Tempered fractional differential equation: variational approach

In this paper, we are concerned with the existence of ground state solution for the following fractional differential equations with tempered fractional derivative: FDD − α , λ ( D + α , λ u ( t ) ) = f ( t , u ( t ) ) ,t ∈ Ru ∈ W α , 2 ( R ) ,where α ∈(1/2,1), λ >0,D ± α , λ u are the left and right tempered fractional derivatives,W α , 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.