nonlocal problem: { M ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) ⁢ ( - Δ ) s ⁢ u = λ u γ + u 2 s * - 1 in ⁢ Ω , u > 0 in ⁢ Ω , u = 0 in ⁢ ℝ N ∖ Ω , \left\{\begin{aligned} &\displaystyle M\bigg{(}\iint_{\mathbb{R}^{2N}}\frac{% \lvert u(x)-u(y)\rvert^{2}}{\lvert x-y\rvert^{N+2s}}\,dx\,dy\biggr{)}(-\Delta)% ^{s}u=\frac{\lambda}{u^{\gamma}}+u^{2^{*}_{s}-1}&&\displaystyle\phantom{}\text% {in }\Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% N}\setminus\Omega,\end{aligned}\right. where Ω is an open bounded subset of ℝ N {\mathbb{R}^{N}} with continuous boundary, dimension N > 2 ⁢ s {N>2s} with parameter s ∈ ( 0 , 1 ) {s\in(0,1)} , 2 s * = 2 ⁢ N / ( N - 2 ⁢ s ) {2^{*}_{s}=2N/(N-2s)} is the fractional critical Sobolev exponent, λ > 0 {\lambda>0} is a real parameter, γ ∈ ( 0 , 1 ) {\gamma\in(0,1)} and M models a Kirchhoff-type coefficient, while ( - Δ ) s {(-\Delta)^{s}} is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.

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

View Paper: https://www.semanticscholar.org/paper/181985f64b9f5cc3c0e33f30dedc4e98dfb31091

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In Applied Mathematics

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Citations in the corpus

On double phase Kirchhoff problems with singular nonlinearity (2021, methodology): “A. Fiscella realized the manuscript within the auspices of the INdAM-GNAMPA project titled “Equazioni alle derivate parziali: problemi e modelli” (Prot_20191219-143223-545) and of the FAPESP Thematic Project titled “Systems and partial differential equations” (2019/02512-5).”