An indefinite concave-convex equation under a Neumann boundary condition I

Abstract

Abstract. We investigate the problemwhere Ω is a bounded smooth domain in IR N (N ≥ 2), 1 < q < 2 < p, λ ∈ IR, and a, b ∈ C α (Ω) with 0 < α < 1. Under some indefinite type conditions on a and b we prove the existence of two nontrivial non-negative solutions for |λ| small. We characterize then the asymptotic profiles of these solutions as λ → 0, which implies in some cases the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type subcontinuum in the non-negative solutions set. We prove in some cases the existence of such subcontinuum via a bifurcation and topological analysis of a regularized version of (P λ ).

Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems

“…In addition, in [10], the authors allowed a to change sign and proved the existence of two nontrivial nonnegative solutions of (P B ) for λ > 0 small. We refer to [24] for a discussion on concaveconvex problems under the Neumann boundary condition.…”

“…Furthermore, the asymptotic profile of nonnegative solutions as λ → 0 + enables one to deduce in some cases their positivity for λ > 0 small, cf. [24,Corollary 1.3].…”

“…In this case, although the loop type subcontinuum C 0 satisfies (1.6), C 0 \ {(0, 0)} appears in λ > 0. This means that C 0 never meets the vertical line {(0, u) : 0 ≡ u ≥ 0}, see [24,Lemma 6.8(1)]. Thus, the approach used in the proof of Theorem 1.2(i) does not work for excluding the possibility that C 0 = {(0, 0)}, see the argument in Subsection 5.1.…”

“…From (1.2), we see that f is not differentiable at s = 0, so that we can not directly apply the usual bifurcation theory from simple eigenvalues to (P B ). To overcome this difficulty, we proceed as in [24,26], 'regularizing' (P B ) at u = 0, using ε > 0. We refer to [20, Section 5] for a similar approach introducing a new parameter for a different regular problem.…”

An indefinite concave-convex equation under a Neumann boundary condition II

“…In this article, we proceed with the investigation of (P λ ) made in [13]. We are now concerned with the case where b ≥ 0 and we investigate the existence of a unbounded subcontinuum C 0 = {(λ, u)} of nontrivial non-negative solutions of (P λ ), bifurcating from the trivial line {(λ, 0)}.…”