Existence and asymptotic profiles of positive solutions of quasilinear Schrödinger equations in R3

We study the quasilinear Schrodinger equation arising from the nonlinear dynamics of the superfluid condensate −Δu+λu+κ2(Δu2)u=β[1α3−1(α+u2)3]u, x ∈ R3, where λ, κ, α, and β are positive constants. By developing perturbation arguments, we prove that for each λ, θ, M > 0 with ακ = θ and βα−3κ = M, there exists κ0 > 0 such that for κ ∈ (0, κ0), the equation has a positive classical radial solution uκ satisfying maxx∈R3|κμuκ(x)|→0 for any μ≥12 as κ → 0+. Moreover, up to a subsequence, it follows that uκ → u0 in H2(R3) ∩ C2(R3) as κ → 0+, where u0 is the least energy solution of problem −Δu + λu = 3Mθ−1u3, x ∈ R3. Our existence result generalizes the previous result in one-dimensional space obtained by Brull and Lange in 1986.