Existence and regularity of weak solutions for mixed local and nonlocal semilinear elliptic equations
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| Název: | Existence and regularity of weak solutions for mixed local and nonlocal semilinear elliptic equations |
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| Autoři: | Cheng, Fuwei, Su, Xifeng, Zhang, Jiwen |
| Zdroj: | Discrete and Continuous Dynamical Systems. 47:341-367 |
| Publication Status: | Preprint |
| Informace o vydavateli: | American Institute of Mathematical Sciences (AIMS), 2026. |
| Rok vydání: | 2026 |
| Témata: | Analysis of PDEs, FOS: Mathematics, Analysis of PDEs (math.AP) |
| Popis: | We study the existence, multiplicity and regularity results of weak solutions for the Dirichlet problem of a semi-linear elliptic equation driven by the mixture of the usual Laplacian and fractional Laplacian \begin{equation*} \left\{% \begin{array}{ll} -Δu + (-Δ)^{s} u+ a(x)\ u =f(x,u) & \hbox{in $Ω$,} u=0 & \hbox{in $\mathbb{R}^n\backslashΩ$} \end{array}% \right. \end{equation*} where $s \in (0,1)$, $Ω\subset \mathbb{R}^{n}$ is a bounded domain, the coefficient $a$ is a function of $x$ and the subcritical nonlinearity $f(x,u)$ has superlinear growth at zero and infinity. We show the existence of a non-trivial weak solution by Linking Theorem and Mountain Pass Theorem respectively for $λ_{1} \leqslant 0$ and $λ_{1} > 0$, where $λ_{1}$ denotes the first eigenvalue of $-Δ+ (-Δ)^{s} +a(x)$. In particular, adding a symmetric condition to $f$, we obtain infinitely many solutions via Fountain Theorem. Moreover, for the regularity part, we first prove the $L^{\infty}$-boundedness of weak solutions and then establish up to $C^{2, α}$-regularity up to boundary. To appear in DCDS |
| Druh dokumentu: | Article |
| ISSN: | 1553-5231 1078-0947 |
| DOI: | 10.3934/dcds.2025122 |
| DOI: | 10.48550/arxiv.2508.01162 |
| Přístupová URL adresa: | http://arxiv.org/abs/2508.01162 |
| Rights: | arXiv Non-Exclusive Distribution |
| Přístupové číslo: | edsair.doi.dedup.....612ead0e522f5e1d4cf06ca8593ebe39 |
| Databáze: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstrakt: | We study the existence, multiplicity and regularity results of weak solutions for the Dirichlet problem of a semi-linear elliptic equation driven by the mixture of the usual Laplacian and fractional Laplacian \begin{equation*} \left\{% \begin{array}{ll} -Δu + (-Δ)^{s} u+ a(x)\ u =f(x,u) & \hbox{in $Ω$,} u=0 & \hbox{in $\mathbb{R}^n\backslashΩ$} \end{array}% \right. \end{equation*} where $s \in (0,1)$, $Ω\subset \mathbb{R}^{n}$ is a bounded domain, the coefficient $a$ is a function of $x$ and the subcritical nonlinearity $f(x,u)$ has superlinear growth at zero and infinity. We show the existence of a non-trivial weak solution by Linking Theorem and Mountain Pass Theorem respectively for $λ_{1} \leqslant 0$ and $λ_{1} > 0$, where $λ_{1}$ denotes the first eigenvalue of $-Δ+ (-Δ)^{s} +a(x)$. In particular, adding a symmetric condition to $f$, we obtain infinitely many solutions via Fountain Theorem. Moreover, for the regularity part, we first prove the $L^{\infty}$-boundedness of weak solutions and then establish up to $C^{2, α}$-regularity up to boundary.<br />To appear in DCDS |
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| ISSN: | 15535231 10780947 |
| DOI: | 10.3934/dcds.2025122 |