tailieunhanh - Liouville type theorem for stable solutions to elliptic equations involving the Grushin operator

In this paper, we will extend the result of Farina to the general case a0 and look for the effect of the degeneracy on the range of the exponent p on Liouville-type theorem. It seems that the presence of the weight term |x|a makes the problem more challenging. The main difficulty is that the Grushin operator is nonautonomous. This requires suitable scaled test functions in the integral estimate. | HNUE JOURNAL OF SCIENCE DOI Natural Science 2019 Volume 64 Issue 6 pp. 12-22 This paper is available online at http LIOUVILLE TYPE THEOREM FOR STABLE SOLUTIONS TO ELLIPTIC EQUATIONS INVOLVING THE GRUSHIN OPERATOR Nguyen Thi Quynh Faculty of Fundamental Science Hanoi University of Industry Abstract. We study a Liouville type theorem for stable solutions of the following semilinear equation involving Grushin operators x u a2 x 2α y u u p 1 u x y RN RN1 RN2 where p gt 1 α gt 0 and a 6 0. Basing on the technique of Farina 1 we establish the nonexistence of nontrivial stable solutions under the range p lt pc Nα where Nα N1 1 α N2 and pc Nα is a certain explicitly given positive constant depending on Nα . Keywords Liouville type theorem stable solution degenerate elliptic equations Grushin operators. 1. Introduction In this paper we study the semilinear degenerate partial differential equation of the form Gα u u p 1u where Gα x a2 x 2α y is the Grushin operator x and y are Laplace operators with respect to x RN1 and y RN2 . Here we always assume that a 6 0 α gt 0 p gt 1 and N1 N2 1. Recall that Gα is elliptic for x 6 0 and degenerates on the N2 manifold 0 R . This operator belongs to the wide class of subelliptic operators studied by Franchi et al. in 2 . In the special case α 1 problem is close related to the Heisenberg Laplacian equation H u f u in H n Cn R where H is the Heisenberg Laplacian see . 3 4 . Problem has recently attracted much attention in variety of mathematics directions. The most interesting questions are about the existence and non-existence 5 the multiplicity of solutions 6 the symmetry properties 7 the asymptotic behaviour 8 the regularity estimates 9-11 . Received April 19 2019. Revised June 19 2019. Accepted June 26 2019. Contact Nguyen Thi Quynh e-mail address 12 Liouville type theorem for stable solutions to elliptic equations involving the Grushin operator Let