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Positive normalized solutions to nonlinear elliptic systems in $\R^4$ with critical Sobolev exponent
Analysis of PDEs · Submitted 19 Jul 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behavior of the normalized solutions to Schr\"{o}dinger systems with Sobolev critical exponent.
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In this paper, we consider the existence and asymptotic behavior on mass of the positive solutions to the following system: \begin{equation}\label{eqA0.1}\nonumber \begin{cases} -\Delta u+\lambda_1u=\mu_1u^3+\alpha_1|u|^{p-2}u+\beta v^2u\quad&\hbox{in}~\R^4,\\ -\Delta v+\lambda_2v=\mu_2v^3+\alpha_2|v|^{p-2}v+\beta u^2v\quad&\hbox{in}~\R^4,\\ \end{cases} \end{equation} under the mass constraint $$\int_{\R^4}u^2=a_1^2\quad\text{and}\quad\int_{\R^4}v^2=a_2^2,$$ where $a_1,a_2$ are prescribed, $\mu_1,\mu_2,\beta>0$; $\alpha_1,\alpha_2\in \R$, $p\!\in\! (2,4)$ and $\lambda_1,\lambda_2\!\in\!\R$ appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., $\alpha_i<0(i=1,2)$. Then turning to the case of $\alpha_i>0 (i=1,2)$, if $2<p<3$, we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Moreover, we give a precise asymptotic behavior of these two solutions as $(a_1,a_2)\to (0,0)$ and $a_1\sim a_2$. This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behavior of the normalized solutions to Schr\"{o}dinger systems with Sobolev critical exponent. When $3\leq p<4$, we prove an existence as well as non-existence ($p=3$) results of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. Furthermore, precise asymptotic behaviors of the ground states are obtained when the masses of whose two components vanish and cluster to a upper bound (or infinity), respectively.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we consider the existence and asymptotic behavior on mass of the positive solutions to the following system: \begin{equation}\label{eqA0.1}\nonumber \begin{cases} -\Delta u+\lambda_1u=\mu_1u^3+\alpha_1|u|^{p-2}u+\beta v^2u\quad&\hbox{in}~\R^4,\\ -\Delta v+\lambda_2v=\mu_2v^3+\alpha_2|v|^{p-2}v+\beta u^2v\quad&\hbox{in}~\R^4,\\ \end{cases} \end{equation} under the mass constraint $$\int_{\R^4}u^2=a_1^2\quad\text{and}\quad\int_{\R^4}v^2=a_2^2,$$ where $a_1,a_2$ are prescribed, $\mu_1,\mu_2,\beta>0$; $\alpha_1,\alpha_2\in \R$, $p\!\in\! | background 0.62 | 0.48 | 0.04 | asserted 0.56 |
| s1 | (2,4)$ and $\lambda_1,\lambda_2\!\in\!\R$ appear as Lagrange multipliers. | background 0.98 | 0.13 | 0.03 | asserted 0.84 |
| s2 | Firstly, we establish a non-existence result for the repulsive interaction case, i.e., $\alpha_i<0(i=1,2)$. | result 0.97 | 0.93 | 0.08 | asserted 1.00 |
| s3 | Then turning to the case of $\alpha_i>0 (i=1,2)$, if $2<p<3$, we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. | result 0.93 | 0.97 | 0.29 | asserted 1.00 |
| s4 | Moreover, we give a precise asymptotic behavior of these two solutions as $(a_1,a_2)\to (0,0)$ and $a_1\sim a_2$. | result 0.93 | 0.97 | 0.12 | asserted 1.00 |
| s5 | This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behavior of the normalized solutions to Schr\"{o}dinger systems with Sobolev critical exponent. | contribution 0.84 | 0.62 | 0.21 | hedged possibility 0.79 |
| s6 | When $3\leq p<4$, we prove an existence as well as non-existence ($p=3$) results of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. | result 0.91 | 0.96 | 0.14 | asserted 1.00 |
| s7 | Furthermore, precise asymptotic behaviors of the ground states are obtained when the masses of whose two components vanish and cluster to a upper bound (or infinity), respectively. | result 0.89 | 0.95 | 0.12 | asserted 1.00 |
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The paper
Original abstract
In this paper, we consider the existence and asymptotic behavior on mass of the positive solutions to the following system: \begin{equation}\label{eqA0.1}\nonumber \begin{cases} -\Delta u+\lambda_1u=\mu_1u^3+\alpha_1|u|^{p-2}u+\beta v^2u\quad&\hbox{in}~\R^4,\\ -\Delta v+\lambda_2v=\mu_2v^3+\alpha_2|v|^{p-2}v+\beta u^2v\quad&\hbox{in}~\R^4,\\ \end{cases} \end{equation} under the mass constraint $$\int_{\R^4}u^2=a_1^2\quad\text{and}\quad\int_{\R^4}v^2=a_2^2,$$ where $a_1,a_2$ are prescribed, $\mu_1,\mu_2,\beta>0$; $\alpha_1,\alpha_2\in \R$, $p\!\in\! (2,4)$ and $\lambda_1,\lambda_2\!\in\!\R$ appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., $\alpha_i<0(i=1,2)$. Then turning to the case of $\alpha_i>0 (i=1,2)$, if $2<p<3$, we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Moreover, we give a precise asymptotic behavior of these two solutions as $(a_1,a_2)\to (0,0)$ and $a_1\sim a_2$. This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behavior of the normalized solutions to Schr\"{o}dinger systems with Sobolev critical exponent. When $3\leq p<4$, we prove an existence as well as non-existence ($p=3$) results of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. Furthermore, precise asymptotic behaviors of the ground states are obtained when the masses of whose two components vanish and cluster to a upper bound (or infinity), respectively.
