Abstract We consider the equation - div ⁡ ( a ⁢ ( x ) ⁢ ∇ ⁡ u ) = b ⁢ ( x ) ⁢ | u | q - 2 ⁢ u + c ⁢ ( x ) ⁢ | u | p - 2 ⁢ u , u ∈ H 0 1 ⁢ ( Ω ) , $-\operatorname{div}(a(x)\nabla u)=b(x)|u|^{q-2}u+c(x)|u|^{p-2}u,\quad u\in H_{% 0}^{1}(\Omega),$ where Ω ⊂ ℝ N ${\Omega\subset\mathbb{R}^{N}}$ is a bounded smooth domain and N ≥ 4 ${N\geq 4}$ . The functions a, b and c satisfy some hypotheses which provide a variational structure for the problem. For 1 < q < 2 < p ≤ 2 ⁢ N / ( N - 2 ) ${1<q<2<p\leq 2N/(N-2)}$ we obtain the existence of two nonzero solutions if the function b has small Lebesgue norm. The proof is based on minimization arguments and the Mountain Pass Theorem.