Abstract
In this paper, we show the existence of solutions for an indefinite fractional Schrödinger equation driven by the variable-order fractional magnetic Laplace operator involving variable exponents and steep potential. By using the decomposition of the Nehari manifold and variational method, we obtain the existence results of nontrivial solutions to the equation under suitable conditions.
Highlights
We investigate the existence of solutions of the following concave-convex fractional elliptic equation driven by the variable-order fractional magnetic Laplace operator involving variable exponents: ð−ΔÞsAð·Þu + VλðxÞu = f ðxÞu ∣ qðxÞ−2u + gðxÞupðxÞ−2u in RN, ð1Þ
As far as we know, this is the first time to study the multiplicity of nontrivial solutions of the indefinite fractional elliptic equation driven by the variable-order fractional magnetic Laplace operator with variable exponents and steep potential in RN
It is worth mentioning that in this paper, we obtain the existence and multiplicity results of nontrivial solutions of the variable-order fractional magnetic Schrodinger equation with variable growth and steep well potential in RN and our main results are based on the study for the decomposition of Nehari manifolds
Summary
Authors studied the multiplicity and concentration of solutions for a Hamiltonian system driven by the fractional Laplace operator with variable-order derivative in [3]. For sð·Þ, pðxÞ, qðxÞ ≡ constant, and A = 0, in [5], under appropriate assumptions, Peng et al obtained the existence, multiplicity, and concentration of nontrivial solutions for the following indefinite fractional elliptic equation by using the Nehari manifold decomposition:
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