Existence and multiplicity of normalized solutions for the generalized Kadomtsev-Petviashvili equation in $\mathbb{R}^2$
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| Formato: | Preprint |
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2025
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| author | Alves, Claudianor O. Ding, Rui Ji, Chao |
| author_facet | Alves, Claudianor O. Ding, Rui Ji, Chao |
| contents | 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{$\mathscr E_λ$} \end{equation*} where $a>0$ and $λ\in \mathbb{R}$ 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 $f(t)=μ|t|^{q-2}t+|t|^{p-2}t$, with $2<q<\frac{10}{3}<p<6$ and $μ>0$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04967 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and multiplicity of normalized solutions for the generalized Kadomtsev-Petviashvili equation in $\mathbb{R}^2$ Alves, Claudianor O. Ding, Rui Ji, Chao Analysis of PDEs 35A15, 35A18 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{$\mathscr E_λ$} \end{equation*} where $a>0$ and $λ\in \mathbb{R}$ 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 $f(t)=μ|t|^{q-2}t+|t|^{p-2}t$, with $2<q<\frac{10}{3}<p<6$ and $μ>0$, 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. |
| title | Existence and multiplicity of normalized solutions for the generalized Kadomtsev-Petviashvili equation in $\mathbb{R}^2$ |
| topic | Analysis of PDEs 35A15, 35A18 |
| url | https://arxiv.org/abs/2506.04967 |