Multiple solutions for elliptic equations driven by higher order fractional Laplacian

TL;DR

This paper studies elliptic equations involving higher order fractional Laplacians, establishing multiple solutions using variational methods under various nonlinear conditions, including superlinear, concave-convex, and symmetric cases.

Contribution

It provides new existence and multiplicity results for solutions of higher order fractional Laplacian equations using advanced variational techniques.

Findings

Existence of two non-trivial solutions via Mountain Pass and Ekeland principles.

Infinitely many solutions with positive and negative energies.

Solutions under superlinear, concave-convex, and symmetric nonlinearities.

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 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.