Dumitru Motreanu

Université de Perpignan | UPVD · Faculty of Exact and Experimental Sciences

The aim of the paper is to study a Dirichlet problem whose equation is driven by a degenerate p-Laplacian with a weight depending on the solution and whose reaction is a convection term, thus depending on the solution and its gradient. The existence of a weak solution is proven by arguing through a truncated auxiliary problem. A major part of the p...

The main result of the paper establishes the existence of a bounded weak solution for a nonlinear Dirichlet problem exhibiting full dependence on the solution u and its gradient ∇u in the reaction term, which is driven by a p-Laplacian-type operator with a coefficient G(u) that can be unbounded. Through a special Moser iteration procedure, it is sh...

This paper deals with the energy functional associated with a quasilinear elliptic equation in RN which is driven by the p-Laplacian operator. It is shown for such functional that any C1(RN) local minimizer in an appropriate sense is a W1,p(RN) local minimizer. This extends to RN the celebrated property of Brezis-Nirenberg type known for bounded do...

We study a nonlinear evolutionary quasi–variational–hemivariational inequality (in short, (QVHVI)) involving a set-valued pseudo-monotone map. The central idea of our approach consists of introducing a parametric variational problem that defines a variational selection associated with (QVHVI). We prove the solvability of the parametric variational...

The paper establishes the existence of infinitely many large energy solutions for a nonlocal elliptic problem involving a variable exponent fractional p(⋅)-Laplacian and a singularity, provided a positive parameter incorporated in the problem is sufficiently small. A variational method can be implemented for an associated problem obtained by trunca...

This paper investigates the inverse problem of estimating a discontinuous parameter in a quasi-variational inequality involving multi-valued terms. We prove that a well-defined parameter-to-solution map admits weakly compact values under some quite general assumptions. The Kakutani-Ky Fan fixed point principle for multi-valued maps is the primary t...

The present paper develops an approximation approach for solving a quasilinear Dirichlet boundary value problem that exhibits a degenerated \begin{document}$ p $\end{document}-Laplacian and full dependence on the solution and its gradient (convection term). The results establish that the solution set is nonempty and bounded. The principal part of t...

We consider Dirichlet boundary value problem where the elliptic equation is driven by a (p, q)-Laplacian with weights and that contains a convection term (i.e., it depends on the solution and its gradient). The notion of (p, q)-Laplacian with weights is considered for the first time. We present an existence result whose proof is based on the theory...

This paper focuses on two Dirichlet boundary value problems whose differential operators in the principal part exhibit a lack of ellipticity and contain a convection term (depending on the solution and its gradient). They are driven by a degenerated (p,q)-Laplacian with weights and a competing (p,q)-Laplacian with weights, respectively. The notion...

"The paper focuses on a nonstandard Dirichlet problem driven by the operator $-\Delta_p +\mu\Delta_q$, which is a competing $(p,q)$-Laplacian with lack of ellipticity if $\mu>0$, and exhibiting a reaction term in the form of a convection (i.e., it depends on the solution and its gradient) composed with the convolution of the solution with an integr...

The paper develops a sub-supersolution approach for quasilinear elliptic equations driven by degenerated p-Laplacian and containing a convection term. The presence of the degenerated operator forces a substantial change to the functional setting of previous works. The existence and location of solutions through a sub-supersolution is established. T...

The paper deals with a quasilinear Dirichlet problem involving a competing ( p , q )-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a gene...

In this paper, we consider the regularization of a class of elliptic variational‐hemivariational inequalities driven by the fractional Laplace operator. First, we demonstrate characterizations of nonlocal elliptic variational‐hemivariational inequalities. Next, we provide coercivity conditions that guarantee the existence and uniqueness of solution...

The paper is devoted to a new kind of implicit obstacle problem given by a fractional Laplacian-type operator and a set-valued term, which is described by a generalized gradient. An existence theorem for the considered implicit obstacle problem is established, using a surjectivity theorem for set-valued mappings, Kluge’s fixed point principle and n...

The chapter focuses on a Kirchhoff-type elliptic inclusion problem driven by a generalized nonlocal fractional p-Laplacian whose nonlocal term vanishes at finitely many points and for which the multivalued term is in the form of the generalized gradient of a locally Lipschitz function. The corresponding elliptic equation has been treated in (Liu et...

Existence and location of solutions to a Neumann problem driven by an nonhomogeneous differential operator and with gradient dependence are established developing a non-variational approach based on an adequate method of sub-supersolution. The abstract theorem is applied to prove the existence of finitely many positive solutions or even infinitely...

