Petru Jebelean

West University of Timisoara · Department of Mathematics

We obtain the existence of multiple pairs of periodic solutions for difference equations of type −ΔΔu(n−1)1−|Δu(n−1)|2=λg(u(n))(n∈Z), where g:R→R is a continuous odd function with anticoercive primitive, and λ>0 is a real parameter. The approach is variational and relies on the critical point theory for convex, lower semicontinuous perturbations of...

We are concerned with the existence of periodic solutions for potential differential inclusions involving the p-relativistic operator $$\begin{aligned} {\mathcal {R}}_pu:= \left( \frac{|u'|^{p-2}u'}{(1-|u'|^p)^{1-1/p}} \right) ' \end{aligned}$$and an (possible) unbounded discontinuous gradient. The approach relies on critical point theory for local...

We provide a complete description of the existence/non-existence and multiplicity of distinct pairs of nontrivial solutions to the problem with Minkowski operator $$ -\mbox{div} \left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= \lambda u(1-a |u|^q) \quad \mbox{ in } \Omega, \; \; u|_{\partial \Omega}=0, \quad (a\geq0

We prove the existence of m+1 geometrically distinct periodic solutions for N-dimensional systems involving the p-relativistic operator Rpu≔|u′|p−2u′(1−|u′|p)1−1∕p′and an m-periodic potential; here m<N. This is obtained by reduction to an equivalent non-singular problem for which the existence of multiple critical orbits can be inferred.

We prove that the four-point boundary value problem -[ϕ(u′)]′=f(t,u,u′),u(0)=αu(ξ),u(T)=βu(η),where f: [ 0 , T] × R ² → R is continuous, α,β∈[0,1), 0 < ξ< η< T, and ϕ: (- a, a) → R (0 < a< ∞) is an increasing homeomorphism, which is always solvable. When instead of f is some g: [ 0 , T] × [ 0 , ∞) → [ 0 , ∞) , we obtain existence, localization, and...

The localization of positive symmetric solutions to the Dirichlet problem for second‐order ordinary differential equations involving a singular ϕ ‐Laplacian is established in a conical annular set, via Ekeland's variational principle, compression type conditions, and a Harnack type inequality. An application to a one‐parameter problem is provided a...

We deal with Dirichlet systems involving the mean curvature operator in Minkowski space M(w)=div([Formula presented]) in a ball in RN. Using the fixed point index and the lower and upper solutions method, we first obtain the existence of positive solutions for a class of differential systems with a singular φ-Laplacian, subjected to homogeneous mix...

We are concerned with the existence of multiple periodic solutions for differential equations involving Fisher-Kolmogorov perturbations of the relativistic operator of the form (Formula Presented) as well as for difference equations, of type (Formula Presented) here q > 0 is fixed, Δ is the forward difference operator, λ > 0 is a real parameter and...

Using a variational approach we obtain the existence of at least three periodic solutions for discontinuous perturbations of the vector p -Laplacian operator .

In this paper, we use the critical point theory for convex, lower semicontinuous perturbations of

A multiplicity result for periodic problems of the form when ψ: ℝN → ℝN belongs to a suitable class of homeomorphisms, V is Ti-periodic in each component ui of u ∈ ℝN, and e has mean value zero on [0, T] is proved, and applied, by a modification technique, to obtain the same multiplicity for the solutions of the relativistic system.

We prove the existence of at least (Formula presented.) geometrically distinct (Formula presented.)-periodic solutions for a differential inclusions system of the form (Formula presented.) Here, (Formula presented.) is a monotone homeomorphism, (Formula presented.) is periodic with respect to each component of the second variable and (Formula prese...

