Maria-Magdalena Boureanu

Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent

We establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic equations involving the anisotropic -Laplace operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev spaces and our main tool is the symmetric mountain-pass theorem of Ambrosetti and Rabinowitz.

Existence of solutions for anisotropic quasilinear elliptic equations with variable exponent

Abstract We study the existence of solutions for a class of quasilinear elliptic equations involving the anisotropic -Laplace operator, on a bounded domain with smooth boundary. Since our differential operator involves partial derivatives with different variable exponents, we work on the anisotropic variable exponent Sobolev spaces. Using the Ekelands variational principle and the mountain-pass theorem of Ambrosetti and Rabinowitz, we establish two existence results.

On a p()-biharmonic problem with no-flux boundary condition

The study of fourth order partial differential equations has flourished in the last years, however, a p()-biharmonic problem with no-flux boundary condition has never been considered before, not even for constant p. This is an important step further, since surfaces that are impermeable to some contaminants are appearing quite often in nature, hence the significance of such boundary condition. By relying on several variational arguments, we obtain the existence and the multiplicity of weak solutions to our problem. We point out that, although we use a mountain pass type theorem in order to establish the multiplicity result, we do not impose an AmbrosettiRabinowitz type condition, nor a symmetry condition, on our nonlinearity f.

Nonlinear Problems with p(·)-Growth Conditions and Applications to Antiplane Contact Models

Abstract We consider a general boundary value problem involving operators of the form div(a(·, ∇u(·)) in which a is a Carathéodory function satisfying a p(·)-growth condition. We are interested on the weak solvability of the problem and, to this end, we start by introducing the Lebesgue and Sobolev spaces with variable exponent, together with their main properties. Then, we state and prove our main existence and uniqueness result, Theorem 3.1. The proof is based on a Weierstrass-type argument. We also consider two antiplane contact problems for nonhomogenous elastic materials of Hencky-type. The contact is frictional and it is modelled with a regularized version of Tresca’s friction law and a power-law friction, respectively. We prove that the problems cast in the abstract setting, then we use Theorem 3.1 to deduce their unique weak solvability.

A new class of nonhomogeneous differential operator and applications to anisotropic systems

We introduce a new class of operators that extend both generalized Laplace operators and generalized mean curvature operators. We start the discussion on general anisotropic systems with variable exponents that involve our operators, then we focus on a specific example of such system, we show that it admits a unique weak solution and we complete our work with some comments on other related systems. The newly introduced operators are appropriate for the study conducted in the anisotropic spaces with variable exponents, but at the end of the paper we also provide their versions corresponding to the studies conducted in the anisotropic Sobolev spaces with constant exponents, or in the isotropic variable exponent Sobolev spaces, since, to the best of our knowledge, they represent a novelty even for the classical Sobolev spaces.

Singular and Degenerate Boundary Value Problems Related to the Electricity Theory

The present paper draws attention to the weak solvability of a class of singular and degenerate problems with nonlinear boundary conditions. These problems derive from the electricity theory serving as mathematical models for physical phenomena related to the anisotropic media with “perfect” insulators or “perfect” conductors points. By introducing an appropriate weighted Sobolev space to the mathematical literature, we establish an existence and uniqueness result.