The paper sets forth a new type of variational problem without any ellipticity or monotonicity condition. A prototype is a differential inclusion whose driving operator is the competing weighted $(p,q)$-Laplacian $-\Delta_p u+\mu\Delta_q u$ with $\mu\in \mathbb{R}$. Local and nonlocal boundary value problems fitting into this nonstandard setting ar...

In this paper, the existence of smooth positive solutions to a Robin boundary-value problem with non-homogeneous differential operator and reaction given by a nonlinear convection term plus a singular one is established. Proofs chiefly exploit sub-super-solution and truncation techniques, set-valued analysis, recursive methods, nonlinear regularity...

In this paper, with a fixed p∈(1+∞) and a bounded domain , whose boundary ∂Ω fulfills the Lipschitz regularity, we study the following boundary value problem where are Carathéodory functions, a > 0 is a constant, is an extension operator related to Ω, and ρ is an integrable function on ℝN. This is a novel problem that involves the nonlocal operato...

The paper investigates a nonlinear elliptic problem with a Robin boundary condition, which exhibits a convection term with full dependence on the solution and its gradient. A sub- supersolution approach is developed for this type of problems. The main result establishes the existence of a solution enclosed in the ordered interval formed by a sub-su...

The paper focuses on a nonlocal Dirichlet problem with asymmetric nonlinearities. The equation is driven by the fractional Laplacian (−Δ)s for s∈(0,1) and exhibits a sublinear term containing a parameter λ, a linear term interfering with the spectrum of (−Δ)s and a superlinear term with fractional critical growth. The corresponding local problem go...

Our objective is to study a new type of Dirichlet boundary value problem consisting of a system of equations with parameters, where the reaction terms depend on both the solution and its gradient (i.e., they are convection terms) and incorporate the effects of convolutions. We present results on existence, uniqueness and dependence of solutions wit...

The paper introduces a new type of nonlinear elliptic Dirichlet problem driven by the (p, q)-Laplacian where the reaction term is in the convection form (meaning that it exhibits dependence on the solution and its gradient) composed with a (possibly nonlinear) general map called intrinsic operator on the Sobolev space. Under verifiable hypotheses,...

In this paper, the existence of smooth positive solutions to a Robin boundary-value problem with non-homogeneous differential operator and reaction given by a nonlinear convection term plus a singular one is established. Proofs chiefly exploit sub-super-solution and truncation techniques, set-valued analysis, recursive methods, nonlinear regularity...

In this paper we study from a qualitative point of view the nonlinear singular Dirichlet problem depending on a parameter λ > 0 that was considered in [32]. Denoting by S λ the set of positive solutions of the problem corresponding to the parameter λ , we establish the following essential properties of S λ : there exists a smallest element $\begin{...

The existence of solutions of opposite constant sign is proved for a Dirichlet problem driven by the weighted (p,q)-Laplacian with q<p and exhibiting a (q−1)-order term as well as a convection term. The approach is based on the method of sub–supersolution. Extremal solutions in relevant ordered intervals are obtained as well.

The purpose of this paper is to study a class of semilinear differential variational systems with nonlocal boundary conditions, which are obtained by mixing semilinear evolution equations and generalized variational inequalities. First we prove essential properties of the solution set for generalized variational inequalities. Then without requiring...

In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution set. Our approach is based on Leray–Schauder alternative principle, method of sub-supersolution, nonlinear regul...

We consider an initial–boundary value problem for a quasilinear parabolic system of hemivariational inequalities which is not necessarily coercive. The system exhibits full dependence on the gradient of the solution and is doubly coupled on both the source and multivalued terms. Based on sub-supersolutions, truncation functions, and nonsmooth analy...

In this paper, we consider the existence of multiple solutions of the homogeneous Dirichlet problem for a (\(p,q\))-elliptic system with nonlinear product term as follows: $$ \textstyle\begin{cases} {-}\Delta_{p}u=\lambda \alpha (x)\vert u\vert ^{ \alpha (x)-2}u\vert v\vert ^{\beta (x)}+F_{u}(x,u,v)&\text{in }\Omega, \\ {-}\Delta_{q}v=\lambda \beta...

We develop an approach based on the subsolution–supersolution method for an elliptic system of hemivariational inequalities. The system exhibits full dependence on the gradient of the solution and is doubly coupled on both the source and multivalued terms. We prove the existence of solutions in a prescribed trapping region and, as an application, p...

