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 a⃗(ξ)=|ξ|q−2ξ $ \overrightarrow {a}\left(\xi \right)=\vert \xi {\vert }^{q-2}\xi $ corresponding to the q-Laplacian is not strictly monotone when q 2.

Keywords

• (2, q)-laplacian
• normalized solutions
• ground state solutions
• existence and nonexistence results
• 35a15
• 35b38
• 35b40
• 35j60
• 35j20