We consider the system of difference equations $\Delta\bigg{(}\frac{\Delta u_{n-1}}{\sqrt{1-|\Delta u_{n-1}|^{2}}}\bigg{)}=% \nabla V_{n}(u_{n})+h_{n},\quad u_{n}=u_{n+T}\quad(n\in\mathbb{Z}),$ with ${\Delta u_{n}=u_{n+1}-u_{n}\in{\mathbb{R}}^{N}}$ , ${V_{n}=V_{n}(x)\in C^{2}({\mathbb{R}}^{N},\mathbb{R})}$ , ${V_{n+T}=V_{n}}$ , ${h_{n+T}=h_{n}}$ fo...

The aim of this paper is to present an existence result of two positive solutions for a nonlinear difference problem by variational methods. The conclusion is achieved by assuming, together with the super-linearity at infinity, a suitable algebraic condition on the nonlinear term, which is more general than the sub-linearity at zero.

We show that the periodically perturbed N-dimensional relativistic pendulum equation has at least N + 1 geometrically distinct periodic solutions Also we obtain the existence of infinitely many solutions for systems with oscillating potential Both results are obtained by reduction to an equivalent non-singular problem using classical critical point...

Systems of differential inclusions of the form -(ϕ(u'))'∈ ∂F(t,u), t ∈ [0,T], where ϕ = ∇Φ, with Φ strictly convex, is a homeomorphism of the ball Ba ⊂RN onto RN, are considered under Dirichlet, periodic and Neumann boundary conditions. Here, ∂F(t, x) stands for the generalized Clarke gradient of F(t, {dot operator}) at x ∈ RN. Using nonsmooth crit...

We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable "sub-linear" growth then the system has at least one solution. The approach relies on the application, in a suitable Fr\'echet...

In this paper we study the existence and multiplicity of periodic solutions for discontinuous perturbations of the operator . The results are obtained by reduction to an equivalent non-singular problem and using the non-smooth critical point theory. Some illustrative examples concerning Filippov type solutions are also provided.

Using the critical point theory for convex, lower semicontinuous perturbations of locally Lipschitz functionals, we prove the solvability of the discontinuous Dirichlet problem involving the operator $u\mapsto{div} (\frac{\nabla u}{\sqrt{1-|\nabla u|^2}})$.

We use the critical point theory for convex, lower semicontinuous perturbations of C 1 −functionals to establish existence of multiple radial solutions near resonance for some Neumann problems involving the mean extrinsic curvature operator. MSC 2010 Classification : 35J20; 35J62;35J93; 35J87.

In this paper we consider the Dirichlet problem with mean curvature operator in Minkowski space : where Ω ⊂ ℝ

We are concerned with extremal solutions for the mixed boundary value problem $$-\left(r^{N-1}\phi(u')\right)' = r^{N-1} g(r, u), \quad u'(0) = 0 = u(R),$$where \({g : [0, R] \times \mathbb{R} \to \mathbb{R}}\) is a continuous function and \({\phi : (-\eta, \eta) \to \mathbb{R}}\) is an increasing homeomorphism with \({\phi(0) = 0.}\) We prove the...

In this note we are concerned with numerical solutions to Dirichlet problem $$[\phi(u')]' =f(x) \quad \mbox{in} [\alpha, \beta]; \quad u(\alpha)=A, \; u(\beta)=B

We study the Dirichlet problem with mean curvature operator in Minkowski spacediv(∇v1−|∇v|2)+λ[μ(|x|)vq]=0in B(R),v=0on ∂B(R), where λ>0λ>0 is a parameter, q>1q>1, R>0R>0, μ:[0,∞)→Rμ:[0,∞)→R is continuous, strictly positive on (0,∞)(0,∞) and B(R)={x∈RN:|x|<R}B(R)={x∈RN:|x|<R}. Using upper and lower solutions and Leray–Schauder degree type arguments...

Using critical point theory, we study the existence of at least three solutions for some periodic and Neumann boundary value problems involving the discrete p(·)-Laplacian operator. 2010 AMS Subject Classification. 39A12; 39A23; 39A70; 65Q10

Using critical point theory, we obtain the existence of solutions for some periodic boundary value problems involving the discrete p(⋅)p(⋅)-Laplacian. These extend and improve known results for similar problems with discrete pp-Laplacian. Similar results for Neumann problems are also provided. As applications we prove upper and lower solutions theo...

