Existence of Nonnegative Solutions for a Class of Systems Involving Fractional(p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional(p, q)-Laplacian operators:{(-△)_p~su=λa(x)|u|+~(p-2)u+λb(x)|u|~(α-2)|u|~βu+μ(x)/αδ|u|~(γ-2)|v|~δu in Ω,(-△)_p~su=λc(x)|v|+~(q-2)v+λb(x)|u|~α|u|~(β-2)v+μ(x)/βγ|u|~γ|v|~(δ-2)v in Ω,u=v=0 on R~N\Ω where λ 0 is a real parameter, ? is a bounded domain in RN, with boundary ?? Lipschitz continuous, s ∈(0, 1), 1 p ≤ q ∞, sq N, while(-?)s pu is the fractional p-Laplacian operator of u and, similarly,(-?)s qv is the fractional q-Laplacian operator of v. Since possibly p = q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalueλ_1 for a related system, they prove that there exists a positive solution for the problem when λ λ_1 by the modified definitions. Moreover, the authors obtain the bifurcation property when λ→λ_1~-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ≥λ_1.     

Existence of Nonnegative Solutions for a Class of Systems Involving Fractional (p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional (p, q)-Laplacian operators:

$$\left\{ {\begin{array}{*{20}{c}} {\left( { - \Delta } \right)_p^su = \lambda a\left( x \right){{\left| u \right|}^{p - 2}}u + \lambda b\left( x \right){{\left| u \right|}^{\alpha - 2}}{{\left| v \right|}^\beta }u + \frac{{\mu \left( x \right)}}{{\alpha \delta }}{{\left| u \right|}^{\gamma - 2}}{{\left| v \right|}^\delta }uin\Omega ,} \\ {\left( { - \Delta } \right)_q^sv = \lambda c\left( x \right){{\left| v \right|}^{q - 2}}v + \lambda b\left( x \right){{\left| u \right|}^\alpha }{{\left| v \right|}^{\beta - 2}}v + \frac{{\mu \left( x \right)}}{{\beta \gamma }}{{\left| u \right|}^\gamma }{{\left| v \right|}^{\delta - 2}}vin\Omega ,} \\ {u = v = 0on{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$

where λ > 0 is a real parameter, Ω is a bounded domain in R ^N , with boundary ?Ω Lipschitz continuous, s ∈ (0, 1), 1 < p ≤ q < ∞, sq < N, while (?Δ) _p ^s u is the fractional p-Laplacian operator of u and, similarly, (?Δ) _q ^s v is the fractional q-Laplacian operator of v. Since possibly p ≠ q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalue λ₁ for a related system, they prove that there exists a positive solution for the problem when λ < λ₁ by the modified definitions. Moreover, the authors obtain the bifurcation property when λ → λ₁^-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ ≥ λ₁.