# Publications of Radulescu, Vicentiu

Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term

The aim of this paper is to establish the existence of at least one solution for a general inequality of quasi-hemivariational type, whose solution is sought in a subset K of a real Banach space E. First, we prove the existence of solutions in the case of compact convex subsets and the case of bounded closed and convex subsets. Finally, the case when K is the whole space is analyzed and necessary and sufficient conditions for the existence of solutions are stated. Our proofs rely essentially on the Schauder’s fixed point theorem and a version of the KKM principle due to Ky Fan (Math Ann 266:519–537, 1984).

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.

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.

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.

Hartman-Stampacchia results for stably pseudomonotone operators and non-linear hemivariational inequalities

We are concerned with two classes of non-standard hemivariational inequalities. In the first case we establish a Hartman–Stampacchia type existence result in the framework of stably pseudomonotone operators. Next, we prove an existence result for a class of non-linear perturbations of canonical hemivariational inequalities. Our analysis includes both the cases of compact sets nd of closed convex sets in Banach spaces. Applications to non-coercive hemivariational and variational-hemivariational inequalities illustrate the abstract results of this article.

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.

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.

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).

Existence results for hemivariational inequalities involving relaxed $\eta - \alpha$ monotone mappings

We establish some existence results for hemivariational inequalities with relaxed $\eta-\alpha$ monotone mappings on bounded, closed and convex subsets in reflexive Banach spaces. Our proofs rely essentially on a fixed point theorem for set valued mappings which is due to Tarafdar. We also give a sufficient condition for the existence of solutions in the case of unbounded subsets.

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.

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. .

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.

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.

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.