Multiplicity results of nonlinear fractional magnetic Schrödinger equation with steep potential

Multiplicity Results for Variable-Order Nonlinear Fractional Magnetic Schrödinger Equation with Variable Growth

“…It is worth noting that in this paper, we not only prove that there exist two different nontrivial solutions to problem (1), but we also show that the two nontrivial solutions of problem (1) converge to two solutions of the limit problem for problem (1). The novelty of this paper is that, compared with [1], we write the original fractional magnetic Schrödinger equation without variable growth to a variable-order fractional magnetic Schrödinger equation with variable growth. In addition, compared with [2], we write the variable order fractional Schrödinger equation to a variable order fractional magnetic Schrödinger equation.…”

“…Secondly, we define variable-order fractional magnetic Sobolev spaces and prove the compact conditions between them. Finally, we give the variational setting for problem (1) and theorems that will be used later.…”

“…In this paper, we consider the multiplicity of nontrivial solutions of the following concave-convex elliptic equation involving variable-order nonlinear fractional magnetic Schrödinger equation: A is the variable-order fractional magnetic Laplace operator; the potential V λ ðxÞ = λV + ðxÞ − V − ðxÞ with V ± = max f±V, 0g; λ > 0 is a parameter; magnetic field A ∈ C 0,α ðℝ N , ℝ N Þ with α ∈ ð0, 1; f , g > 0 are two bounded nonnegative measurable function; p, q ∈ CðΩÞ; and u : ℝ N ⟶ ℂ. In [1], the fractional magnetic Laplacian has been defined as for x ∈ ℝ N . In [2], the variable-order fractional magnetic Laplace ð−ΔÞ sð•Þ is defined as follows:…”

“…For sð•Þ = s, pðxÞ, qðxÞ ≡ constant, in [1], the authors study the existence of solutions for the following equation on the whole space by using the method of Nehari manifold decomposition,and obtain some sufficient conditions for the existence of nontrivial solutions of the following equation:…”

“…Based on the hypothesis ðS 2 Þ, we can give the following definition of weak solutions for problem (1).…”