Existence and Asymptotic Behaviors of Ground States for a Fourth-Order Nonlinear Schrödinger Equations with a Potential
In this paper, we study the existence and asymptotic behaviors of ground state solutions to a fourth-order nonlinear Schrödinger equation with mass-critical exponent, where the fourth-order term appears as a perturbation with <inline-formula><math xmlns="http://www.w3.org/1998/Math/Mat...
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| Format: | Article |
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MDPI AG
2022-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/15/2736 |
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| author | Jintao He Tingjian Luo |
| author_facet | Jintao He Tingjian Luo |
| author_sort | Jintao He |
| collection | DOAJ |
| description | In this paper, we study the existence and asymptotic behaviors of ground state solutions to a fourth-order nonlinear Schrödinger equation with mass-critical exponent, where the fourth-order term appears as a perturbation with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. By considering a constrained variational problem, we first establish the existence of ground state solutions. Then, we prove the asymptotic behaviors of the solutions as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></semantics></math></inline-formula>. The main ingredients of the proofs are some energy estimate arguments. Our results improve somewhat the ones in the existing reference. |
| first_indexed | 2024-03-09T10:06:42Z |
| format | Article |
| id | doaj.art-96a5fe43080c4478a6a26d63dd4b940c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T10:06:42Z |
| publishDate | 2022-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-96a5fe43080c4478a6a26d63dd4b940c2023-12-01T23:02:25ZengMDPI AGMathematics2227-73902022-08-011015273610.3390/math10152736Existence and Asymptotic Behaviors of Ground States for a Fourth-Order Nonlinear Schrödinger Equations with a PotentialJintao He0Tingjian Luo1School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, ChinaSchool of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, ChinaIn this paper, we study the existence and asymptotic behaviors of ground state solutions to a fourth-order nonlinear Schrödinger equation with mass-critical exponent, where the fourth-order term appears as a perturbation with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. By considering a constrained variational problem, we first establish the existence of ground state solutions. Then, we prove the asymptotic behaviors of the solutions as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></semantics></math></inline-formula>. The main ingredients of the proofs are some energy estimate arguments. Our results improve somewhat the ones in the existing reference.https://www.mdpi.com/2227-7390/10/15/2736fourth-order equationground statesasymptotic behaviorsenergy estimates |
| spellingShingle | Jintao He Tingjian Luo Existence and Asymptotic Behaviors of Ground States for a Fourth-Order Nonlinear Schrödinger Equations with a Potential Mathematics fourth-order equation ground states asymptotic behaviors energy estimates |
| title | Existence and Asymptotic Behaviors of Ground States for a Fourth-Order Nonlinear Schrödinger Equations with a Potential |
| title_full | Existence and Asymptotic Behaviors of Ground States for a Fourth-Order Nonlinear Schrödinger Equations with a Potential |
| title_fullStr | Existence and Asymptotic Behaviors of Ground States for a Fourth-Order Nonlinear Schrödinger Equations with a Potential |
| title_full_unstemmed | Existence and Asymptotic Behaviors of Ground States for a Fourth-Order Nonlinear Schrödinger Equations with a Potential |
| title_short | Existence and Asymptotic Behaviors of Ground States for a Fourth-Order Nonlinear Schrödinger Equations with a Potential |
| title_sort | existence and asymptotic behaviors of ground states for a fourth order nonlinear schrodinger equations with a potential |
| topic | fourth-order equation ground states asymptotic behaviors energy estimates |
| url | https://www.mdpi.com/2227-7390/10/15/2736 |
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