Blow-Up Phenomena and Asymptotic Profiles Passing from H 1-Critical to Super-Critical Quasilinear Schrödinger Equations

Abstract

Abstract We study the asymptotic profile, as ℏ → 0 {\hbar\rightarrow 0} , of positive solutions to - ℏ 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u - ℏ 2 + γ ⁢ u ⁢ Δ ⁢ u 2 = K ⁢ ( x ) ⁢ | u | p - 2 ⁢ u , x ∈ ℝ N , -\hbar^{2}\Delta u+V(x)u-\hbar^{2+\gamma}u\Delta u^{2}=K(x)\lvert u\rvert^{p-2% }u,\quad x\in\mathbb{R}^{N}, where γ ⩾ 0 {\gamma\geqslant 0} is a parameter with relevant physical interpretations, V and K are given potentials and the dimension N is greater than or equal to 5, as we look for finite L 2 {L^{2}} -energy solutions. We investigate the concentrating behavior of solutions when γ > 0 {\gamma>0} and, differently from the case γ = 0 {\gamma=0} where the leading potential is V, the concentration is here localized by the source potential K. Moreover, surprisingly for γ > 0 {\gamma>0} we find a different concentration behavior of solutions in the case p = 2 ⁢ N N - 2 {p=\frac{2N}{N-2}} and when 2 ⁢ N N - 2 < p < 4 ⁢ N N - 2 {\frac{2N}{N-2}<p<\frac{4N}{N-2}} . This phenomenon does not occur when γ = 0 {\gamma=0} .