We establish the existence of at least three nontrivial solutions for a nonvariational quasilinear elliptic system with homogeneous Dirichlet boundary condition. Two of these solutions are of opposite constant sign and the third one is nodal in an appropriate sense provided that a suitable location occurs. The approach combines the methods of sub-s...

Existence and regularity results for quasilinear elliptic equations driven by $(p,q)$-Laplacian and with gradient dependence are presented. A location principle for nodal (i.e., sign changing) solutions is obtained by means of constant sign solutions whose existence is also derived. Criteria for the existence of extremal solutions are finally estab...

In this paper we investigate the system obtained by mixing a nonlinear evolutionary equation and a mixed variational inequality ((EEVI), for short) on Banach spaces in the case where the set of constraints is not necessarily compact and the problem is driven by a ϕ-pseudomonotone operator which is not necessarily monotone. In this way, we extend th...

In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting...

Existence and location of solutions to a Dirichlet problem driven by (p, q)-Laplacian and containing a (convection) term fully depending on the solution and its gradient are established through the method of subsolution-supersolution. Here we substantially improve the growth condition used in preceding works. The abstract theorem is applied to get...

Existence and location of solutions to a Dirichlet problem driven by (p, q)-Laplacian and containing a (convection) multivalued term fully depending on the solution and its gradient are established through the method of subsolution–supersolution. This result extends preceding works, in particular improving the growth condition for the lower order t...

This paper is devoted to the study of the differential systems in arbitrary Banach spaces that are obtained by mixing nonlinear evolutionary equations and generalized quasi-hemivariational inequalities (EEQHVI). We start by showing that the solution set of the quasi-hemivariational inequality associated to problem EEQHVI is nonempty, closed, and co...

We discuss the well-posedness and the well-posedness in the generalized sense of differential mixed quasi-variational inequalities ((DMQVIs), for short) in Hilbert spaces. This gives us an outlook to the convergence analysis of approximating sequences of solutions for (DMQVIs). Using these concepts we point out the relation between metric character...

For the homogeneous Dirichlet problem involving a system of equations driven by (p, q)-Laplacian operators and general gradient dependence we prove the existence of solutions in the ordered rectangle determined by a subsolution-supersolution. This extends the preceding results based on the method of subsolution-supersolution for systems of elliptic...

We prove the existence and regularity of solutions for a quasilinear elliptic system with convection terms that can be singular in the solution and its gradient. Comparison properties and a priori estimates are also obtained. Our approach relies on invariance, regularity, strong maximum principle, and fixed point arguments.

We consider the Dirichlet boundary value problem for quasilinear elliptic systems in a bounded domain (Formula presented.) with a diagonal (Formula presented.)-Laplacian as leading differential operator of the form (Formula presented.)where the component functions (Formula presented.) ((Formula presented.)) of the lower order vector field may also...

We prove the existence of positive solutions for the equation −∑i=1N∂∂xiai(x,u,∇u)=f(x,u,∇u)inΩ under the Dirichlet boundary condition, where the essential point is the dependence of the terms of the elliptic equation on the solution u and its gradient ∇u. We develop an approach based on approximate solutions and on a new strong maximum principle.

The paper focuses on the homoclinic solutions of a general second order Hamiltonian system. By applying an abstract parametric transversality result, it is shown that generically the problem admits finitely many homoclinic solutions. These solutions are nondegenerate in the sense that they correspond to nondegenerate critical points of the associat...

The aim of this paper is to study the Dirichlet boundary value problem for systems of equations involving the (pi, qi)-Laplacian operators and parameters μi ≥ 0 (i = 1, 2) in the principal part. Another main point is that the nonlinearities in the reaction terms are allowed to depend on both the solution and its gradient. We prove results ensuring...

The aim of this paper is to introduce and study a new class of problems called partial differential hemivariational inequalities that combines evolution equations and hemivariational inequalities. First, we introduce the concept of strong well-posedness for mixed variational quasi hemivariational inequalities and establish metric characterizations...

In this paper, we investigate boundary blow-up solutions of the problem where Δp(x)u = div (|∇u|p(x)-2∇u) is called p(x)-Laplacian. Our results extend the previous work of J. García-Melián, A. Suárez [23] from the case where p(.) ≡ 2, without gradient term, to the case where p(.) is a function, with gradient term. It also extends the previous work...

The generic existence of Morse functions in a prescribed family of smooth functionals is investigated. The approach is based on arguments involving the transversality theory. The abstract result is applied to semilinear elliptic boundary value problems. One obtains qualitative information concerning the set of solutions.

