Normalized solutions to a class of (2, q)-Laplacian equations

Abstract

This paper is concerned with the existence of normalized solutions to a class of (2, q)-Laplacian equations in all the possible cases with respect to the mass critical exponents 2(1 + 2/N), q(1 + 2/N). In the mass subcritical cases, we study a global minimization problem and obtain a ground state solution. While in the mass critical cases, we prove several nonexistence results. At last, we derive a ground state and infinitely many radial solutions in the mass supercritical case. Compared with the classical Schrödinger equation, the (2, q)-Laplacian equation possesses a quasi-linear term, which brings in some new difficulties and requires a more subtle analysis technique. Moreover, the vector field → a ( ξ ) = | ξ | q − 2 ξ corresponding to the q-Laplacian is not strictly monotone when q < 2, so we shall consider separately the case q < 2 and the case q > 2.