# Publications of Mihailescu, Mihai

An existence result for a nonhomogeneous problem in R^2 related to nonlinear Hencky-type materials

This paper investigates a nonlinear and non-homogeneous system of partial differential equations. The motivation comes from the fact that in a particular case the problem discussed here can be used in modeling the behavior of nonlinear Hencky-type materials. The main result of the paper establishes the existence of a nontrivial solution in an adequate functional space of Orlicz-Sobolev type by using Schauder’s fixed point theorem combined with adequate variational techniques.

On a PDE involving the Ap(⋅)-Laplace operator

This paper establishes the existence of solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present. This operator represents a generalization of the p(⋅)-Laplace operator, i.e. View the MathML source, where p(⋅) is a continuous function. The proof of the main result is based on Schauder’s fixed point theorem combined with adequate variational arguments. The function space setting used here makes appeal to the variable exponent Lebesgue and Sobolev spaces.

Eigenvalues of the Laplace operator with nonlinear boundary conditions

An eigenvalue problem on a bounded domain for the Laplacian with a nonlinear Robin-like boundary condition is investigated. We prove the existence, isolation and simplicity of the ﬁrst two eigenvalues.

Equations involving a variable exponent Grushin-type operator

In this paper we define a Grushin-type operator with a variable exponent growth and establish existence results for an equation involving such an operator. A suitable function space setting is introduced. Regarding the tools used in proving the existence of solutions for the equation analysed here, they rely on the critical point theory combined with adequate variational techniques.

Homoclinic solutions of difference equations with variable exponents

We study the existence of homoclinic solutions for a class of non-homogeneous difference equation with periodic coefficients. Our proofs rely on the critical point theory combined with adequate variational techniques, which are mainly based on the mountain-pass lemma.

An eigenvalue problem involving a degenerate and singular elliptic operator

We study an eigenvalue problem involving a degenerate and singular elliptic operator on the whole space RN. We prove the existence of an unbounded and increasing sequence of eigenvalues. Our study generalizes to the case of degenerate and singular operators a result of A. Szulkin and M. Willem.

Multiplicity results for some elliptic problems with nonlinear boundary conditions involving variable exponents

In this paper we analyze an elliptic partial differential equation involving variable exponent growth conditions coupled with a nonlinear boundary condition. We show the existence of infinitely many bounded weak solutions provided there is a suitable oscillatory behavior of the nonlinearity either at infinity or at zero. Our proofs rely on a method due to Saint Raymond.

Sublinear eigenvalue problems associated to the Laplace operator revisited

Eigenvalue problems involving the Laplace operator on bounded domains lead to a discrete or a continuous set of eigenvalues. In this paper we highlight the case of an eigenvalue problem involving the Laplace operator which possesses, on the one hand, a continuous family of eigenvalues and, on the other hand, at least one more eigenvalue which is isolated in the set of eigenvalues of that problem.

Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces

We study a non-homogeneous boundary value problem in a smooth bounded domain in RN. We prove the existence of at least two non-negative and non-trivial weak solutions. Our approach relies on Orlicz–Sobolev spaces theory combined with adequate variational methods and a variant of Mountain Pass Lemma.

An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue

In this paper we analyze an eigenvalue problem, involving a homogeneous Neumann boundary condition, in a smooth bounded domain. We show that the problem possesses, on the one hand, a continuous family of eigenvalues and, on the other hand, exactly one more eigenvalue which is isolated in the set of eigenvalues of the problem.

Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions

We study a boundary value problem of the type in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in (N≥ 3) with smooth boundary and the functions are of the type with , (i = 1, …, N). Combining the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle we show that under suitable conditions the problem has two non-trivial weak solutions.

Eigenvalue problems for anisotropic elliptic equations: an Orlicz-Sobolev space setting

The paper studies a class of anisotropic eigenvalue problems involving an elliptic, nonhomogeneous di®erential operator on a bounded domain from RN with smooth boundary. Some results regarding the existence or non-existence of eigenvalues are obtained. In each case the competition between the growth rates of the anisotropic coe±cients plays an essential role in the description of the set of eigenvalues.

Spectral estimates for a nonhomogeneous difference problem

We study an eigenvalue problem in the framework of difference equations. We show that there exist two positive constants λ0 and λ1 verifying λ0 ≤ λ1 such that any λ ∈ (0, λ0) is not an eigenvalue of the problem, while any λ ∈ [λ1,∞) is an eigenvalue of the problem. Some estimates for λ0 and λ1 are also given.

Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential

In this paper we establish the concentration of the spectrum in an unbounded interval for a class of eigenvalue problems involving variable growth conditions and a sign-changing potential. We also study the optimization problem for the particular eigenvalue given by the infimum of the associated Rayleigh quotient when the variable potential lies in a bounded, closed and convex subset of acertain variable exponent Lebesgue space.

