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Two Positive Normalized Solutions on Star-shaped Bounded Domains to the Br\'ezis-Nirenberg Problem, I: Existence
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We develop a new framework to prove the existence of two positive solutions with prescribed mass on star-shaped bounded domains: one is the normalized ground state and another is of M-P type. We merely address the Sobolev critical cases since the Sobolev subcritical ones can be addressed by following similar arguments and are easier. Our framework is based on some important observations, that, to the best of our knowledge, have not appeared in previous literatures. Using these observations, we firstly establish the existence of a normalized ground state solution, whose existence is unknown so for. Then we use some novel ideas to obtain the second positive normalized solution, which is of M-P type. It seems to be the first time in the literatures to get two positive solutions under our settings, even in the Sobolev subcritical cases. We further remark that our framework is applicable to many other equations.
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Cited by 7 Pith papers
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Normalized solutions for the NLS equation with potential in higher dimension: the purely Sobolev critical case
For N≥6, the normalized Sobolev-critical NLS with potential has a positive mountain-pass solution; for N≥3, a negative-energy local minimizer exists under weaker conditions than in the cited preprint.
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Multiple sign-changing and semi-nodal normalized solutions for a Gross-Pitaevskii type system on bounded domains: the $L^2$-supercritical case
For small prescribed masses, the m-coupled Gross-Pitaevskii system on a bounded domain in R^3 or R^4 admits arbitrarily many sign-changing and semi-nodal normalized solutions, for all signs of the coupling constants.
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For supercritical power combinations, the NLS with a small potential on large bounded domains admits a negative-energy minimizer and a positive-energy mountain pass solution; in the Sobolev critical case, a ground sta...
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Normalized solutions for fractional Choquard equation with critical growth on bounded domain
For the critical fractional Choquard equation with a perturbation on a bounded star-shaped domain, at least two positive normalized solutions exist when the prescribed mass lies below an explicit threshold.
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Existence and local uniqueness of multi-spike solutions for Br\'{e}zis-Nirenberg problem with prescribed mass
The authors prove the existence and local uniqueness of multi-peak solutions with prescribed L2 mass for the critical Brézis-Nirenberg problem in dimensions at least six.
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Normalized solutions on large smooth domains to the Schr\"{o}dinger equations with potential and combined nonlinearities: The Sobolev critical case
A variational existence proof for normalized critical NLS solutions on large domains contains a false Liouville step and sign errors, so the main theorems are not established.
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Normalized Solutions on large smooth domains to the Schr\"{o}dinger equation with potential and general nonlinearity: Mass super-critical case
For mass-supercritical nonlinearities with an external potential, positive normalized solutions exist on sufficiently large star-shaped domains and, under a radial condition on the potential, in R^N.
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