Salvato in:
| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2407.11415 |
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Sommario:
- We are devoted to the study of the following nonlinear $p$-Laplacian Schrödinger equation with $L^{p}$-norm constraint \begin{align*} \begin{cases} &-Δ_{p} u=λ|u|^{p-2}u +|u|^{r-2}u\quad\mbox{in}\quadΩ,\\ &u=0\quad\mbox{on}\quad \partialΩ,\\ &\int_Ω|u|^{p}dx=a, \end{cases} \end{align*} where $Δ_{p}u=\text{div} (|\nabla u|^{p-2}\nabla u)$, $Ω\subset\mathbb{R}^{N}$ is an exterior domain with smooth boundary $\partialΩ\neq\emptyset$ satisfying that $\R^{N}\setminusΩ$ is bounded, $N\geq3$, $2\leq p<N$, $p<r<p+\frac{p^{2}}{N}$, $a>0$ and $λ\in\R$ is an unknown Lagrange multiplier. First, by using the splitting techniques and the Gagliardo-Nirenberg inequality, the compactness of Palais-Smale sequence of the above problem at higher energy level is established. Then, exploiting barycentric function methods, Brouwer degree and minimax principle, we obtain a solution $(u,\la)$ with $u>0$ in $\R^{N}$ and $\la<0$ when $\R^{N}\setminusΩ$ is contained in a small ball. Moreover, we give a similar result if we remove the restriction on $Ω$ and assume $a>0$ small enough. Last, with the symmetric assumption on $Ω$, we use genus theory to consider infinite many solutions.