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 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).
Highlights
Let us consider a bounded sequence uf ng exists in a sWub12se(qu)e.ncSeinfcuentkhgesuimchbetdhdaitnugnWk ; 12!(
It follows from fact that, a continuous, proper and odd map of the form I + C is suriective.[2]
Let g satis es the assumptions of theorem
Summary
) if N = 2, r > 1, above problem has f is decreasing for p>1+ solutions. T 0, we or shall show that, 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). We will consider the following nonlinear Dirichlet problem u + u ; f(u) u = g in u = 0 on R is a continuous, even function satisfying the following condition
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