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

Existence and Relaxation Results for Second Order Multivalued Systems

We consider nonlinear systems driven by a general nonhomogeneous differential operator with various types of boundary conditions and with a reaction in which we have the combined effects of a maximal monotone term A(x) and of a multivalued perturbation F(t, x, y) which can be convex or nonconvex valued. We consider the cases where D(A) ≠ RN and D(A) = RN and prove existence and relaxation theorems. Applications to differential variational inequalities and control systems are discussed

Positive solutions for parametric nonlinear periodic problems with competing nonlinearities

We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator plus an indefinite potential and a reaction having the competing effects of concave and convex terms. For the superlinear (concave) term we do not employ the usual in such cases Ambrosetti-Rabinowitz condition. Using variational methods together with truncation, perturbation and comparison techniques, we prove a bifurcation-type theorem describing the set of positive solutions as the parameter varies

Constant sign and nodal solutions for nonlinear elliptic equations with combined nonlinearities

We study a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which is ”concave” (i.e., (p − 1)− sublinear) near zero and ”convex” (i.e., (p − 1)− superlinear) near ±1. Using variational methods combined with truncation and comparison techniques, we show that for all small values of the parameter > 0, the problem has at least five nontrivial smooth solutions (four of constant sign and the fifth nodal). In the Hilbert space case (p = 2), using Morse theory, we produce a sixth nontrivial smooth solution but we do not determine its sign

Periodic problems with a reaction of arbitrary growth

We consider nonlinear periodic equations driven by the scalar p-Laplacian and with a Carath eodory reaction which does not satisfy a global growth condition. Using truncation-perurbation techniques, variational methods and Morse theory, we prove a "three solutions theorem", providing sign information for all the solutions. In the semilinear case (p = 2), we produce a second nodal solution, for a total of four nontrivial solutions. We also cover problems which are resonant at zero

Comparison of resin modified glass ionomer cement and composite resin in class II primary molar restorations: a 2-year parallel randomised clinical trial

AIM To compare the 2-year success rates of a Resin Modified Glass Ionomer Cement (RMGIC) with a composite resin in class II primary molar restorations. METHODS Healthy, cooperative children aged 4-7.5 years with at least one carious primary molar requiring a class II restoration were included in this parallel randomised trial and allocated on a 1:1 basis to composite resin (Z250, 3M ESPE) or RMGIC (Vitremer, 3M ESPE). Restorations were assessed semiannually up to 2 years clinically and radiographically using modified United States Public Health Service criteria, with the primary outcome being all-cause failure. Data were analysed per protocol by binomial linear regression with Relative Risks (RR) and their 95% confidence intervals (CI). RESULTS 55 patients were randomly allocated to either group and 44 analysed at 2 years; with 49 teeth in the Z250 and 55 teeth in the Vitremer group. The all-cause failure rate for both materials was 3% after 1 year (4 and 2% for Z250 and Vitremer, respectively) and 16% after 2 years (16% for both Z250 and Vitremer). Overall, no difference between materials could be found at 2 years (RR = 1.4; 95% CI 0.8, 2.4; P = 0.30). However, Vitremer was associated with more favourable gingival health compared to composite (RR = 0.2; 95% CI 0.1, 0.9; P = 0.03), but also occlusal wear, which was observed exclusively for Vitremer. CONCLUSION No significant difference was found in the overall performance of the two materials, making them suitable for class II primary molar restorations, although RMGIC presented more pronounced occlusal wear of limited clinical importance after 2 years