Normalized ground states of nonlinear biharmonic Schrodinger equations with Sobolev critical growth and combined nonlinearities

Article Mathematics, Applied

Orbital stability of ground states for a Sobolev critical Schrodinger equation

Louis Jeanjean et al.

Summary: In this study, we investigate the existence of ground state standing waves, of prescribed mass, for the nonlinear Schrodinger equation with mixed power nonlinearities. We prove that all ground states correspond to local minima of the associated Energy functional, and despite the fact that the nonlinearity is Sobolev critical, the set of ground states is orbitally stable.

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES (2022)

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Article Mathematics

Non-homogeneous Gagliardo-Nirenberg inequalities in RN and application to a biharmonic non-linear Schrodinger equation

Antonio J. Fernandez et al.

Summary: This paper presents a new method based on the Tomas-Stein inequality to establish non-homogeneous Gagliardo-Nirenberg-type inequalities in R-N. These inequalities are then used to study the energy-minimizing standing waves for a fourth-order Schrodinger equation with mixed dispersion, while keeping the L-2 norm (mass) fixed. The authors provide optimal results on the existence of minimizers in the mass-subcritical and mass-critical cases, while in the mass-supercritical case global minimizers do not exist. However, local minimizers can be found if the Laplacian and the bi-Laplacian have the same sign.

JOURNAL OF DIFFERENTIAL EQUATIONS (2022)

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Article Mathematics

Normalized solutions for Schrdinger equations with critical Sobolev exponent and mixed nonlinearities

Juncheng Wei et al.

Summary: In this paper, we study the nonlinear Schrodinger equations with mixed nonlinearities. We establish the existence of solutions and investigate the existence and nonexistence of ground states in different parameter ranges. Additionally, we provide precise asymptotic behaviors of the ground states and mountain-pass solutions as the parameters approach specific values.

JOURNAL OF FUNCTIONAL ANALYSIS (2022)

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Article Mathematics