Mihai Mihăilescu

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

Abstract. We study the boundary value problem −div(a(x,∇u)) = λ(u − u) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R and div(a(x,∇u)) is a p(x)-Laplace type operator, with 1 < β < γ < infx∈Ω p(x). We prove that if λ 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. 2000 Mathematics Subject Classification: 35D05, 35J60, 35J70, 58E05, 68T40, 76A02.

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

We consider the nonlinear eigenvalue problem -div (|∇ u | p(x) - 2 ∇ u ) = λ( u ( q(x) - 2 u in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R N with smooth boundary and p, q are continuous functions on Ω such that 1 0 sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekelands variational principle.

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 deserve as models in different fields of mathematical physics.

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. 2000 Mathematics Subject Classification: 35D05, 35J60, 35J70, 58E05, 68T40, 76A02.

Discrete boundary value problems involving oscillatory nonlinearities: small and large solutions

We consider the discrete boundary value problem (P): where the nonlinear term has an oscillatory behaviour near the origin or at infinity. By a direct variational method, we show that (P) has a sequence of non-negative, distinct solutions which converges to 0 (respectively ) in the sup-norm whenever f oscillates at the origin (respectively at infinity).

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 Ekelands variational principle we show that under suitable conditions the problem has two non-trivial weak solutions.

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

Abstract We study a non-homogeneous boundary value problem in a smooth bounded domain in R N . 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.We study a non-homogeneous boundary value problem in a smooth bounded domain in R . We prove the existence of at least two nonnegative 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. 2010 Mathematics Subject Classification: 35D30, 35J60, 35J70, 46N20, 58E05.

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.

EIGENVALUE PROBLEMS ASSOCIATED WITH NONHOMOGENEOUS DIFFERENTIAL OPERATORS, IN ORLICZ–SOBOLEV SPACES

We study the boundary value problem -div((a1(|∇ u|) + a2(|∇ u|))∇ u) = λ|u|q(x)-2u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝN (N ≥ 3) with smooth boundary, λ is a positive real number, q is a continuous function and a1, a2 are two mappings such that a1(|t|)t, a2(|t|)t are increasing homeomorphisms from ℝ to ℝ. We establish the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that any λ ∈ [λ1, ∞) is an eigenvalue, while any λ ∈ (0, λ0) is not an eigenvalue of the above problem.

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.