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.
Farina, On the classification of solutions of the Lane–Emden equation on unbounded domains of RN, J
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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.