Existence of Multiple Solutions for a -Kirchhoff Problem with Nonlinear Boundary Conditions

Abstract

The paper considers the existence of multiple solutions of the singular nonlocal elliptic problem −M(∫Ω‍ | x|−ap | ∇u|p)div(|x|−ap | ∇u|p−2∇u) = λh(x) | u|r−2 u, x ∈ Ω, M(∫Ω‍ | x|−ap | ∇u|p) | x|−ap | ∇u|p−2 (∂u/∂ν) = g(x) | u|q−2 u, on ∂Ω, where 1 < (N + 1)/2 < p < N, a < (N − p)/p. By the variational method on the Nehari manifold, we prove that the problem has at least two positive solutions when some conditions are satisfied.