Nonnegative solutions for the degenerate logistic indefinite sublinear equation

Abstract

The goal of this paper is to study the nonnegative steady-states solutions of the degenerate logistic indefinite sublinear problem. We combine bifurcation method and linking local subsupersolution technique to show the existence and multiplicity of nonnegative solutions. We employ a change of variable already used in a different context and the spectral singular theory to prove uniqueness results.

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

“…For other works dealing with non-negative solutions of indefinite concave-convex problems under Dirichlet boundary conditions we refer to [12,23,28]. Several differences between (P λ ) and (1.2) may be observed.…”

“…In [19] he extended these results to a problem with a non-autonomous concave-convex nonlinearity. Delgado and Suárez [12] considered the problem Lu = λ|u| q−2 u + a(x)|u| p−2 u in Ω,…”

On the uniqueness of positive solution of an elliptic equation

“…with different values of q and p, and a, b ∈ L ∞ (Ω). This nonlinearity arises from the study of the population density of a species whose mobility depends upon its density, see [10] and [11]. Some uniqueness results were obtained in [12] and [13].…”

Periodic parabolic problems with concave and convex nonlinearities

“…On the other hand, for a non-selfadjoint operator, some existence results have been proved recently for 0 ≤ a, b ∈ C 1 (Ω) and allowing these functions to have a singularity in the boundary in [12], Section 3, and also in [4] assuming that a ≡ 1 and b ∈ C α Ω , α ∈ (0, 1). In the parabolic case, when b ≤ 0 (logistic-type equation), existence of positive periodic solutions was shown in [9], but no results seem to be known for (1.3) when b > 0 in a set of positive measure.…”

“…In both [1] and [6], existence of at least two (positive) solutions for (1.3) is proved under some growth restriction on q (namely, q ≤ (N + 2) / (N − 2)) using variational arguments, which of course are not eligible in our case. On the other hand, also under some restrictions on b and q, in [4] a second (positive) solution is found in a different way assuming that L is selfadjoint and making use of some spectral theory with singular potential 446 T. Godoy and U. Kaufmann NoDEA (see [12]) which is not available for the periodic parabolic problem. We believe that at least some of the above mentioned multiplicity results should still be true in the parabolic case, but we are not able to give a proof.…”