Existence and concentration of normalized solutions for $p$-Laplacian equations with logarithmic nonlinearity
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arXiv
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| Format: | Preprint |
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2024
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| author | Shen, Liejun Squassina, Marco |
| author_facet | Shen, Liejun Squassina, Marco |
| contents | We investigate the existence and concentration of normalized solutions for a $p$-Laplacian problem with logarithmic nonlinearity of type \[
\left\{
\begin{array}{ll}
\displaystyle -\varepsilon^pΔ_p u+V(x)|u|^{p-2}u=λ|u|^{p-2}u+|u|^{p-2}u\log|u|^p ~\text{in}~\mathbb R^N,\newline
\displaystyle \int_{\mathbb R^N}|u|^pdx=a^p\varepsilon^N,
\end{array}
\right. \] where $a,\varepsilon> 0$, $λ\in\mathbb R$ is known as the Lagrange multiplier, $Δ_p\cdot =\text{div} (|\nabla \cdot|^{p-2}\nabla \cdot)$ denotes the usual $p$-Laplacian operator with $2\leq p < N$ and $V \in \mathcal{C}^0(\mathbb R^N)$ is the potential which satisfies some suitable assumptions. We prove that the number of positive solutions depends on the profile of $V$ and each solution concentrates around its corresponding global minimum point of $V$ in the semiclassical limit when $\varepsilon\to0^+$ using variational method.
Moreover, we also get the existence of normalized solutions for some logarithmic $p$-Laplacian equations involving mass-supercritical nonlinearities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09366 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and concentration of normalized solutions for $p$-Laplacian equations with logarithmic nonlinearity Shen, Liejun Squassina, Marco Analysis of PDEs 35J10, 35J20, 35B06 We investigate the existence and concentration of normalized solutions for a $p$-Laplacian problem with logarithmic nonlinearity of type \[ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^pΔ_p u+V(x)|u|^{p-2}u=λ|u|^{p-2}u+|u|^{p-2}u\log|u|^p ~\text{in}~\mathbb R^N,\newline \displaystyle \int_{\mathbb R^N}|u|^pdx=a^p\varepsilon^N, \end{array} \right. \] where $a,\varepsilon> 0$, $λ\in\mathbb R$ is known as the Lagrange multiplier, $Δ_p\cdot =\text{div} (|\nabla \cdot|^{p-2}\nabla \cdot)$ denotes the usual $p$-Laplacian operator with $2\leq p < N$ and $V \in \mathcal{C}^0(\mathbb R^N)$ is the potential which satisfies some suitable assumptions. We prove that the number of positive solutions depends on the profile of $V$ and each solution concentrates around its corresponding global minimum point of $V$ in the semiclassical limit when $\varepsilon\to0^+$ using variational method. Moreover, we also get the existence of normalized solutions for some logarithmic $p$-Laplacian equations involving mass-supercritical nonlinearities. |
| title | Existence and concentration of normalized solutions for $p$-Laplacian equations with logarithmic nonlinearity |
| topic | Analysis of PDEs 35J10, 35J20, 35B06 |
| url | https://arxiv.org/abs/2403.09366 |