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Volume 38, Issue 2
Existence and Multiplicity Results for a Class of Nonlinear Schrödinger Equations with Magnetic Potential Involving Sign-Changing Nonlinearity

Francisco Odair de Paiva, Sandra Machado de Souza Lima & Olimpio Hiroshi Miyagaki

Anal. Theory Appl., 38 (2022), pp. 148-177.

Published online: 2022-07

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  • Abstract

In this work we consider the following class of elliptic problems $\begin{cases}  −∆_Au + u = a(x)|u|^{q−2}u + b(x)|u|^{p−2}u & {\rm in} & \mathbb{R}^N, \\u ∈ H^1_A (\mathbb{R}^N),   \tag{P} \end{cases}$ with $2 < q < p < 2^∗ = \frac{2N}{N−2},$ $a(x)$ and $b(x)$ are functions that can change sign and satisfy some additional conditions; $u \in H^1_A (\mathbb{R}^N)$ and $A : \mathbb{R}^N → \mathbb{R}^N$ is a magnetic potential. Also using the Nehari method in combination with other complementary arguments, we discuss the existence of infinitely many solutions to the problem in question, varying the assumptions about the weight functions.

  • Keywords

Magnetic potential, sign-changing weight functions, Nehari manifold, Fibering map.

  • AMS Subject Headings

35Q60, 35Q55, 35B38, 35B33

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-38-148, author = {Francisco Odair and de Paiva and and 24073 and and Francisco Odair de Paiva and Sandra Machado de and Souza Lima and and 24074 and and Sandra Machado de Souza Lima and Olimpio Hiroshi and Miyagaki and and 24075 and and Olimpio Hiroshi Miyagaki}, title = {Existence and Multiplicity Results for a Class of Nonlinear Schrödinger Equations with Magnetic Potential Involving Sign-Changing Nonlinearity}, journal = {Analysis in Theory and Applications}, year = {2022}, volume = {38}, number = {2}, pages = {148--177}, abstract = {

In this work we consider the following class of elliptic problems $\begin{cases}  −∆_Au + u = a(x)|u|^{q−2}u + b(x)|u|^{p−2}u & {\rm in} & \mathbb{R}^N, \\u ∈ H^1_A (\mathbb{R}^N),   \tag{P} \end{cases}$ with $2 < q < p < 2^∗ = \frac{2N}{N−2},$ $a(x)$ and $b(x)$ are functions that can change sign and satisfy some additional conditions; $u \in H^1_A (\mathbb{R}^N)$ and $A : \mathbb{R}^N → \mathbb{R}^N$ is a magnetic potential. Also using the Nehari method in combination with other complementary arguments, we discuss the existence of infinitely many solutions to the problem in question, varying the assumptions about the weight functions.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2021-0001}, url = {http://global-sci.org/intro/article_detail/ata/20797.html} }
TY - JOUR T1 - Existence and Multiplicity Results for a Class of Nonlinear Schrödinger Equations with Magnetic Potential Involving Sign-Changing Nonlinearity AU - de Paiva , Francisco Odair AU - Souza Lima , Sandra Machado de AU - Miyagaki , Olimpio Hiroshi JO - Analysis in Theory and Applications VL - 2 SP - 148 EP - 177 PY - 2022 DA - 2022/07 SN - 38 DO - http://doi.org/10.4208/ata.OA-2021-0001 UR - https://global-sci.org/intro/article_detail/ata/20797.html KW - Magnetic potential, sign-changing weight functions, Nehari manifold, Fibering map. AB -

In this work we consider the following class of elliptic problems $\begin{cases}  −∆_Au + u = a(x)|u|^{q−2}u + b(x)|u|^{p−2}u & {\rm in} & \mathbb{R}^N, \\u ∈ H^1_A (\mathbb{R}^N),   \tag{P} \end{cases}$ with $2 < q < p < 2^∗ = \frac{2N}{N−2},$ $a(x)$ and $b(x)$ are functions that can change sign and satisfy some additional conditions; $u \in H^1_A (\mathbb{R}^N)$ and $A : \mathbb{R}^N → \mathbb{R}^N$ is a magnetic potential. Also using the Nehari method in combination with other complementary arguments, we discuss the existence of infinitely many solutions to the problem in question, varying the assumptions about the weight functions.

Francisco Odair de Paiva, Sandra Machado de Souza Lima & Olimpio Hiroshi Miyagaki. (2022). Existence and Multiplicity Results for a Class of Nonlinear Schrödinger Equations with Magnetic Potential Involving Sign-Changing Nonlinearity. Analysis in Theory and Applications. 38 (2). 148-177. doi:10.4208/ata.OA-2021-0001
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