The paper focuses on a Dirichlet problem driven by the -Laplacian containing a parameter in the principal part of the elliptic equation and a (convection) term fully depending on the solution and its gradient. Existence of solutions, uniqueness, a priori estimates, and asymptotic properties as and are established under suitable conditions.

In this paper we introduce the differential system obtained by mixing an evolution equation and a variational inequality ((EEVI), for short). First, by using KKM theorem and monotonicity arguments, we prove the superpositional measurability and upper semicontinuity for the solution set of a general variational inequality. Then we establish that the...

This paper provides existence and non-existence results for a positive solution of the quasilinear elliptic equation driven by the nonhomogeneous operator (p, q)-Laplacian under Dirichlet boundary condition, with μ > 0 and 1 < q < p < ∞. We show that in the case where μ > 0 the results are completely different from those for the usual eigenvalue pr...

This paper gives new existence results for elliptic and evolutionary variational and quasi-variational inequalities. Specifically, we give an existence theorem for evolutionary variational inequalities involving different types of pseudo-monotone operators. Another existence result embarks on elliptic variational inequalities driven by maximal mono...

The aim of this paper is to present a coincidence point theorem for sequentially weakly continuous maps. Moreover, as a consequence, a critical point theorem for functionals possibly containing a nonsmooth part is obtained. Finally, as an application, existence results for nonlinear differential problems depending also on the derivative of the solu...

In the present paper we prove a multiplicity theorem for a quasi-linear elliptic problem with dependence on the gradient ensuring the existence of a positive solution and of a negative solution. In addition, we show the existence of the extremal constant-sign solutions: the smallest positive solution and the biggest negative solution. Our approach...

An ill-posed quasi-variational inequality with contaminated data can be stabilized by employing the elliptic regularization. Under suitable conditions, a sequence of bounded regularized solutions converges strongly to a solution of the original quasi-variational inequality. Moreover, the conditions that ensure the bounded-ness of regularized soluti...

The chapter presents a general method, based on approximation of spaces and operators, to solve certain nonsmooth problems. The method allows us to obtain location properties of the solutions, for instance the inclusion of the solutions in prescribed sets. This is achieved through an approximation approach by means of sequences of associated proble...

Here we study the approximate controllability for control problems driven by a claß of nonlinear evolution hemivariational inequalities in Hilbert spaces. Actually, our results cover a broader claß of inclusion problems involving time-dependent operators. First, by using a fixed point approach and nonsmooth analysis, we show the existence of mild s...

This paper deals with the Dirichlet boundary value problem for quasilinear elliptic systems in a bounded domain with a diagonal -Laplacian as leading differential operator and a Carathéodory right-hand side vector field . Only by imposing certain growth conditions on , , near zero we are able to prove the existence of multiple, nontrivial solutions...

The paper presents existence and multiplicity results for non-linear boundary value problems on possibly non-smooth and unbounded domains under possibly non-homogeneous Dirichlet boundary conditions. We develop here an appropriate functional setting based on weighted Sobolev spaces. Our results are obtained by using global minimization and a minima...

In this paper we establish existence and regularity of positive solutions for a singular quasilinear elliptic system with competitive structure. The approach is based on comparison properties, a priori estimates and the Schauder’s fixed point theorem.

This chapter is concerned with parametric Dirichlet boundary value problems involving the p-Laplacian operator. Specifically, this chapter gives an account of recent results that establish the existence and multiplicity of solutions according to different types of nonlinearities in the problem. More precisely, we focus on problems exhibiting nonlin...

The aim of this paper is to prove the existence of a positive solution for a quasi-linear elliptic problem involving the (p,q)-Laplacian and a convection term, which means an expression that is not in the principal part and depends on the solution and its gradient. The solution is constructed through an approximating process based on gradient bound...

This chapter examines the existence and multiplicity of periodic solutions for nonlinear ordinary differential equations. The first section of the chapter investigates a nonlinear periodic problem involving the scalar p-Laplacian for 1 p ∞ in the principal part and a smooth potential. The results cover cases of resonance at any eigenvalue of the pr...

This chapter focuses on important classes of nonlinear operators stating abstract results that offer powerful tools for establishing the existence of solutions to nonlinear equations. Specifically, they are useful in the study of nonlinear elliptic boundary value problems as demonstrated in the final three chapters of the present book. The first se...

