Existence of weak solutions for a class of nonuniformly nonlinear elliptic equations in unbounded domains

The goal of this paper is to study the existence of non-trivial weak solutions for the nonuniformly nonlinear elliptic equation −div(h(x)∇u)+q(x)u=f(x,u) in an unbounded domain Ω⊂RN(N≥3)Ω⊂RN(N≥3), where h(x)∈Lloc1(Ω). The solutions will be obtained in a subspace of the Sobolev space H01(Ω) and the proofs rely essentially on a variation of the Mountain pass theorem in [D.M. Duc, Nonlinear singular elliptic equations, J. London. Math. Soc. 40 (2) (1989) 420–440].