Existence of nonnegative solutions for quasilinear elliptic equations with indefinite critical nonlinearities

We study the quasilinear elliptic equation −div(ϕ(|∇u|)∇u)=a(x)f(u)+b(x)g(u)inΩ with Dirichlet boundary condition u=0u=0 on ∂Ω∂Ω, where ΩΩ is a bounded domain in RN, a(x)a(x), b(x)b(x) are sign-changing continuous functions, and g(u)g(u) has critical growth at infinity with respect to the principal part ϕϕ. A nonnegative, nontrivial solution is given under appropriate growth conditions on f(u),g(u) at 0 and ∞∞.