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Normalized solutions for Sobolev critical Schr\"{o}dinger equations on bounded domains
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abstract
We study the existence and multiplicity of positive solutions with prescribed $L^2$-norm for the Sobolev critical Schr\"odinger equation on a bounded domain $\Omega\subset\mathbb{R}^N$, $N\ge3$: \[ -\Delta U = \lambda U + U^{2^{*}-1},\qquad U\in H^1_0(\Omega),\qquad \int_\Omega U^2\,dx = \rho^{2}, \] where $2^*=\frac{2N}{N-2}$. First, we consider a general bounded domain $\Omega$ in dimension $N\ge3$, with a restriction, only in dimension $N=3$, involving its inradius and first Dirichlet eigenvalue. In this general case we show the existence of a mountain pass solution on the $L^2$-sphere, for $\rho$ belonging to a subset of positive measure of the interval $(0,\rho^{**})$, for a suitable threshold $\rho^{**}>0$. Next, assuming that $\Omega$ is star-shaped, we extend the previous result to all values $\rho\in(0,\rho^{**})$. With respect to that of local minimizers, already known in the literature, the existence of mountain pass solutions in the Sobolev critical case is much more elusive. In particular, our proofs are based on the sharp analysis of the bounded Palais-Smale sequences, provided by a nonstandard adaptation of the Struwe monotonicity trick, that we develop.
Forward citations
Cited by 4 Pith papers
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An action approach to nodal and least energy normalized solutions for nonlinear Schr\"odinger equations
The masses of action and nodal action ground states form a full interval, which yields normalized nodal solutions and least-energy/action-ground-state identifications on bounded domains.
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Normalized solutions for NLS equations with potential on bounded domains: Ground states and multiplicity
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 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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