Bifurcation for some quasilinear operators

Abstract

This paper deals with existence, uniqueness and multiplicity results of positive solutions for the quasilinear elliptic boundary-value problemwhere « is a bounded open domain in R N with smooth boundary. Under suitable assumptions on the matrix A(x; s), and depending on the behaviour of the function f near u = 0 and near u = +1 , we can use bifurcation theory in order to give a quite complete analysis on the set of positive solutions. We will generalize in di® erent directions some of the results in the papers by Ambrosetti et al., Ambrosetti and Hess, and Artola and Boccardo.

Combining linear and fast diffusion in a nonlinear elliptic equation

“…2. In view of (12) and since f (λ, x, s) := λq(x, s) satisfies f (0, x, s) ≡ 0 for all x ∈ Ω and s ≥ 0, we can apply the Theorem 3.4 of [3] and get the results.…”

“…Case Ω a0 = ∅ In view of (11), we can apply Theorem 4.4 of [3] and obtain that λ a0 is a bifurcation point from the trivial solution of positive solutions, and it is the only one in IR + 0 . Furthermore, there exists an unbounded component Σ 0 ⊂ Σ meeting λ a0 .…”

“…If Ω a0 = ∅, by (11), we can apply the Theorem 4.4 of [3] to obtain that λ a0 is the only bifurcation point from the trivial solution. To conclude the direction of bifurcation we will apply the paragraphs (i) and (ii) of Theorem 4.4 of [3] and argue as follows. Denote…”

Nonnegative solutions for the degenerate logistic indefinite sublinear equation

“…In this section we adapt the results of [5], see also [6] and [15], to show that a bifurcation from the trivial solution of (1.4) occurs at λ = 0. We include them for the reader's convenience and send to [15] for details.…”

“…When 1 < m < 2 (q < 1 < p) and a(x) ≡ a 0 with a 0 a positive constant, (1.4) was studied in [4] in the particular case L = −∆ and in [6] when L is a quasilinear operator. When a changes sign, (1.4) was analyzed in [24] in the particular case λ ≤ 0.…”