Multiple solutions for elliptic equations driven by higher order fractional Laplacian
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913823644975104 |
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| author | Cheng, Fuwei Su, Xifeng Zhang, Jiwen |
| author_facet | Cheng, Fuwei Su, Xifeng Zhang, Jiwen |
| contents | We consider an elliptic partial differential equation driven by higher order fractional Laplacian $(-Δ)^{s}$, $s \in (1,2)$ with homogeneous Dirichlet boundary condition
\begin{equation*}
\left\{%
\begin{array}{ll}
(-Δ)^{s} u=f(x,u) & \text{ in }Ω,
u=0 & \text{ in } \mathbb{R}^n \setminus Ω.
\end{array}%
\right.
\end{equation*}
The above equation has a variational nature, and we investigate the existence and multiplicity results for its weak solutions under various conditions on the nonlinear term $f$: superlinear growth, concave-convex and symmetric conditions and their combinations.
The existence of two different non-trivial weak solutions is established by Mountain Pass Theorem and Ekeland's variational principle, respectively.
Furthermore, due to Fountain Theorem and its dual form, both infinitely many weak solutions with positive energy and infinitely many weak solutions with negative energy are obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02065 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple solutions for elliptic equations driven by higher order fractional Laplacian Cheng, Fuwei Su, Xifeng Zhang, Jiwen Analysis of PDEs 35A15, 35J35, 35R09, 35R11, 45K05 We consider an elliptic partial differential equation driven by higher order fractional Laplacian $(-Δ)^{s}$, $s \in (1,2)$ with homogeneous Dirichlet boundary condition \begin{equation*} \left\{% \begin{array}{ll} (-Δ)^{s} u=f(x,u) & \text{ in }Ω, u=0 & \text{ in } \mathbb{R}^n \setminus Ω. \end{array}% \right. \end{equation*} The above equation has a variational nature, and we investigate the existence and multiplicity results for its weak solutions under various conditions on the nonlinear term $f$: superlinear growth, concave-convex and symmetric conditions and their combinations. The existence of two different non-trivial weak solutions is established by Mountain Pass Theorem and Ekeland's variational principle, respectively. Furthermore, due to Fountain Theorem and its dual form, both infinitely many weak solutions with positive energy and infinitely many weak solutions with negative energy are obtained. |
| title | Multiple solutions for elliptic equations driven by higher order fractional Laplacian |
| topic | Analysis of PDEs 35A15, 35J35, 35R09, 35R11, 45K05 |
| url | https://arxiv.org/abs/2505.02065 |