Multiple positive solutions for image-Laplace elliptic equations involving concave–convex nonlinearities and a Hardy-type term Original Research Article

In this paper, the following problem is considered: View the MathML source{−Δpu−μ|u|p−2u|x|p=λf(x)|u|q−2u+g(x)|u|p∗−2u,x∈Ω,u=0,x∈∂Ω, Turn MathJax on where Ω⊂RNΩ⊂RN is a bounded domain such that 0∈Ω0∈Ω, 10λ>0, View the MathML sourceμ<μ̄,ff and gg are nonnegative functions, View the MathML sourceμ̄=(N−pp)p is the best Hardy constant and View the MathML sourcep∗=NpN−p is the critical Sobolev exponent. By extracting the Palais–Smale sequence in the Nehari manifold, the existence of multiple positive solutions to this equation is verified.