School of Mathematics and Statistics
Southwest University
Chongqing 400715, China
email: shbdeng@swu.edu.cn
School of Mathematics and Statistics
Southwest University
Chongqing 400715, China
email: yichen_ling01@163.com
Shengbing Deng, Yichen Ling
Abstract:
Let \(m,n\geq1\), \(\gamma>0\), and \(N_\gamma=m+(1+\gamma)n >2\).
Let \(\Omega\subset\mathbb R^m\times\mathbb R^n\) be a bounded smooth domain
intersecting \(\{0\}\times\mathbb R^n\). We study the boundary-value problem
$$\displaylines{
-\Delta_\gamma u=\lambda g(z,u)+\Big(\int_\Omega \dfrac{|u(w)|^p}{d(z-w)^\mu}\,dw\Big)
|u|^{p-2}u \quad \text{in }\Omega,\cr
u=0 \quad \text{on }\partial\Omega,
}$$
where \(z=(x,y)\), \(\Delta_\gamma=\Delta_x+(1+\gamma)^2|x|^{2\gamma}\Delta_y\),
\(d(z)=(|x|^{2(1+\gamma)}+|y|^2)^{1/[2(1+\gamma)]}\),
\(0 < \mu< N_\gamma\), and \(p=(2N_\gamma-\mu)/(N_\gamma-2)\). We use a nonlocal Brezis-Lieb splitting and
variational methods to prove compactness below the critical energy.
For \(1< q< 2\) and small \(\lambda >0\), the problem with \(g(z,u)=|u|^{q-2}u\) has a nontrivial
nonnegative local minimizer of negative energy. A second nontrivial nonnegative solution
is obtained under a quantitative concentration condition.
We verify this condition in an explicit range of \(q\) under a whole-space extremal and
decay assumption. For \(2< q< 2^*_\gamma=2N_\gamma/(N_\gamma-2)\) and large \(\lambda\),
there is a nontrivial nonnegative mountain-pass solution whose energy and norm tend to
zero as \(\lambda\to\infty\). The large-parameter existence result extends to general
subcritical Caratheodory nonlinearities satisfying the stated growth and
superlinearity assumptions.
Submitted September 21, 2026. Published October 23, 2026.
Math Subject Classifications: 35J70, 35H20, 35A15, 35B33, 35B38.
Key Words: Grushin operator; Choquard equation; concave perturbation;
Hardy-Littlewood-Sobolev critical exponent; superlinear perturbation.
DOI: 10.58997/ejde.2026.84
Show me the PDF file (428 KB), TEX file for this article.
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Shengbing Deng School of Mathematics and Statistics Southwest University Chongqing 400715, China email: shbdeng@swu.edu.cn |
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Yichen Ling School of Mathematics and Statistics Southwest University Chongqing 400715, China email: yichen_ling01@163.com |
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