Multiplicity and concentration of nontrivial solutions for generalized extensible beam equations in R^N

Abstract

In this article, we study a class of generalized extensible beam equations with a superlinear nonlinearity Δ2u - M(∥∇u∥2L2) Δu + λV(x)u = ƒ(x, u) in ℝN, u ∈ H2 (ℝN), where N ≥ 3, M(t) = αtδ + b with α, δ > 0 and b ∈ ℝ, λ > 0 is a parameter, V ∈ C(ℝN, ℝ) and ƒ ∈ C(ℝN x ℝ, ℝ). Unlike most other papers on this problem, we allow the constant b to be non-positive, which has the physical significance. Under some suitable assumptions on V(x) and ƒ(x, u), when α is small and λ is large enough, we prove the existence of two nontrivial solutions u(1)α,λ and u(2)α,λ, one of which will blow up as the nonlocal term vanishes. Moreover, u(1)α,λ → u(1)∞ and u(2)α,λ → u(2)∞ strongly in H2(ℝN) as λ → ∞, where u(1)∞ ≠ u(2)∞ ∈ H20(Ω) are nontrivial solutions of Dirichlet BVPs on the bounded domain Ω. Also, the nonexistence of nontrivial solutions is also obtained for α large enough.