Abstract
Abstract We establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a nonregular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn’s topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved by us, and extend previous results established in the powerlike case.
Highlights
Introduction and main resultsLet Ω ⊂ RN (N ≥ 1) be a bounded domain with smooth boundary ∂Ω
We focus on the case f (s) = sq, q ∈ (0, 1)
We describe here the explicit formula for the growth condition Hb of b+ in a neighborhood of ∂Ωb+ used in Theorems 1.1 and 1.2, which originates from Amann and Lopez-Gomez [2, Theorem 4.3]:
Summary
Let Ω ⊂ RN (N ≥ 1) be a bounded domain with smooth boundary ∂Ω. In this paper, we consider nonnegative solutions of the problem (PB ). We consider the Neumann problem under the condition b < 0 We are able to discuss the positivity of (nontrivial) nonnegative solutions for (PB) with (1.1), assuming It is known [26, Theorem 1] that if Ωa+ is connected, (PB) possesses a loop type subcontinuum C∗ in R × C(Ω) of nonnegative solutions which satisfies (1.6). We state our main results for the Neumann problem, which are given in a similar way as in Theorem 1.1, and where condition (1.16) provides us with a loop type subcontinuum bifurcating both subcritically and supercritically at (0, 0).
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