Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential

Abstract

In this article, we prove the existence, multiplicity and concentration of non-trivial solutions for the following indefinite fractional elliptic equation with concave-convex nonlinearities: (−∆) α u + V λ (x)u = a(x)|u| q−2 u + b(x)|u| p−2 u in R N , u ≥ 0 in R N , where 0 < α < 1, N > 2α, 1 < q < 2 < p < 2 * α with 2 * α = 2N/(N − 2α), the potential V λ (x) = λV + (x)−V − (x) with V ± = max{±V, 0} and the parameter λ > 0. Our multiplicity results are based on studying the decomposition of the Nehari manifold.

Variational Approach for the Variable-Order Fractional Magnetic Schrödinger Equation with Variable Growth and Steep Potential in ℝ N ∗

“…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:…”

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

“…For sð•Þ = α, pðxÞ, qðxÞ ≡ constant, and A = 0; in [4], the authors obtained the existence, multiplicity, and concentration of nontrivial solutions for the following indefinite fractional elliptic equation by using Nehari manifold decomposition:…”

On the semilinear fractional elliptic equations with singular weight functions

“…Recently, the concave-convex problems involving the fractional Laplacian for equation (E λ ) have been attracted many attentions (cf. [6,7,25,37,41,42,45]). In particular, Peng and Xia [41] extended Cheng-Wu's results [18] to fractional Laplacian case with 0 < α < 1.…”

“…In this paper, we want to generalized the results of [18] and [41] with conditions (1) relaxed. Instead, we assume that b satisfies condition (V 3) and control f and g by a.…”

“…The next theorem shows that the concentration phenomena holds for the nontrivial solutions obtained in Theorem 1.1. Because, the proof is essential same as that of Theorem 1.3 in [18] (or see [41,Theorem 1.3]) and is not repeated here. Theorem 1.3.…”