ON A CLASS OF SEMILINEAR ELLIPTIC SYSTEMS INVOLVING GRUSHIN TYPE OPERATOR

In recent years, more and more mathematicians have studied the existence of solutions for degenerate elliptic problems. This comes from the fact that they arise in many areas of applied physics, including nuclear physics, field theory, solid waves and problems of false vacuum. These problems are introduced as models for several physical phenomena related to equilibrium of continuous media which somewhere be perfect insulators (see [8, 19]). However, the study have essentially based on the Caffarelli-Kohn-Nirenberg inequalities and their variants, see for example [6, 7, 9, 11, 14, 26] and the references therein. In this paper, we will study the existence of solutions for degenerate elliptic problems involving Grushin type operator Gs = ∆x + |x| ∆y for s ≥ 0. To our knowledge, the Grushin type operators were firstly introduced in [10], and developed in [13, 15, 17, 22, 23, 24, 25]. Let Ω ⊂ R = R1 ×R2 be a bounded domain with smooth boundary ∂Ω, and 0 ∈ Ω. In this paper, we are interested in the semilinear elliptic system with Grushin type operator