A weighted eigenvalue problem for mixed local and nonlocal operators with potential
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914934086959104 |
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| author | Lakshmi, R. Giri, Ratan Kr. Ghosh, Sekhar |
| author_facet | Lakshmi, R. Giri, Ratan Kr. Ghosh, Sekhar |
| contents | We study an {\it indefinite weighted eigenvalue problem} for an operator of {\it mixed-type} (that includes both the classical {\it $p$-Laplacian} and the {\it fractional $p$-Laplacian}) in a bounded open subset $Ω\subset \mathbb{R}^N \,(N\geq2)$ with {\it Lipschitz boundary} $\partial Ω$, which is given by \begin{align*}
-Δ_p u + (-Δ_p)^su+V(x)|u|^{p-2}u&=λg(x)|u|^{p-2}u~\text{in}~Ω,
u&=0~\text{in}~\mathbb{R}^N\setminusΩ, \end{align*} where $λ>0$ is a parameter, exponents $0<s<1<p<N$, and $V, g\in L^q(Ω)$ for $q\in \left(\frac{N}{sp}, \infty\right)$ with $V\geq 0, g > 0$ a.e. in $Ω$. Using the variational tools together with a {\it weak comparison} and {\it strong maximum principles}, we investigate the existence and uniqueness of {\it principal eigenvalue} and discuss its qualitative properties. Moreover, with the help of {\it Ljusternik-Schnirelman category theory}, it is proved that there exists a {\it nondecreasing sequence of positive eigenvalues} which goes to infinity. Further, we show that {\it the set of all positive eigenvalues is closed}, and {\it eigenfunctions} associated with every {\it positive eigenvalue} are bounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01349 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A weighted eigenvalue problem for mixed local and nonlocal operators with potential Lakshmi, R. Giri, Ratan Kr. Ghosh, Sekhar Analysis of PDEs 35P15, 35P30, 35R11, 35J20 We study an {\it indefinite weighted eigenvalue problem} for an operator of {\it mixed-type} (that includes both the classical {\it $p$-Laplacian} and the {\it fractional $p$-Laplacian}) in a bounded open subset $Ω\subset \mathbb{R}^N \,(N\geq2)$ with {\it Lipschitz boundary} $\partial Ω$, which is given by \begin{align*} -Δ_p u + (-Δ_p)^su+V(x)|u|^{p-2}u&=λg(x)|u|^{p-2}u~\text{in}~Ω, u&=0~\text{in}~\mathbb{R}^N\setminusΩ, \end{align*} where $λ>0$ is a parameter, exponents $0<s<1<p<N$, and $V, g\in L^q(Ω)$ for $q\in \left(\frac{N}{sp}, \infty\right)$ with $V\geq 0, g > 0$ a.e. in $Ω$. Using the variational tools together with a {\it weak comparison} and {\it strong maximum principles}, we investigate the existence and uniqueness of {\it principal eigenvalue} and discuss its qualitative properties. Moreover, with the help of {\it Ljusternik-Schnirelman category theory}, it is proved that there exists a {\it nondecreasing sequence of positive eigenvalues} which goes to infinity. Further, we show that {\it the set of all positive eigenvalues is closed}, and {\it eigenfunctions} associated with every {\it positive eigenvalue} are bounded. |
| title | A weighted eigenvalue problem for mixed local and nonlocal operators with potential |
| topic | Analysis of PDEs 35P15, 35P30, 35R11, 35J20 |
| url | https://arxiv.org/abs/2409.01349 |