Cristian Bereanu

Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces

In this paper, using the Schauder fixed point theorem, we prove existence results of radial solutions for Dirichlet problems in the unit ball and in an annular domain, associated to mean curvature operators in Euclidean and Minkowski spaces.

Existence and multiplicity results for periodic solutions of nonlinear difference equations

We use Brouwer degree to prove existence and multiplicity results for the periodic solutions of some nonlinear second-order and first-order difference equations. We obtain, in particular upper and lower solutions theorems, Ambrosetti–Prodi type results and sharp existence conditions for nonlinearities which are bounded from below or above.

Boundary value problems for second-order nonlinear difference equations with discrete phi-Laplacian and singular phi

In this article, we study the existence and multiplicity of solutions for boundary value problems of the type where denotes an increasing homeomorphism such that and denotes the Dirichlet, periodic or Neumann boundary conditions and f k are continuous functions. Our main tool is Brouwer degree together with fixed point reformulations of the above problems.

Existence of at least two periodic solutions of the forced relativistic pendulum

Using Szulkin’s critical point theory, we prove that the relativistic forced pendulum with periodic boundary value conditions ( u′ √ 1− u′2 )′ + μ sinu = h(t), u(0)− u(T ) = 0 = u′(0)− u′(T ), has at least two solutions not differing by a multiple of 2π for any continuous function h : [0, T ] → R with ∫ T 0 h(t)dt = 0 and any μ = 0. The existence of at least one solution has been recently proved by Brezis and Mawhin.

Variational methods for nonlinear perturbations of singular ϕ-Laplacians

Motivated by the existence of radial solutions to the Neumann problem involving the mean extrinsic curvature operator in Minkowski space div(EQUATION PRESANT) where 0 ≤ R 1 2 , A = {x a R N : R 1 ≤ |x| a R 2 } and g: [R 1 ; R 2 ]x R → R is continuous, we study the more general problem [r N - 1 φ{u)] = r N - 1 g(r;u); u(R 1 ) = 0 = u{R 2 ); where φ:= φ′: (-a;a) → R is an increasing homeomorphism with φ(0) = 0 and the continuous function φ: [-a; a] → R is of class C 1 on (-a; a). The associated functional in the space of continuous functions over [R1; R 2 ] is the sum of a convex lower semicontinuous functional and of a functional of class C 1 . Using the critical point theory of Szulkin, we obtain various existence and multiplicity results for several classes of nonlinearities. We also discuss the case of the periodic problem.

The Dirichlet problem with mean curvature operator in Minkowski space - A variational approach

Abstract In this paper we consider the Dirichlet problem with mean curvature operator in Minkowski space : where Ω ⊂ ℝN is a smooth open bounded set and f : Ω × ℝ → ℝ is a Carathéodory function. We show that u is a solution of the above problem iff it is a critical point of the corresponding non-smooth action functional. Applications concerning nontrivial solutions for a class of such Dirichet problems depending on a parameter are provided.

Ground state and mountain pass solutions for discrete p(⋅)-Laplacian

In this paper we study the existence of solutions for discrete p(⋅)-Laplacian equations subjected to a potential type boundary condition. Our approach relies on Szulkin’s critical point theory and enables us to obtain the existence of ground state as well as mountain pass type solutions.MSC: 39A12, 39A70, 49J40, 65Q10.

PERIODIC SOLUTIONS FOR SINGULAR PERTURBATIONS OF THE SINGULAR ϕ-LAPLACIAN OPERATOR

In this paper, using Leray–Schauder degree arguments and the method of lower and upper solutions, we give existence and multiplicity results for periodic problems with singular nonlinearities of the type where r, n, e : [0, T] → ℝ are continuous functions and λ > 0. We also consider some singular nonlinearities arising in nonlinear elasticity or of Rayleigh–Plesset type.

Corrigendum to: The Dirichlet problem with mean curvature operator in Minkowski space

We essentially use Lemma 2.1 which is proved in [1]. Accordingly, the set Ω must be “a bounded domain in RN (N ≥ 2) with boundary ∂Ω of class C2”, instead of “an open bounded set in RN with boundary ∂Ω of class C2”. This change does not affect the validity of the results stated in the paper. But, it affects the proof of Lemma 3.1. In this view, the only necessary modification is the following. Proof of Lemma 3.1. It suffices to show that u ≥ 0 in Ω. From (2.2), (3.4) and the integration by parts formula it follows −∫ Ω ∇u ⋅ ∇v √1 − |∇u|2 = ∫ Ω μ(x)|u|q−1uv − λ∫ Ω g(x, u)v, (1)

Existence and multiplicity of entire radial spacelike graphs with prescribed mean curvature function in certain Friedmann–Lemaître–Robertson–Walker spacetimes

We provide sufficient conditions for the existence of a uniparametric family of entire spacelike graphs with prescribed mean curvature in a Friedmann–Lemaitre–Robertson–Walker spacetime with flat fiber. The proof is based on the analysis of the associated homogeneous Dirichlet problem on a Euclidean ball together with suitable bounds for the gradient which permit the prolongability of the solution to the whole space.