Abstract
We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter $\lambda > 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_\lambda$ and determine the monotonicity and continuity properties of the map $\lambda \mapsto \tilde{u}_\lambda$.
Highlights
Let Ω ⊆ R (N ≥ 2) be a bounded domain with a C 2 -boundary ∂Ω
In this paper we study the following parametric Robin problem
We proved a multiplicity result for all small values of the parameter λ > 0, producing five nontrivial smooth solutions, four of which have constant sign
Summary
Let Ω ⊆ R (N ≥ 2) be a bounded domain with a C 2 -boundary ∂Ω. In this paper we study the following parametric Robin problem REPOVŠ was initiated with the well-known work of Ambrosetti, Brezis and Cerami [2], who dealt with a Dirichlet problem with zero potential (that is, ξ ≡ 0) and the reaction had the form λxq−1 + xr−1 for all x ≥ 0 with 1 < q < 2 < r < 2∗ They proved a bifurcation-type result for small values of the parameter λ > 0. We establish the existence of a minimal positive solution ũλ and determine the monotonicity and continuity properties of the map λ 7→ ũλ
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