Denisa Stancu-Dumitru

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.

A Caffarelli–Kohn–Nirenberg-type inequality with variable exponent and applications to PDEs

Given Ω ⊂ ℝ N (N ≥ 2) a bounded smooth domain we establish a Caffarelli–Kohn–Nirenberg type inequality on Ω in the case when a variable exponent p(x), of class C 1, is involved. Our main result is proved under the assumption that there exists a smooth vector function , satisfying and for any x ∈ Ω. Particularly, we supplement a result by Fan et al. [X. Fan, Q. Zhang, and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), pp. 306–317] regarding the positivity of the first eigenvalue of the p(x)-Laplace operator. Moreover, we provide an application of our result to the study of degenerate PDEs involving variable exponent growth conditions.

The limiting Behavior of Solutions to Inhomogeneous Eigenvalue Problems in Orlicz-Sobolev Spaces

Abstract The asymptotic behavior of the sequence {un} of positive first eigenfunctions for a class of inhomogeneous eigenvalue problems is studied in the setting of Orlicz-Sobolev spaces. After possibly extracting a subsequence, we prove that un → u∞ uniformly in Ω as n→∞, where u∞ is a nontrivial viscosity solution of a nonlinear PDE involving the ∞-Laplacian.

Two nontrivial weak solutions for the Dirichlet problem on the Sierpinski Gasket

We study a Dirichlet problem involving the weak Laplacian on Sierpinski gasket and we prove the existence of at least two distinct nontrivial weak solutions using Ekelands Variational Principle and standard tools in the critical point theory combined with corresponding variational techniques. DOI: 10.1017/S000497271100298X

On the spectrum of a nontypical eigenvalue problem

We study a nontypical eigenvalue problem in a bounded domain from the Euclidian space R2 subject to the homogeneous Dirichlet boundary condition. We show that the spectrum of the problem contains two distinct intervals separated by an interval where there are no other eigenvalues.

Variational treatment of nonlinear equations on the Sierpiński gasket

We study two Dirichlet boundary value problems involving the weak Laplacian on the Sierpiński gasket. Using variational methods we prove the existence and multiplicity of weak solutions for these problems.

An Existence Result for a PDE Involving a Grushin‐type Operator and Variable Exponents

We define a Grushin‐type operator with a variable exponent and establish existence results for an equation involving such an operator in a suitable function space. The tools used in proving our existence result rely on the critical point theory combined with adequate variational techniques.