This chapter provides a self-contained account of the spectral properties of the following fundamental differential operators: Laplacian, p-Laplacian, and p-Laplacian plus an indefinite potential, with any 1 p ∞. The first section of the chapter examines the spectrum of the Laplacian separately under Dirichlet and Neumann boundary conditions, takin...

This chapter addresses variational principles and critical point theory that will be applied later in the book for setting up variational methods in the case of nonlinear elliptic boundary value problems. The first section of the chapter illustrates the connection between the variational principles of Ekeland and Zhong and compactness-type conditio...

This chapter provides the fundamental elements of degree theory used later in the book for showing abstract results of critical point theory or bifurcation theory as well as for the study of the existence and multiplicity of solutions to nonlinear problems. The first section of the chapter introduces Brouwer’s degree and its important applications...

This chapter provides a comprehensive survey of the mathematical background of Sobolev spaces that is needed in the rest of the book. In addition to the standard notions, results, and calculus rules, various other useful topics, such as Green’s identity, the Poincaré–Wirtinger inequality, and nodal domains, are also discussed. A careful distinction...

This chapter studies nonlinear Dirichlet boundary value problems through various methods such as degree theory, variational methods, lower and upper solutions, Morse theory, and nonlinear operators techniques. The combined application of these methods enables us to handle, under suitable hypotheses, a large variety of cases: sublinear, asymptotical...

This chapter represents a self-contained presentation of basic results and techniques of Morse theory that are useful for studying the multiplicity of solutions of nonlinear elliptic boundary value problems with a variational structure. The first section of the chapter contains the needed preliminaries of algebraic topology. The second section focu...

This chapter examines the bifurcation points of parametric equations, that is, values of a parameter from which the set of solutions splits into several branches. The deep connection between bifurcation points and the spectrum of linear operators involved in problems is pointed out. The presentation consists of two parts regarding the used approach...

The present survey aims to report on recent advances in the study of non-linear elliptic problems whose differential part is expressed by a general operator in divergence form. The pattern of such differential operator is the p-Laplacian Δp with \(1<p<+\infty\). More general operators can be considered, possibly having completely different properti...

In this article, we investigate the existence of positive solutions of a singular quasilinear elliptic system for which the cooperative structure is not required. The approach is based on the Schauder fixed point theorem combined with perturbation arguments that involve the singular terms.

Using variational methods based on the critical point theory and suitable truncation and comparison techniques, we study existence, multiplicity and nonexistence of positive solutions for a parametric nonlinear Neumann problem driven by the p-Laplacian. Our hypotheses cover the case of nonlinearities of concave-convex type whose exponents depend on...

The existence of positive solutions for nonlinear elliptic problems under Dirichlet boundary condition is studied as well as the compactness and directness of the solution set. The main novelties consist in the presence of a Leray–Lions operator in the differential part and in the dependence of the reaction term on the gradient of the solution. Our...

This article sets forth results on the existence,a priori estimates and boundedness of positive solutions of a singular quasilinear system of elliptic equations. The systems studied here have in the principal part differentpLaplacians with Dirichlet boundary condition on a bounded domain. The approach is based on the sub-supersolution methods for s...

The paper presents existence results for nonlinear elliptic problems under a nonhomogeneous Dirichlet boundary condition. The considered elliptic equations exhibit nonlinearities containing derivatives of the solution.

Through variational methods, sub-supersolution and truncation techniques we prove the existence of three nontrivial solutions for a quasilinear elliptic equation with Neumann boundary condition. We provide sign information for each of these solutions: two of them are of opposite constant sign and the third one is sign changing.

This chapter provides a comprehensive presentation of regularity theorems and maximum principles that are essential for the subsequent study of nonlinear elliptic boundary value problems. In addition to the presentation of fundamental results, the chapter offers, to a large extent, a novel approach with clarification of tedious arguments and simpli...

This chapter aims to present relevant knowledge regarding recent progress on nonlinear elliptic equations with Neumann boundary conditions. In fact, all the results presented here bring novelties with respect to the available literature. We emphasize the specific functional setting and techniques involved in handling the Neumann problems, which are...

This chapter offers a systematic presentation of nonsmooth analysis containing all that is necessary in this direction for the rest of the book. The first section of the chapter gathers significant results of convex analysis, especially related to the convex subdifferential such as its property of being a maximal monotone operator. The second secti...

We prove the solvability of a unilateral dynamic problem driven by a wave equation with nonconstant coefficients in the principal part and containing a nonlinear reaction term and constraints of obstacle type on the boundary.