Positive solutions for nonhomogeneous elliptic equations with critical growth on R2

Let f(x, t) : R-2 x R --> R be a C-2-function with respect to t is an element of R, f(x, 0) = 0, f(x, t) similar to e(bt2) as t --> +infinity for some b > 0. Under suitable conditions on f(x, t), author shows that for g is an element of L-2 (R-2), g(x) greater than or equal to (=)0, the following semilinear elliptic problem: [GRAPHICS] has at least two distinct positive solutions for any lambda is an element of (0, lambda*), at least one positive solution for any lambda is an element of [lambda*, lambda*] and has no positive solution for all lambda > lambda*. It is also proved that lambda* less than or equal to lambda* < +infinity.