Year: 2025
Type: article
Abstract: We consider an elliptic partial differential equation driven by higher-order fractional Laplacian $ (-\Delta)^{s} $, $ s \in (1, 2) $ with homogeneous Dirichlet boundary condition$ \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u = f(x, u) & \text{ in }\Omega, \\ u = 0 & \text{ in } \mathbb{R}^n \setminus \Omega. \ \end{array} \right. \end{equation*} $The above equation has a variational nature, and we investigated 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 was established by mountain pass theorem and Ekeland's variational principle. 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 were obtained.
DOI: 10.3934/cpaa.2025067
Authors: Fuwei Cheng, Xifeng Su, Jiwen Zhang
Institution: Beijing Normal University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Award: G5683399595
OpenAlex ID: W4410305166