Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.19946 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916028315860992 |
|---|---|
| author | Xia, Jiankang Zhong, Chao |
| author_facet | Xia, Jiankang Zhong, Chao |
| contents | We investigate normalized groundstates for mixed $(p,2)$-Laplacian equations \begin{align*}
\begin{cases}
-Δ_p u-Δu+λu=f(u) & \text{in } \mathbb{R}^2,
\displaystyle \int_{\mathbb{R}^2}|u|^2\,\mathrm{d}x=m,
u\in H^1(\mathbb{R}^2)\cap D^{1,p}(\mathbb{R}^2),
\end{cases} \end{align*} where $Δ_p$ denotes the $p$-Laplacian with $1<p<2$, $λ\in\mathbb{R}$ represents a Lagrange multiplier and the nonlinerity $f$ exhibits exponential critical growth. Compared to the single-Laplacian case, the lack of regularity here precludes the Pohozaev identity, and the exponential critical growth severely compromises the restoration of compactness. To address these issues, we introduce a refined Moser iteration technique adapted to exponential critical growth, which establishes the Pohozaev identity for weak solutions under the mere assumption of $C_{\mathrm{loc}}^{1,α}$-regularity. By combining constrained minimization on the Pohozaev manifold within a closed $L^2$-ball with a minimax characterization, we prove the existence of normalized groundstates for any prescribed mass $m>0$. Notably, our approach works independently of the sign of the Lagrange multiplier $λ$, thereby surmounting the fundamental barrier in recovering compactness for mixed Laplacian problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19946 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Normalized groundstates for mixed $(p,2)$-Laplacian equations in $\mathbb R^2$ with exponential critical growth Xia, Jiankang Zhong, Chao Analysis of PDEs 35A15, 35J92, 35B33, 35M10, 35B45 We investigate normalized groundstates for mixed $(p,2)$-Laplacian equations \begin{align*} \begin{cases} -Δ_p u-Δu+λu=f(u) & \text{in } \mathbb{R}^2, \displaystyle \int_{\mathbb{R}^2}|u|^2\,\mathrm{d}x=m, u\in H^1(\mathbb{R}^2)\cap D^{1,p}(\mathbb{R}^2), \end{cases} \end{align*} where $Δ_p$ denotes the $p$-Laplacian with $1<p<2$, $λ\in\mathbb{R}$ represents a Lagrange multiplier and the nonlinerity $f$ exhibits exponential critical growth. Compared to the single-Laplacian case, the lack of regularity here precludes the Pohozaev identity, and the exponential critical growth severely compromises the restoration of compactness. To address these issues, we introduce a refined Moser iteration technique adapted to exponential critical growth, which establishes the Pohozaev identity for weak solutions under the mere assumption of $C_{\mathrm{loc}}^{1,α}$-regularity. By combining constrained minimization on the Pohozaev manifold within a closed $L^2$-ball with a minimax characterization, we prove the existence of normalized groundstates for any prescribed mass $m>0$. Notably, our approach works independently of the sign of the Lagrange multiplier $λ$, thereby surmounting the fundamental barrier in recovering compactness for mixed Laplacian problems. |
| title | Normalized groundstates for mixed $(p,2)$-Laplacian equations in $\mathbb R^2$ with exponential critical growth |
| topic | Analysis of PDEs 35A15, 35J92, 35B33, 35M10, 35B45 |
| url | https://arxiv.org/abs/2605.19946 |