Department of Mathematics and Statistics
Central South University
Changsha 410083, China
email: ayee_sha@qq.com
Department of Mathematics and Statistics
Central South University
Changsha 410083, China
email: 1735881486@qq.com
Ayesha Baig, Zhouxin Li
Abstract:
In this article, we study the existence of standing wave solutions
for the quasilinear Schrodinger system
$$\displaylines{
- \varepsilon^2 \Delta u + W(x) u - \kappa \varepsilon^2 \Delta (u^2) u
= Q_u (u,v) \quad \text{in } \mathbb{R}^N, \cr
- \varepsilon^2 \Delta v + V(x) v - \kappa \varepsilon^2 \Delta (v^2) v
= Q_v (u,v) \quad \text{in } R^N, \cr
u, v > 0 \quad \text{in } R^N, \quad u,v \in H^1 (R^N).
}$$
where \(N \geq 3\), \(\kappa > 0\), \(\varepsilon > 0\),
\(W,V:\mathbb{R}^N \to \mathbb{R}\) are continuous functions that fall
into two classes of potentials. To overcome the lack of differentiability,
we use the dual approach developed by Colin–Jeanjean.
The existence of solutions is obtained using Del Pino–Felmer’s penalization
technique with an adaptation of Alves’ arguments [1].
Submitted April 17, 2024. Published March 3, 2025.
Math Subject Classifications: 35Q55, 58E05, 58E30.
Key Words: Quasilinear Schrodinger system; Cerami sequence; dual approach; positive solution.
DOI: 10.58997/ejde.2025.24
An addendum was posted on March 27, 2025. It indicates the overlap with the Ph.D. thesis by Laila Conceição Fontinele published in 2022. See the last page of this article.
Show me the PDF file (422 KB), TEX file for this article.
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Ayesha Baig Department of Mathematics and Statistics Central South University Changsha 410083, China email: ayee_sha@qq.com |
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Zhouxin Li Department of Mathematics and Statistics Central South University Changsha 410083, China email: 1735881486@qq.com |
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