In this paper the authors investigate a class of \(p\)-Laplace equations with logarithmic nonlinearity, which were considered in Le and Le (Acta Appl. Math. 151:149–169, 2017), where, among other things, global existence and finite time blow-up of solutions were proved when the initial energy is subcritical and critical, that is, initial energy smaller than or equal to the depth of the potential well. Their results are complemented in this paper in the sense that an abstract criterion is given for the existence of global solutions that vanish at infinity or solutions that blow up in finite time, when the initial energy is supercritical. As a byproduct it is shown that the problem admits a finite time blow-up solution for arbitrarily high initial energy.
\(p\)-Laplace Logarithmic nonlinearity High initial energy Vanish at infinity Blow up
Mathematics Subject Classification (2010)
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The authors would like to thank the referees for pointing out two important references which bring us some new ideas. They would also like to express their sincere gratitude to Professor Wenjie Gao for his enthusiastic guidance and constant encouragement.
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