On a degenerate and singular elliptic equation with critical exponent and non-standard growth conditions

In this paper we study a class of degenerate and singular elliptic equations involving critical exponents and non-standard growth conditions in the whole space RN. We show the existence of at least one nontrivial solution using as main argument Ekeland’s variational principle.

On the asymptotic behavior of variable exponent power-law functionals and applications

We study, via -convergence, the asymptotic behavior of several classes of power–law functionals acting on fields belonging to variable exponent Lebesgue spaces and which are subject to constant rank differential constraints. Applications of the -convergence results to the derivation and analysis of several models related to polycrystal plasticity arising as limiting cases of more flexible power–law models are also discussed.

Eigenvalue problems with weight and variable exponent for the Laplace operator

This paper deals with an eigenvalue problem for the Laplace operator on a bounded domain with smooth boundary in RN (N ≥ 3). We establish that there exist two positive constants λ and λ with λ ≤ λ such that any λ ∈ (0, λ) is not an eigenvalue of the problem while any λ ∈ [λ,∞) is an eigenvalue of the problem.

Γ-convergence of power-law functionals with variable exponents

Γ-convergence results for power-law functionals with variable exponents are obtained. The main motivation comes from the study of (first-failure) dielectric breakdown. Some connections with the generalization of the ∞-Laplace equation to the variable exponent setting are also explored.

On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting

In this paper we study a non-homogeneous eigenvalue problem involving variable growth conditions and a potential V . The problem is analyzed in the context of Orlicz–Sobolev spaces. Connected with this problem we also study the optimization problem for the particular eigenvalue given by the infimum of the Rayleigh quotient associated to the problem with respect to the potential V when V lies in a bounded, closed and convex subset of a certain variable exponent Lebesgue space.

Eigenvalue problems in anisotropic Orlicz–Sobolev spaces

We establish sufficient conditions for the existence of solutions to a class of nonlinear eigenvalueproblems involving nonhomogeneous differential operators inOrlicz–Sobolevspaces. To cite this article: M. Mihăilescu et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Nonlinear, degenerate and singular eigenvalue problems on $IR^N$

The goal of this paper is to establish the existence of a continuous family of eigenvalues lying in a neighborhood at the right of the origin for some nonlinear, nonhomogeneous, degenerate and singular elliptic operators on the whole space $IR^N$ . Our proofs rely essentially on the CaffarelliKohnNirenberg inequality combined with the Banach fixed point theorem.

On an eigenvalue problem involving the p(x)-Laplace operator plus a non-local term

We study an eigenvalue problem involving variable exponent growth conditions and a non-local term, on a bounded domain Ω ⊂ RN . Using adequate variational techniques, mainly based on the mountain-pass theorem of A. Ambrosetti and P. H. Rabinowitz, we prove the existence of a continuous family of eigenvalues lying in a neighborhood at the right of the origin.

Eigenvalue Problems for Anisotropic Discrete Boundary Value Problems

In this paper, we prove the existence of a continuous spectrum for a family of discrete boundary value problems. The main existence results are obtained by using critical point theory. The equations studied in the paper represent a discrete variant of some recent anisotropic variable exponent problems which serve as models in di®erent ¯elds of mathematical physics.

Two nontrivial solutions for a non-homogeneous Neumann problem: an Orlicz-Sobolev setting

In this paper we study a non-homogeneous Neumann-type problem which involves a nonlinearity satisfying a non-standard growth condition. By using a recent variational principle of Ricceri, we establish the existence of at least two non-trivial solutions in an appropriate Orlicz–Sobolev space.

Neumann problems associated to nonhomogeneous differentiable operators in Orlicz-Sobolev spaces

We study a nonlinear Neumann boundary value problem associated to a nonhomogeneous differential operator. Taking into account the competition between the nonlinearity and the bifurcation parameter, we establish sufficient conditions for the existence of nontrivial solutions in a related Orlicz--Sobolev space.

Spectrum in an unbounded interval for a class of nonhomogeneous differential operators

The present paper deals with the spectrum of a nonhomogeneous problem involving variable exponents on an exterior domain in RN. The existence of two positive real numbers λ0 and λ1, is established satisfying the condition λ0 λ1, such that the problem has no eigenvalue in the interval (0, λ0) while any number in the interval [λ1,∞) is an eigenvalue.

Neumann problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces

We study a nonlinear Neumann boundary value problem associated to a nonhomogeneous differential operator. Taking into account the competition between the nonlinearity and the bifurcation parameter, we establish sufficient conditions for the existence of nontrivial solution in a related Orlicz-Sobolev space.

Eigenvalue problems for some nonlinear perturbations of the Laplace operator

The goal of this paper is to prove existence results for some eigenvalue problems in which is involved a class of nonlinear operators which perturb the Laplace operator. Our proofs rely essentially on the Banach fixed point theorem and on a minimization technique.

