| Contents |
| 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. |
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