Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian

CDSNS Colloquium
Tuesday, January 7, 2014 - 3:05pm
1 hour (actually 50 minutes)
Skiles 005
Beijing Normal University
We consider the semi-linear elliptic PDE driven by the fractional Laplacian: \begin{equation*}\left\{%\begin{array}{ll}    (-\Delta)^s u=f(x,u) & \hbox{in $\Omega$,} \\    u=0 &  \hbox{in $\mathbb{R}^n\backslash\Omega$.} \\\end{array}% \right.\end{equation*}An $L^{\infty}$ regularity result is given, using De Giorgi-Stampacchia iteration  method.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.