12-102 Xifeng Su and Yuanhong Wei
Multiplicity of solutions for non-local elliptic equations driven by fractional Laplacian (300K, pdf) Sep 21, 12
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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.

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