For sufficiently small prescribed mass, normalized solutions exist, and often many of them exist, for a large class of elliptic boundary value problems with general nonlinearities and several boundary condition types.
Existence and Multiplicity of Normalized Solutions to Schr\"{o}dinger Equations with General Nonlinearities in Bounded Domains
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abstract
This paper focuses on the existence of multiple normalized solutions to Schr\"{o}dinger equations with general nonlinearities in bounded domains via variational methods. We first obtain two positive normalized solutions, one is a normalized ground state by searching for a local minimizer, and the other one is a mountain pass solution. Secondly, using a version of Linking theorems for normalized solutions, we prove the multiplicity of solutions to Schr\"{o}dinger equations in a star-shaped bounded domain. Moreover, we arrive at the existence of nonradial normalized solutions to Schr\"odinger equations in a ball.
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Existence and multiplicity of normalized solutions to a large class of elliptic equations on bounded domains with general boundary conditions
For sufficiently small prescribed mass, normalized solutions exist, and often many of them exist, for a large class of elliptic boundary value problems with general nonlinearities and several boundary condition types.