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

TL;DR

This paper investigates the existence and nonexistence of normalized solutions for a class of $(2,q)$-Laplacian equations across different critical regimes, revealing new solutions and challenges due to the quasi-linear nature of the equations.

Contribution

It provides a comprehensive analysis of normalized solutions for $(2,q)$-Laplacian equations, including existence, nonexistence, and multiplicity results in various critical cases, addressing the complexities introduced by the quasi-linear term.

Findings

Ground state solutions in the $L^2$-subcritical case

Nonexistence results in the $L^2$-critical case

Infinitely many radial solutions in the $L^2$-supercritical case

Abstract

This paper concerns the existence of normalized solutions to a class of $(2,q)$-Laplacian equations in all the possible cases according to the value of $p$ with respect to the critical exponent $2(1+2/N)$. In the $L^2$-subcritical case, we study a global minimization problem and obtain a ground state solution. While in the $L^2$-critical case, we prove several nonexistence results, extended also in the $L^q$-critical case. At last, we derive a ground state and infinitely many radial solutions in the $L^2$-supercritical case. Compared with the classical Schr\"{o}dinger 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 $\vec{a}(\xi)=|\xi|^{q-2}\xi$ corresponding to the $q$-Laplacian is not strictly monotone when $q<2$, so we shall consider separately the case $q<2$ and the case $q>2$.