In this paper, by using Leray-Schauder degree arguments and critical point theory for convex, lower semicontinuous perturbations of C-1-functionals, we obtain existence of classical positive radial solutions for Dirichlet problems of type div(del v-root 1 - vertical bar del v vertical bar(2)) + f(vertical bar x vertical bar, v) = 0 in B(R), v = 0 o...

We deal with a class of functionals I on a Banach space X; having the structure I = ψ+G; with : ψ: X → (-∞, +∞] proper, convex, lower semi-continuous and G : X → ℝ of class C 1: Also, I is G-invariant with respect to a discrete subgroup G ⊂ X with dim (span G) = N. Under some appropriate additional assumptions we prove that I has at least N + 1 cri...

In this paper we study the existence of solutions for discrete -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.

We use the critical point theory for convex, lower semicontinuous perturbations of C1-functionals to establish existence of multiple radial solutions for some one parameter Neumann problems involving the operator v↦div(∇v1−|∇v|2). Similar results for periodic problems are also provided.

We consider nonlinear periodic systems driven by the vector p-Laplacian. An existence and a multiplicity theorem are proved. In the existence theorem the potential function is p-superlinear, but in general does not satisfy the AR-condition. In the multiplicity theorem the problem is strongly resonant with respect to the principal eigenvalue λ 0 = 0...

We use the critical point theory to establish existence results for periodic solutions of some nonlinear boundary value problems involving the discrete p-Laplacian operator. As an application we give an alternative proof to the upper and lower solutions theorem.

We present an abstract result concerning Poincaré inequalities in cones. Some examples in Sobolev spaces are provided. We also discuss an application to a priori bounds of solutions for a general boundary value problem involving the vector p-Laplacian operator.

We show that if A Ì \mathbbRN{{\mathcal A} \subset \mathbb{R}^N} is an annulus or a ball centered at zero, the homogeneous Neumann problem on A{{\mathcal A}} for the equation with continuous data Ñ(\fracÑvÖ{1 - |Ñv|2} ) = g(|x|,v) + h(|x|)\nabla \cdot \left(\frac{\nabla v}{\sqrt{1 - |\nabla v|^2}} \right) = g(|x|,v) + h(|x|) has at least one ra...

Motivated by the existence of radial solutions to the Neumann problem involving the mean extrinsic curvature operator in Minkowski space div ∇v p 1 − |∇v| 2 ! = g(|x|, v) in A, ∂v ∂ν = 0 on ∂A, where 0 ≤ R1 < R2, A = {x ∈ R N : R1 ≤ |x| ≤ R2} and g : [R1, R2]×R → R is continuous, we study the more general problem [r N−1 φ(u ′)] ′ = r N−1 g(r, u), u...

We study the existence and multiplicity of radial solutions for Neumann problems in a ball and in an annular domain, associated to pendulum-like perturbations of mean curvature operators in Euclidean and Minkowski spaces and of the p-Laplacian operator. Our approach relies on the Leray-Schauder degree and the upper and lower solutions method.

In this paper we study the existence and multiplicity of periodic solutions of pendulum-like perturbations of bounded or singular f{\phi}-Laplacians. Our approach relies on the Leray-Schauder degree and the upper and lower solutions method. KeywordsCurvature operators-Periodic problem-Pendulum-like non linearities-Leray-Schauder degree-Upper and l...

In this paper we study the existence of radial solutions for Neumann problems in a ball and in an annular domain, associated to mean curvature operators in Euclidean and Minkowski spaces. Our approach relies on the Leray-Schauder degree together with some fixed point reformulations of our nonlinear Neumann boundary value problems (© 2010 WILEY-VCH...

