Robin problems with indefinite linear part and competition phenomena

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 $\lambda \mapsto \tilde{u}_\lambda$

Concave-Convex Problems for the Robin p-Laplacian Plus an Indefinite Potential

“…In this paper, we have investigated a very general version of the "convex-concave problem". As it was illustrated in earlier works (see, for example, [2,[4][5][6]8,10]), such problems exhibit interesting mathematical features. In the past, most of the works deal with Dirichlet problems with no potential term.…”

“…In addition, we show that, for every admissible parameter λ ∈ L = (0, λ * ], problem (1) has a smallest positive solution u * λ , and we also examine the monotonicity and continuity properties of the map L λ → u * λ . Our work here extends to nonlinear problems driven by the p-Laplacian the recent semilinear work of Papageorgiou-Rȃdulescu-Repovš [2]. An inspection of their method of proof reveals that it is heavily dependent on the fact that the Sobolev space H 1 (Ω) can be written as the orthogonal direct sum of the eigenspaces of −∆ + ξ(z)I with Robin boundary condition.…”

“…In contrast, in the nonlinear case, it is much more difficult to come up with strong comparison principles and stronger conditions are needed. We point out that our conditions on the two nonlinearities g(z, x) and f (z, x), are in general less restrictive than those in [2]. Finally, we mention the recent work of Marano-Marino-Papageorgiou [3] on Dirichlet problems with no potential term and a more restrictive reaction.…”

“…In the literature, equations having a (p − 1)-superlinear reaction are usually treated using the AR-condition, which says that there exist [19]). Actually, this is a unilateral version of the AR-condition since we have assumed (2) and 3. Integrating (4a) and using (4b), we obtain a weaker condition, namely that…”

Nodal solutions for the Robin <i>p</i>-Laplacian plus an indefinite potential and a general reaction term

“…More recently, Li and Wang [6] produced infinitely many nodal solutions for semilinear Schrödinger equations. We also refer to our recent papers [11,12], which deal with the qualitative analysis of nonlinear Robin problems.…”