Positive and nodal solutions for nonlinear nonhomogeneous parametric neumann problems

We consider a parametric Neumann problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential term. The reaction term is superlinear but does not satisfy the Ambrosetti-Rabinowitz condition. First we prove a bifurcation-type result describing in a precise way the dependence of the set of positive solutions on the parameter λ > 0. We also show the existence of a smallest positive solution. Similar results hold for the negative solutions and in this case we have a biggest negative solution. Finally using the extremal constant sign solutions we produce a smooth nodal solution

Robin problems with indefinite linear part and competition phenomena

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

Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $\lambda<\widehat{\lambda}_{1}$ ($\widehat{\lambda}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. In the superlinear case, for $\lambda<\widehat{\lambda}_{1}$ we have at least two positive solutions and for $\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $\bar{u}_{\lambda}$ and we investigate the properties of the map $\lambda\mapsto\bar{u}_{\lambda}$

Confounders in epidemiological associations

The following description of a retrospective non-randomized study is given within a publication in a scientific journal: “A total of 54 consecutive patients (24 male / 28 female; mean age: 14.1 years) treated with sequential thermoplastic aligners during the last 12 months were identified from the archive of a private practice. They were compared to 52 consecutive patients treated with conventional fixed appliances during the same period, who were matched for age, sex, and case severity with the patients in the aligner group. The total duration of treatment in months was extracted from the patient files by a third party not involved in any way with their treatment. Initially, descriptive statistics were calculated for all patient characteristics and for the study’s primary outcome (treatment duration), consisting of means and Standard Deviations (SDs). Student’s t-tests for independent samples and chi-square tests were performed on patient age, sex, and case severity to confirm that the two groups were matched. Finally, a t-test for independent samples was performed on treatment duration to assess any differences between the aligner and the fixed appliance group at the 5% level." The authors of the study give the following table (Table 1) in their Results section and conclude that (i) the two groups were adequately matched, since no statistically significant difference was found for any baseline difference and (ii) aligners and braces are equally efficient, since no statistically significant difference was found for treatment duration