The aim of this article is to establish a general existence result for the p,q-Laplacian system -[h p (u ' )] ' =f(t,u,v)+α(t)-[h q v ' )] ' =g(t,u,v)+β(t)(t∈(0,T)),(1) subjected to the boundary conditions (h p (u ' )(0),-h p (u ' )(T))∈∂j(u(0),u(T)),(h q (v ' )(0),-h q (v ' )(T))∈∂k(v(0),v(T))·(2) Here h p , h q are homeomorphisms of ℝ n and ℝ m d...

Dedicated to the memory of Ehrard Schmidt In this paper we study the existence of radial solutions for Neumann problems in a ball and in an annular domain, associated to mean curvature operators in Euclidean and Minkowski spaces. Our approach relies on the Leray-Schauder degree together with some fixed point reformulations of our nonlinear Neumann...

In this paper, using 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 Euclidian and Minkowski spaces.

In this paper, using Schauder fixed point theorem, we prove exis-tence results concerning radial solutions for Dirichlet problems associated with some systems involving mean curvature operators in Euclidian and Minkowski spaces. The p-Laplacian case is also considered.

We consider a nonlinear nonvariational periodic problem with a nonsmooth potential. Using the spectrum of the asymptotic (as |x| → ∞) differential operator and degree theoretic methods based on the degree map for multivalued perturbations of (S) +  operators, we establish the existence of a nontrivial smooth solution.

In this paper we establish the solvability and approximation of a general inequality problem by means of a sequence of problems satisfying some compatibility conditions with respect to the initial one. The setting allows to unify and extend various existence results in the smooth and nonsmooth analysis. The approach mainly relies on Galerkin like a...

This paper deals with the existence of infinitely many solutions for the boundary value problem {-(vertical bar u'vertical bar(p-2)u')' + epsilon vertical bar u vertical bar(p-2)u = del F (t, u), in (0, T), ((vertical bar u'vertical bar(p-2)u')(0), -(vertical bar u'vertical bar(p-2)u')(T)) is an element of partial derivative j (u(0), u(T)), where e...

This paper deals with the solvability of the boundary value problem {ll -(|u¢|p-2u¢)¢ = f(t,u) + l(t),in (0, T),((|u¢|p-2u¢)(0), -(|u¢|p-2u¢)(T)) Î ¶j(u(0), u(T)), \left\{\begin{array}{ll} -(|u^{\prime}|^{p-2}u^{\prime})^{\prime}= f(t,u) + l(t),\,\,{\rm in} (0, T),\\ ((|u^{\prime}|^{p-2}u^{\prime})(0), -(|u^{\prime}|^{p-2}u^{\prime})(T)) \in {\par...

We present existence results for ordinary p-Laplacian systems of the form -({norm of matrix}u'{norm of matrix}p-2u')' = f(t, u), in [O, T] submitted to the general potential boundary condition ({norm of matrix}u'{norm of matrix}p-2u')(0), -({norm of matrix}u'{norm of matrix}p-2u')(T)) ∈ ∂j (u (0); u (T)). Here, p ∈ (1, ∞) is fixed, j: ℝN,× ℝN → (∞;...

This paper deals with the existence of infinitely many solutions for the boundary value problem -(|u ' | p-2 u ' ) ' +ε|u| p-2 u=∇F(t,u),in(0,T),((|u ' | p-2 u ' )(0)-(|u ' | p-2 u ' )(T))∈∂ j (u(0),u(T)), where ε≥0, p∈(1,∞) are fixed, the convex function j:ℝ N ×ℝ N →(-∞,+∞] is proper, even, lower semicontinuous and F:(0,T)×ℝ N →ℝ is a Carathéodory...

In this paper we discuss some existence results for the boundary value problem

This paper is concerned with the existence of solutions for the boundary value problem where ɛ⩾0, p∈(1,∞) are fixed, is a proper, convex and lower semicontinuous function and is a Carathéodory mapping, continuously differentiable with respect to the second variable and satisfies some usual growth conditions. Our approach is a variational one and re...

