Multiple solutions for elliptic equations driven by higher order fractional Laplacian

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.