A Priori Bounds and Multiple Solutions for Superlinear Indefinite Elliptic Problems

Abstract

In this work we study existence and multiplicity questions for positive solutions of second-order semilinear elliptic boundary value problems, where the nonlinearity is multiplied by a weight function which is allowed to change sign and vanish on sets of positive measure. We do not impose a variational structure, thus techniques from the calculus of variations are not applicable. Under various qualitative assumptions on the nonlinearity we establish a priori bounds and employ bifurcation and fixed point index theory to prove existence and multiplicity results for positive solutions. In an appendix we derive interior L p -estimates for general elliptic systems of arbitrary order under minimal smoothness hypotheses. Special instances of these results are used in the derivation of a priori bounds. Academic Press

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

“…By using assertion (1), assertion (2) is a direct consequence of the global bifurcation theory [9]. The following a priori upper bound on the uniform norm of nonnegative solutions of (Q λ,ǫ ) is obtained using a blow up technique from Gidas and Spruck [6] and follows from Amann and López-Gómez [2] and López-Gómez, Molina-Meyer and Tellini [7]: Proposition 4.2. Assume (H 1 ) and (H 2 ).…”

“…Proof. The proof is carried out with a minor modification of that of Proposition 2.1 (2). Assume that u n is a positive solution of (Q λn,ǫn ) such that max Ω u n → 0, ǫ n → 0 + , and λ n ≤ −Λ.…”

“…To prove Λ 0 = ∞, we establish an a priori bound for positive solutions of (P λ ) in a similar way as Proposition 2.1 (2). For the sake of a contradiction we may assume |λ n | ≤ Λ, u n → ∞, and u n is a positive solution for λ = λ n .…”

Positive solutions for some indefinite nonlinear eigenvalue elliptic problems with Robin boundary conditions

“…For this step, we use (3.14), an adequate rescaling Gidas-Spruck argument and a Liouville type theorem, see exactly Lemma 4.2 and Theorem 4.3 of [4].…”

Nonnegative solutions for the degenerate logistic indefinite sublinear equation

“…In the last years the case m = 1 (q = 1 and p = 2) has attracted much attention, see [2], [3], [9], [10], [17], [22], [26] and references therein.…”