Existence and multiplicity of normalized solutions for the generalized Kadomtsev-Petviashvili equation in ${\mathbb{R}}^2$

Abstract

In this paper, we study the existence and {multiplicity} of nontrivial solitary waves for the generalized Kadomtsev-Petviashvili equation with prescribed {$L^2$-norm} \begin{equation*}\label{Equation1} \left\{\begin{array}{l} \left(-u_{x x}+D_x^{-2} u_{y y}+λu-f(u)\right)_x=0,{\quad x \in \mathbb{R}^2, } \\[10pt] \displaystyle \int_{\mathbb{R}^2}u^2 d x=a^2, \end{array}\right.%\tag{${\mathcal{E}}_{\hat{I}}»$} \end{equation*} where $a>0$ and $\hat{I}»\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier. For the case $f(t)=|t{|}^{q-2}t$, with $2<q<\frac{10}{3}$ ($L^2$-subcritical case) and $\frac{10}{3}<q<6$ ($L^2$-supercritical case), we establish the existence of normalized ground state solutions for the above equation. Moreover, when $f(t)=\hat{I}¼|t{|}^{q-2}t+|t{|}^{p-2}t$, with $2<q<\frac{10}{3}<p<6$ and $\hat{I}¼>0$, we prove the existence of normalized ground state solutions which corresponds to a local minimum of the associated energy functional. In this case, we further show that there exists a sequence $(a_n)\subset (0,a_0)$ with $a_n\to 0$ as $n\to +\infty$, such that for each $a=a_n$, the problem admits a second solution with positive energy. To the best of our knowledge, this is the first work that studies the existence of solutions for the generalized Kadomtsev-Petviashvili equations under the $L^2$-constraint, which we refer to them as the normalized solutions.

26 pages