Non-Ljusternik--Schnirelman eigenvalues of the pure $p$-Laplacian exist
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| Format: | Preprint |
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2026
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| _version_ | 1866914437555814400 |
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| author | Bobkov, Vladimir |
| author_facet | Bobkov, Vladimir |
| contents | An old and well-known open problem in the critical point theory asks whether, for some $p \neq 2$ and some bounded domain $Ω$, there exists a critical value of the $p$-Dirichlet energy $\|\nabla u\|_p^p$ over an $L^p(Ω)$-sphere in $W_0^{1,p}(Ω)$ lying outside of a Ljusternik--Schnirelman type sequence of critical values, the latter will be called LS eigenvalues of the $p$-Laplacian. In this work, we provide a positive answer by showing the existence of a non-LS eigenvalue when $p>2$ is sufficiently close to $2$ and $Ω$ is just a planar rectangle close to the square. The arguments pursue the observation that a simple eigenvalue of the Laplacian can be a meeting point for several branches of eigenvalues of the $p$-Laplacian as $p$ varies. Since LS eigenvalues are continuous with respect to $p$ and exhaust the whole spectrum when $p=2$, we deduce that at least one of the branches must contain non-LS eigenvalues. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_01138 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-Ljusternik--Schnirelman eigenvalues of the pure $p$-Laplacian exist Bobkov, Vladimir Analysis of PDEs Functional Analysis Spectral Theory 35J92, 35P30, 47J10 An old and well-known open problem in the critical point theory asks whether, for some $p \neq 2$ and some bounded domain $Ω$, there exists a critical value of the $p$-Dirichlet energy $\|\nabla u\|_p^p$ over an $L^p(Ω)$-sphere in $W_0^{1,p}(Ω)$ lying outside of a Ljusternik--Schnirelman type sequence of critical values, the latter will be called LS eigenvalues of the $p$-Laplacian. In this work, we provide a positive answer by showing the existence of a non-LS eigenvalue when $p>2$ is sufficiently close to $2$ and $Ω$ is just a planar rectangle close to the square. The arguments pursue the observation that a simple eigenvalue of the Laplacian can be a meeting point for several branches of eigenvalues of the $p$-Laplacian as $p$ varies. Since LS eigenvalues are continuous with respect to $p$ and exhaust the whole spectrum when $p=2$, we deduce that at least one of the branches must contain non-LS eigenvalues. |
| title | Non-Ljusternik--Schnirelman eigenvalues of the pure $p$-Laplacian exist |
| topic | Analysis of PDEs Functional Analysis Spectral Theory 35J92, 35P30, 47J10 |
| url | https://arxiv.org/abs/2604.01138 |