Existence and multiplicity of solutions for fractional Schrödinger–Kirchhoff equations with Trudinger–Moser nonlinearity. (September 2019) - Xiang, Mingqi; Zhang, Binlin; Repovš, Dušan
Abstract: We study the existence and multiplicity of solutions for a class of fractional Schrödinger–Kirchhoff type equations with the Trudinger–Moser nonlinearity. More precisely, we consider M ( ‖ u ‖ N ∕ s ) ( − Δ ) N ∕ s s u + V ( x ) | u | N s − 1 u = f ( x, u ) + λ h ( x ) | u | p − 2 u in R N, ‖ u ‖ = ∬ R 2 N | u ( x ) − u ( y ) | N ∕ s | x − y | 2 N d x d y + ∫ R N V ( x ) | u | N ∕ s d x s ∕ N, where M : [ 0, ∞ ] → [ 0, ∞ ) is a continuous function, s ∈ ( 0, 1 ), N ≥ 2, λ > 0 is a parameter, 1 < p < ∞, ( − Δ ) N ∕ s s is the fractional N ∕ s -Laplacian, V : R N → ( 0, ∞ ) is a continuous function, f : R N × R → R is a continuous function, and h : R N → [ 0, ∞ ) is a measurable function. First, using the mountain pass theorem, a nonnegative solution is obtained when f satisfies exponential growth conditions and λ is large enough, and we prove that the solution converges to zero in W V s, N ∕ s ( R N ) as λ → ∞ . Then, using the Ekeland variational principle, a nonnegative nontrivial solution is obtained when λ is small enough, and we show that the solution converges to zero in W V s, N ∕ s ( R N ) as λ → 0 . Furthermore, using the genus theory, infinitely many solutions are obtained when M is a special function and λ is small enough. We note that our paper covers a novel feature of Kirchhoff problems, that is, the Kirchhoff function M ( 0 ) = 0 .