Elliptic equations with one-sided critical growth

We consider elliptic equations in bounded domains Ω ⊃ ℝN with non-linearities which have critical growth at +∞ and linear growth λ at -∞, with λ > λ1, the first eigenvalue of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms provided N ≥ 6. In dimensions N = 3, 4,5 an additional lower order growth term has to be added to the nonlinearity, similarly as in the famous result of Brezis-Nirenberg for equations with critical growth.