 12102 Xifeng Su and Yuanhong Wei
 Multiplicity of solutions for nonlocal elliptic equations driven by fractional Laplacian
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Sep 21, 12

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Abstract. We consider the semilinear 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 nontrivial solutions for
the above equation are established. The validity of the PalaisSmale
condition without AmbrosettiRabinowitz condition for nonlocal
elliptic equations is proved. Two nontrivial solutions are given
under some weak hypotheses. Nonlocal elliptic equations with
concaveconvex nonlinearities are also studied, and existence of at
least six solutions are obtained.
Moreover, a global result of AmbrosettiBrezisCerami 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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