Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 77, pp. 1-15.
Title: Kirchhoff systems involving fractional p-Laplacian and singular nonlinearity
Author: Mouna Kratou (Univ. of Imam Abdulrahman Bin Faisal, Dammam, Saudi Arabia)
Abstract:
In this work we consider the fractional Kirchhoff equations
with singular nonlinearity,
$$ \displaylines{
M\Big( \int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dx
dy\Big) (-\Delta)^s_p u \cr
= \lambda a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta},
\quad \text{in }\Omega,\cr
M\Big( \int_{\mathbb{R}^{2N}}\frac{|v(x)-v(y)|^p}{|x-y|^{N+sp}}dx
dy\Big) (-\Delta)^s_p v \cr
= \mu b(x)|v|^{q-2}v +\frac{1-\beta}{2-\alpha-\beta} c(x)|u|^{1-\alpha}|v|^{-\beta}, \quad
\text{in }\Omega,\cr
u=v = 0 ,\quad\text{in }\mathbb{R}^N\setminus\Omega,
}$$
where Ω is a bounded domain in RN with smooth boundary,
N> ps, s in (0,1), 0<α<1, 0< β< 1,
2-α-β<p≤ pθ<q<p*s,
&p*s=2N/(N-2s) is the fractional Sobolev exponent,
λ, μ are two parameters, a, b, c in C(overlineΩ)
are non-negative weight functions,
M(t)=k+lt&theta-1 with k>0, l,θ≥1, and
(-Δsp is the fractional p-laplacian operator.
We prove the existence of multiple non-negative solutions by
studying the nature of the Nehari manifold with respect to the parameters
λ and μ.
Submitted September 11, 2022. Published November 21, 2022.
Math Subject Classifications: 34B15, 37C25, 35R20.
Key Words: Kirchhoff-type equations; fractional p-Laplace operator;
Nehari manifold; singular elliptic system; multiple positive solutions,