Tempered fractional differential equation: variational approach

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In this paper, we are concerned with the existence of ground state solution for the following fractional differential equations with tempered fractional derivative: D−α,λ(D+α,λu(t))=f(t,u(t)),t∈Ru∈Wα,2(R),(FD) where α∈(1/2,1), λ>0, D±α,λu are the left and right tempered fractional derivatives, Wα,2(R) is the fractional Sobolev spaces, and f∈C(R×R,R). Assuming that f satisfies the Ambrosetti–Rabinowitz condition and another suitable conditions, by using mountain pass theorem and minimization argument over Nehari manifold, we show that (FD) has a ground state solution. Furthermore, we show that this solution is a radially symmetric solution. Copyright © 2017 John Wiley & Sons, Ltd.