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How the literature uses it
1 arXiv paper cite this in 3 sentences
Papers citing it · 1 sentence without a confident label
Uses its method or tool 1 paper
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The rest of proof is similar to that of Lemma 5.5 in [ref] , and just needs a slight modification..
Normalized solutions for Schr\"{o}dinger system with quadratic and cubic interactions · Proof of Theorems [ref] and [ref] · 0.82
Compares with it 1 paper
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We must point out that compared with [ref] , the problem [ref] in $^4$ is more delicate due to the uncertainty of sign of the quadratic interaction term and inhomogeneity of the coupling term.
Normalized solutions for Schr\"{o}dinger system with quadratic and cubic interactions · Introduction and main results · 0.85
What it cites · 8 sentences without a confident label
Uses the method or tool of 2 papers
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Motivated by [ref] , we choose appropriate functions to prove that $m^-(a_1,a_2)=0$ when $α_1a_1≥w_3_2$ and $α_2a_2≥w_3_2$, and then by contradiction [ref]-[ref] has no normalized ground state solution.
Normalized solutions for Schr\"odinger equations with critical Sobolev exponent and mixed nonlinearities · Introduction · 0.87
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Motivated by [ref] , we choose appropriate functions to prove that $m^-(a_1,a_2)=0$ when $α_1a_1≥w_3_2$ and $α_2a_2≥w_3_2$, and then by contradiction [ref]-[ref] has no normalized ground state solution.
Studies of normalized solutions to Schr\"{o}dinger equations with Sobolev critical exponent and combined nonlinearities · Introduction · 0.87
Compares with 2 papers
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Here, we have to emphasize that the system considered by Li and Zou in [ref] contains only a single critical nonlinearity, and coupled terms of the systems considered by Mederski and Schino in [ref] must be Sobolev subcritical.
Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities · Introduction · 0.98
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Here, we have to emphasize that the system considered by Li and Zou in [ref] contains only a single critical nonlinearity, and coupled terms of the systems considered by Mederski and Schino in [ref] must be Sobolev subcritical.
Least energy solutions to a cooperative system of Schr\"odinger equations with prescribed $L^2$-bounds: at least $L^2$-critical growth · Introduction · 0.98
Cites as background 4 papers
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For single Sobolev critical nonlinear Schrödinger equation [ref] with $q=2+4/N$, there exists a critical mass $a^*>0$ such that if $0<a<a^*$ then [ref] has normalized ground state solutions, while if $a^*≤a$, [ref] has no normalized ground states (see [ref] ).
Normalized solutions for Schr\"odinger equations with critical Sobolev exponent and mixed nonlinearities · Introduction · 0.83
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For single Sobolev critical nonlinear Schrödinger equation [ref] with $q=2+4/N$, there exists a critical mass $a^*>0$ such that if $0<a<a^*$ then [ref] has normalized ground state solutions, while if $a^*≤a$, [ref] has no normalized ground states (see [ref] ).
Studies of normalized solutions to Schr\"{o}dinger equations with Sobolev critical exponent and combined nonlinearities · Introduction · 0.83
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Recently, Li, Zou in [ref] and Mederski, Schino in [ref] considered normalized solutions to coupled Schrödinger systems with a wide class of nonlinearities including the Sobolev critical terms.
Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities · Introduction · 0.86
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