This paper deals with the existence of solutions for the discontinuous boundary value problem � (|u'|p−2u')' + "|u|p−2u 2 @F(t,u), in (0,T), ((|u'|p−2u')(0), (|u'|p−2u')(T)) 2 @j(u(0),u(T)),

Using a recent continuation theorem for periodic solutions of quasilin-ear equations of p-Laplacian type, we extend some results of Habets and Sanchez for positive periodic solutions to equations of the form (|x ′ | p−2 x ′) ′ + f (x)x ′ + g(x) = h(t) with p > 1, |f (x)| ≥ c > 0 and g singular at 0.

This paper is concerned with existence results for inequality problems of type F0(u;v)+Ψ′(u;v)≥0, for all v∈X, where X is a Banach space, F:X→ℝ is locally Lipschitz, and Ψ:X→(−∞+∞] is proper, convex, and lower semicontinuous. Here F0 stands for the generalized directional derivative of F and Ψâ...

Using some recent extensions of upper and lower solutions techniques and continuation theorems to the periodic solutions of quasilinear equations of p-Laplacian type, we prove the existence of positive periodic solutions of equations of the form (|xʹ| with p > 1, f arbitrary and g singular at 0. This extends results of Lazer and Solimini for the un...

This paper is concerned with existence and approximation results for differential inclusions of type Ju ∈ Nu where J is a duality mapping corresponding to the weight function ψ(t) = tp-1, p > 1, and the set-valued operator N is a generalized gradient in the sense of Clarke of some locally Lipschitz functional. The applications which we consider foc...

Existence results for a class of nonlinear equations involving a duality mapping were established. The equations were considered in the real reflexive Banach space and described by hemicontinuous and demicontinuous operators. Solutions of the equation were obtained by a study of eigen values. Theorem proving using the Laplacian operator formed part...

Existence results for the equation Ju = Nu are given; here J : X → X* is a duality mapping on a reflexive Banach space and N : Z → Z* is a demicontinuous operator, Z being a Banach space such that X is compactly imbedded in Z. As examples, the existence of a W1,p0-solution, 1 < p < ∞, is proved in an unitary manner for J being the p-Laplacian, the...

Existence results for the equation Ju = Nu are given; here J: X → X* is a duality mapping on a reflexive Banach space and N:Z → Z* is a demicontinuous operator, Z being a Banach space such that X is compactly imbedded in Z. As examples, the existence of a W01,p -solution, 1 < p < ∞, is proved in an unitary manner for J being the p-Laplacian, the Ap...

Due to the compactness of the operator (-Δp)1 Nƒ, 1 < p < ∞, where Δp is the p-Laplacian and Nƒ is the Nemytskii operator corresponding to a Caratheodory function ƒ: Ω × R → R, which satisfies a particular growth condition, the homotopy invariance of Leray-Schauder degree can be used in order to prove the existence of a W01,p (Ω)-solution for the e...

A finite-dimensional approximation of a multivalued problem with a p-Laplacian operator is presented. The equation is based on the hypothesis that the corresponding Euler-Lagrange function is coercive and attains its infinitum only at one point. Convergence of the finite dimensional approximation is also presented.

We consider the continuous linear finite element approximation of the problem -div(|∇u| p-2 ∇u)∈[a ̲(x,u),a ¯(x,u)]+finΩ,u=0on∂Ω under the hypothesis that the corresponding Euler-Lagrange functional Φ is coercive and it attains its infimum in only one point in W 0 1,p (Ω). The single-valued case, when the nonlinearity a is Carathéodory, is also con...

We study the mixed boundary value problem with singular ϕ-Laplacian [r N-1 ϕ(u ' )] ' =r N-1 [α(r)u q-1 -λp(r,u)]in[0,R],u ' (0)=0=u(R), where λ>0 is a parameter, q>1,α:[0,R]→ℝ is positive on (0,R) and the function p:[0,R]×[0,A]→ℝ is positive on (0,R)×(0,A), with p(r,0)=0=p(r,A) for all r∈[0,R]. Using a variational approach, we provide sufficient c...