Combined effects in nonlinear problems arising in the study of anisotropic continuous media

We are concerned with the Lane–Emden–Fowler equation −Δu=λk(x)uq±h(x)up in Ω, subject to the Dirichlet boundary condition u=0 on ∂Ω, where Ω is a smooth bounded domain in RN, k and h are variable potential functions, and 0<q<1<p. Our analysis combines monotonicity methods with variational arguments.