Normalised solutions for $p$-Laplacian equations with $L^p$-supercritical growth
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908434266324992 |
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| author | Dhara, Raj Narayan Rizzi, Matteo |
| author_facet | Dhara, Raj Narayan Rizzi, Matteo |
| contents | For $N\ge 3$ and $2<p<N$, we find normalised solutions to the equation \begin{align*} -Δ_p u+(1+V(x))|u|^{p-2}u+λu&=|u|^{q-2}u\qquad\text{in $\mathbb{R}^N$}\\ \|u\|_2&=ρ\end{align*} in the mass supercritical and Sobolev subcritical case, that is $q\in(p\frac{N+2}{N},\frac{Np}{N-p})$, at least if $ρ>0$ is small enough. The function $V\in L^{N/p}(\mathbb{R}^N)$, which plays the role of potential, is assumed to be non-positive and vanishing at infinity. Moreover, we will prove the compactness of the embedding of the space of radial functions $W^{1,p}_{rad}(\mathbb{R}^N)\subset L^q(\mathbb{R}^N)$ for $p\in(1,N)$ and $q\in(p\frac{N+2}{N},\frac{Np}{N-p})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03429 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalised solutions for $p$-Laplacian equations with $L^p$-supercritical growth Dhara, Raj Narayan Rizzi, Matteo Analysis of PDEs Primary 46E35, 35J20, 35J92, Secondary 35B06 For $N\ge 3$ and $2<p<N$, we find normalised solutions to the equation \begin{align*} -Δ_p u+(1+V(x))|u|^{p-2}u+λu&=|u|^{q-2}u\qquad\text{in $\mathbb{R}^N$}\\ \|u\|_2&=ρ\end{align*} in the mass supercritical and Sobolev subcritical case, that is $q\in(p\frac{N+2}{N},\frac{Np}{N-p})$, at least if $ρ>0$ is small enough. The function $V\in L^{N/p}(\mathbb{R}^N)$, which plays the role of potential, is assumed to be non-positive and vanishing at infinity. Moreover, we will prove the compactness of the embedding of the space of radial functions $W^{1,p}_{rad}(\mathbb{R}^N)\subset L^q(\mathbb{R}^N)$ for $p\in(1,N)$ and $q\in(p\frac{N+2}{N},\frac{Np}{N-p})$. |
| title | Normalised solutions for $p$-Laplacian equations with $L^p$-supercritical growth |
| topic | Analysis of PDEs Primary 46E35, 35J20, 35J92, Secondary 35B06 |
| url | https://arxiv.org/abs/2507.03429 |