Nonhomogeneous Neumann problems in Orlicz-Sobolev spaces

We establish sufficient conditions for the existence of nontrivial solutions for a class of nonlinear Neumann boundary value problems involving nonhomogeneous differential operators.

Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent

In this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power{like variable reaction term. One of the main features of our work is that the di®erential operator involves partial derivatives with di®erent variable exponents, so that the functional{analytic framework relies upon anisotropic Sobolev and Lebesgue spaces. Existence and nonexistence results are deeply in°uenced by the competition between the growth rates of the anisotropic coe±cients. Our main results point out some striking phenomena related to the existence of a continuous spectrum in several distinct situations. .

An extension of the Hermite-Hadamard inequality through subharmonic functions

In this paper we obtain a Hermite-Hadamard type inequality for a class of subharmonic functions. Our proofs rely essentially on the properties of elliptic partial differential equations of second order. Our study extends some recent results from [1], [2] and [6].

Nonhomogeneous boundary value problems in Orlicz-Sobolev spaces

We study the nonlinear Dirichlet problem −div(log(1+q|∇u|)|∇u|p−2∇u)=−λ|u|p−2u+|u|r−2u in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary, while p, q and r are real numbers satisfying p,q>1, p+q<min{N,r}, r<(Np−N+p)/(N−p). The main result of this Note establishes that for any λ>0 this boundary value problem has infinitely many solutions in the Orlicz–Sobolev space View the MathML source, where View the MathML source. To cite this article: M. Mihăilescu, V. Rădulescu, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting

We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary. We distinguish the cases where either $f(u)=-\lambda|u|^{p-2}u+|u|^{r-2}u$ or $f(u)=\lambda|u|^{p-2}u-|u|^{r-2}u$, with $p$, $q>1$, $p+q<\min\{N,r\}$, and $r<(Np-N+p)/(N-p)$. In the first case we show the existence of infinitely many weak solutions for any $\lambda>0$. In the second case we prove the existence of a nontrivial weak solution if $\lambda$ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.

Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator

In this paper we study an elliptic equation with nonstandard growth conditions and the Neumann boundary condition. We establish the existence of at least three solutions by using as the main tool a variational principle due to Ricceri.

A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids

We study the boundary value problem $-{\rm div}(a(x,\nabla u))=\lambda(u^{\gamma-1}- u^{\beta-1})$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\RR^N$ and ${\rm div}(a(x,\nabla u))$ is a $p(x)$-Laplace type operator, with $1<\beta<\gamma<\inf_{x\in\Omega}p(x)$. We prove that if $\lambda$ is large enough then there exist at least two nonnegative weak solutions. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with adequate variational methods and a variant of Mountain Pass Lemma.

Elliptic problems in variable exponent spaces

In this paper we study a nonlinear elliptic equation involving p(x)-growth conditions on a bounded domain having cylindrical symmetry. We establish existence and multiplicity results using as main tools the mountain pass theorem of Ambosetti and Rabinowitz and Ekeland's variational principle.

Existence and multiplicity of solutions for an elliptic equation with p(x)-growth conditions

We study a partial differential equation on a bounded domain ⊂ N with a p(x)-growth condition in the divergence operator and we establish the existence of at least two nontrivial weak solutions in the generalized Sobolev space W1,p(x)0 (). Such equations have been derived as models of several physical phenomena. Our proofs rely essentially on critical point theory combinedwith corresponding variational techniques.

Existence and multiplicity of weak solutions for a class of degenerate nonlinear elliptic equations

The goal of this paper is to study the existence and the multiplicity of non-trivial weak solutions for some degenerate nonlinear elliptic equations on the whole space RN. The solutions will be obtained in a subspace of the Sobolev space W1,p(RN). The proofs rely essentially on the Mountain Pass theorem and on Ekeland’s Variational principle.

Nonlinear eigenvalue problems for some degenerate elliptic operators on $R^N$

We study two nonlinear degenerate eigenvalue problems on $\mathbb R^N$. For the first problem we prove the existence of a positive eigenvalue while for the second we show the existence of a continuous family of eigenvalues. Our approach is based on standard tools in the critical point theory combined with adequate variational methods. We also apply an idea developed recently by Szulkin and Willem.

Ground state solutions of non-linear singular Schrödinger equations with lack of compactness

We study a class of time-independent non-linear Schrödinger-type equations on the whole space with a repulsive singular potential in the divergence operator and we establish the existence of non-trivial standing wave solutions for this problem in an appropriate weighted Sobolev space. Such equations have been derived as models of several physical phenomena. Our proofs rely essentially on critical point theory tools combined with the Caffarelli–Kohn–Nirenberg inequality.