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arXiv · 2603.26493

Mixed-dispersion Schr\"odinger equations and Gagliardo-Nirenberg inequalities: equivalence between ground states and optimizers

Zhisu Liu·Giulio Romani·Yu Su
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Abstract

We study a nonlinear Schr\"odinger equation with mixed dispersion in the mass competition regime, namely mass-supercritical for the Laplacian and mass-subcritical for the Bilaplacian. In this setting, the existence of a critical value of the mass $c_\varepsilon$, which divides existence and nonexistence of energy ground state solutions, was established in [Bonheure, Cast\'eras, dos Santos, Nascimento, SIAM J. Math. Anal. 50 (2018)]. In this work, we strengthen these results by investigating the relationship between the energy ground states with critical mass, and the optimizers of mixed Gagliardo-Nirenberg-type inequalities. Moreover, we discuss the equivalence between energy and action ground states solutions.

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Zhisu Liu, Giulio Romani, Yu Su. 2026-03-27. Mixed-dispersion Schr\"odinger equations and Gagliardo-Nirenberg inequalities: equivalence between ground states and optimizers. https://arxiv.org/abs/2603.26493

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