Normalized solutions to a class of (2, <i>q</i> )-Laplacian equations
Laura Baldelli, Tao Yang
Abstract
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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mo>⃗</m:mo> </m:mover> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mo stretchy="false">|</m:mo> <m:mi>ξ</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>ξ</m:mi> </m:math> $ \overrightarrow {a}\left(\xi \right)=\vert \xi {\vert }^{q-2}\xi $ 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.