Communications of the
Korean Mathematical Society CKMS

In this paper we prove the existence of nontrivial weak solutions to the boundary value problem \begin{align*} - G_1 u & =u^3 + f(x,y,u) \quad \text{ in } \Omega ,\\ u &\geq 0 \quad \text{ in } \Omega ,\\ u & =0 \quad \text{ on } \partial\Omega , \end{align*} where $\Omega $ is a bounded domain with smooth boundary in $\mathbb{R}^3$, $G_1 $ is a Grushin type operator, and $f(x,y,u)$ is a lower order perturbation of $u^3$ with $f(x,y,0)=0$. The nonlinearity involved is of critical exponent, which differs from the existing results in \cite{Tri:2018,TriLuyen:2020}.

Keywords: Semilinear degenerate elliptic equations, Grushin operator, critical exponent, critical point

MSC numbers: Primary 35J61, 35J70, 35B33, 35B38