Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities
By Jun-cheng Wei, Yuanze Wu. Journal of Functional Analysis, 2021
246 citations recorded by the source corpus.
In this paper, we consider the following nonlinear Schr\"{o}dinger equations with mixed nonlinearities: \begin{eqnarray*} \left\{\aligned&-\Delta u=\lambda u+\mu |u|^{q-2}u+|u|^{2^*-2}u\quad\text{in }\mathbb{R}^N,\\&u\in H^1(\bbr^N),\quad\int_{\bbr^N}u^2=a^2, \endaligned\right. \end{eqnarray*} where $N\geq3$, $\mu>0$, $\lambda\in\mathbb{R}$ and $2<q<2^*$. We prove in this paper \begin{enumerate} \item[$(1)$]\quad Existence of solutions of mountain-pass type for $N=3$ and $2<q<2+\frac{4}{N} $. \item[$(2)$]\quad Existence and nonexistence of ground states for $2+\frac{4}{N}\leq q<2^*$ with $\mu>0$ large. \item[$(3)$]\quad Precisely asymptotic behaviors of ground states and mountain-pass solutions as $\mu\to0$ and $\mu$ goes to its upper bound. \end{enumerate} Our studies answer some questions proposed by Soave in \cite[Remarks~1.1, 1.2 and 8.1]{S20}.
Subfields: Applied Mathematics, Mathematical Physics, Mechanics of Materials
View Paper: https://www.semanticscholar.org/paper/4dbfa2fd85d37a1cab934b837783a1bd3775e5c0
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In Applied Mathematics
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Citations in the corpus
- Multiple normalized solutions for a Sobolev critical Schrödinger equation (2020, methodology): “The choice done in [38] is more in the spirit of the one of [37], see Remark 5.”
- Standing waves to upper critical Choquard equation with a local perturbation: Multiplicity, qualitative properties and stability (2021, methodology): “When N 5 ≥ and p̄ 2 < , the methods used in [29] can not threat the nonlocal term I u u u α p p ̄ ̄ 2 ∣ ∣ ∣ ∣ ( ) ∗ − directly, see Lemma 4.”
- Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems (2022, background): “Similar results have been achieved for scalar equations in [43].”
- A global branch approach to normalized solutions for the Schrödinger equation (2021, background): “We also refer to [19, 35, 39] for the treatment of nonlinearities which are Sobolev critical.”