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We are concerned with the multiplicity of solutions to the system driven by a fractional operator with homogeneous Dirichlet boundary conditions. Namely, using fibering maps and the Nehari manifold, we obtain multiple solutions to the following fractional elliptic system:

where Ω is a smooth bounded set in ℝn , n > 2s, with s ∈ (0, 1); (–Δ)s is the fractional Laplace operator;, λ, μ > 0 are two parameters; the exponent n/(n – 2s) ⩽ q < 2; α > 1, β > 1 satisfy is the fractional critical Sobolev exponent.

A homogeneous Dirichlet problem with p-Laplacian and reaction term depending on a parameter λ > 0 is investigated. At least five solutions—two negative, two positive and one sign-changing (namely, nodal)—are obtained for all λ sufficiently small by chiefly assuming that the involved non-linearity exhibits a concave-convex growth rate. Proofs combine variational methods with truncation techniques.

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Multiple solutions for a critical fractional elliptic system involving concave–convex nonlinearities

Constant-Sign and Nodal Solutions to a Dirichlet Problem with p-Laplacian and Nonlinearity Depending on a Parameter

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2 results

Multiple solutions for a critical fractional elliptic system involving concave–convex nonlinearities

Constant-Sign and Nodal Solutions to a Dirichlet Problem with p-Laplacian and Nonlinearity Depending on a Parameter