Xifeng Su and Yuanhong Wei
Multiplicity of solutions for non-local elliptic equations driven by fractional Laplacian
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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.