Solutions of a nonlinear Dirichlet problem in which the nonlinear part is bounded from above and below by polynomials

Abstract

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 R^N , 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 ∈ L_r (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