Solutions of a nonlinear Dirichlet problem in which the nonlinear part is bounded from above and below by polynomials - Beczek, Jan (Mathematical Modelling and Analysis, 1997)
In this paper we study the existence and multiplicity solutions of nonlinear elliptic problem of the form Here Ω is a smooth and bounded domain in RN , N ≥ 2, λ ∈ R and f : R → R is a continuous, even function satisfying the following condition for some c 1, c 2, c 3, p, α ∈ R, c 1, c 2, c 3, α > 0 and p > 1+ α.  We shall show that, for λ ∈ R, g ∈ Lr (Ω) if N = 2, r > 1, p > 1 + α or  the above problem has solutions. Assuming additionally that, λ ≤ λ1 and f is decreasing for t ≤ 0, we shall show that, this problem have exctly one solution. We take advantage of the fact, that a continuous, proper and odd (injective) map of the form I + C (where C is compact) is suriective (a homeomorphism).First Published Online: 14 Oct 2010