Lihong Bao, Libo Wang, Min Li
Abstract:
In this article, we consider the multiplicity results of normalized solutions of the
fractional \((p,q)\)-Laplacian equation
$$\displaylines{
(-\Delta)_p^{s} u+(-\Delta)_q^{s} u=\lambda |u|^{p-2}u+h(\varepsilon x)f(u)+g(u)
\quad\text{in }\mathbb{R}^N,\cr
\int_{\mathbb {R}^{N}}|u|^p \mathrm{d}x=a^p,
}$$
where \(\varepsilon, a >0\), \(0< s< 1< p< q\), \(sq< N< \frac{spq(q-1)}{q-p}\) and
\(\lambda \in \mathbb {R}\) is a unknown Lagrange multiplier,
\(h\) is a continuous positive function, \(f\) and \(g\) are also continuous and satisfy
some growth conditions.
When \(\varepsilon\) is small enough, we show that the number of normalized solutions is
at least the number of global maximum points of \(h\) according to the Ekeland's
variational principle and the concentration compactness principle.
Submitted July 26, 2025. Published November 13, 2026.
Math Subject Classifications: 35J20, 35J62, 35J92.
Key Words: Ekeland's variational principle; fractional (p,q)-Laplacian equation;
normalized solutions; concentration compactness,
DOI: 10.58997/ejde.2026.14
Show me the PDF file (446 KB), TEX file for this article.
| Lihong Bao School of Mathematics and statistics Beihua University Jilin, Jilin, 132013, China mail: 15246791290@163.com |
| Libo Wang School of Mathematics and statistics Beihua University Jilin, Jilin, 132013, China mail: wlb_math@163.com |
| Min Li School of Mathematics and statistics Beihua University Jilin, Jilin, 132013, China mail: lm_18179579903@163.com |
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