By approximating the log Schrödinger equation with Lane-Emden problems at σ_q=2/(q-1) and passing to the limit q→1+, the paper constructs a solution u with concave log u.
Hofer, A note on the topological degree at a critical point of mountain-pass type , Proc
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Power law convergence and concavity for the Logarithmic Schr\"odinger equation
By approximating the log Schrödinger equation with Lane-Emden problems at σ_q=2/(q-1) and passing to the limit q→1+, the paper constructs a solution u with concave log u.