Normalized solutions of a (2,p)-Laplacian equation

Abstract

In this paper, we are concerned with normalized solutions of a $(2,p)$-Laplacian equation with an $L^p$ constraint in $R^3$, where $2<p<3$. Different from literature previous, we focus on the $L^p$ not $L^2$ constraint for $p>2$. Moreover, an interesting finding is that the non-homogeneity driven by the operators Δ and ${\Delta }_p$ has an important impact on $L^p$ constraint $(2,p)$-Laplacian equations, as reflected in the definition of the $L^p$ critical exponent, and the existence of normalized solutions in both $L^p$ subcritical and supercritical cases. All these new phenomena, which are different from those exhibited by a single p-Laplacian equation, reveal the essential characteristics of $(2,p)$-Laplacian equations.