Blow-Up Phenomena and Asymptotic Profiles Passing from H1-Critical to Super-Critical Quasilinear Schrodinger Equations

作者：Cassani, Daniele^*; Wang, Youjun

来源：ADVANCED NONLINEAR STUDIES, 2021, 21(4): 855-874.

DOI：10.1515/ans-2021-2144

• Semiclassical States
• Concentration Phenomena
• Finite Energy Solutions
• Non-Autonomous Schrodinger Equations
• Critical Growth

We study the asymptotic profile, as h -> 0, of positive solutions to @@@ -h(2)Delta u + V(x)u - h(2+y)u Delta u(2) = K(x)vertical bar u vertical bar(p-2)u, x is an element of R-N, @@@ where gamma >= 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-energy solutions. We investigate the concentrating behavior of solutions when gamma > 0 and, differently from the case gamma = 0 where the leading potential is V, the concentration is here localized by the source potential K. Moreover, surprisingly for gamma > 0 we find a different concentration behavior of solutions in the case p = 2N/N-2 and when 2N/N-2 < p < 4N/N-2 This phenomenon does not occur when gamma = 0.

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