Xifeng Su and Yuanhong Wei
Multiplicity of solutions for non-local elliptic equations driven by fractional Laplacian
(300K, pdf)

ABSTRACT.  We consider the semi-linear elliptic PDEs driven by the fractional 
Laplacian: 
egin{equation*} 
\left\{% 
egin{array}{ll} 
 (-\Delta)^s u=f(x,u), & \hbox{in $\Omega$,} \ 
 u=0, & \hbox{in $\mathbb{R}^nackslash\Omega$.} \ 
\end{array}% 

ight. 
\end{equation*} 
By the Mountain Pass Theorem and some other nonlinear analysis 
methods, the existence and multiplicity of non-trivial solutions for 
the above equation are established. The validity of the Palais-Smale 
condition without Ambrosetti-Rabinowitz condition for non-local 
elliptic equations is proved. Two non-trivial solutions are given 
under some weak hypotheses. Non-local elliptic equations with 
concave-convex nonlinearities are also studied, and existence of at 
least six solutions are obtained. 
Moreover, a global result of Ambrosetti-Brezis-Cerami type is given, 
which shows that the effect of the parameter $\lambda$ in the 
nonlinear term changes considerably the nonexistence, existence and 
multiplicity of solutions.