Le Thi Hong Hanh, Duong Trong Luyen

Existence of Positive Solutions to Perturbed Semilinear Strongly Degenerate Elliptic Problems Involving Critical Growth

abstract:
In this article, we consider the following perturbed semilinear equations involving strongly degenerate elliptic problem with critical growth:
\begin{gather*}
-\varepsilon^2\Delta^{\alpha,\beta}_{\alpha_1,\beta_1} u+V(X)u  =f(X)|u|^{p-2}u+\frac{a}{a+b}\,K(X)|u|^{a-2}u|v|^b, \;\; X\in \mathbb{R}^N,\\
-\varepsilon^2\Delta^{\alpha,\beta}_{\alpha_1,\beta_1} v+V(X)v  =g(X)|v|^{p-2}v+\frac{b}{a+b}\,K(X)|u|^a|v|^{b-2}v, \;\; X\in \mathbb{R}^N,\\
u(X),\;v(X)\to 0 \;\;\text{as}\;\; |X|\to\infty,
\end{gather*}
where $\Delta^{\alpha,\beta}_{\alpha_1,\beta_1}$ is the subelliptic operator of the type
\begin{gather*}
\Delta^{\alpha,\beta}_{\alpha_1,\beta_1}:=\Delta_x+\Delta_y+|x|^{2\alpha}|y|^{2\beta}\big(|x|^{\alpha_1}+|y|^{\beta_1}\big)^2\Delta_z, \;\; x\in\mathbb R^{N_1}, \;\; y\in \mathbb R^{N_2}, \;\; z\in \mathbb R^{N_3}, \\
N=N_1+N_2+N_3, \;\; \alpha, \beta, \alpha_1, \beta_1>0, \;\; X=(x,y,z).
\end{gather*}
Using variational methods, we prove the existence of positive solutions.

Mathematics Subject Classification: 35B33, 35J60, 35J65

Key words and phrases: Semilinear strongly degenerate elliptic equations; critical growth, Palais-Smale condition, variational methods