Multiple solutions for elliptic equations driven by higher-order fractional Laplacian

Fuwei Cheng; Xifeng Su; Jiwen Zhang

doi:10.3934/cpaa.2025067

Article Contents

2025, Volume 24, Issue 10: 1973-1991. Doi: 10.3934/cpaa.2025067

This issue Previous Article On the equivalence of derivatives for maps between Carnot groups Next Article The Wong-Zakai approximations of invariant manifolds for stochastic evolution equations with linear noise

Multiple solutions for elliptic equations driven by higher-order fractional Laplacian

School of Mathematical Sciences, Beijing Normal University, No. 19, XinJieKouWai Street, HaiDian District, Beijing 100875, China

^*Corresponding author: Xifeng Su
^*Corresponding author: Xifeng Su

Received: February 2025

Revised: April 2025

Early access: May 12, 2025

Published online: May 12, 2025

Xifeng Su is supported by National Natural Science Foundation of China (Grant No. 12371186).

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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.

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Mathematics Subject Classification: Primary: 35A15, 35J35, 35R09; Secondary: 35R11, 45K05.

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