Existence of the solution for a class of the semilinear degenerate elliptic equation involving the Grushin operator in R2$\mathbb {R}^2$: The interaction between Grushin's critical exponent and exponential growth

In this work, we study the existence of the solution for a class of semilinear degenerate elliptic equations in the whole space R2$\mathbb {R}^{2}$ involving the Grushin operator and a nonlinearity that involves the Grushin's critical growth and the exponential growth. The existence of at least one nontrivial weak solution is done by combining the mountain‐pass theorem, Trudinger–Moser inequality, and a version of a result due to Lions for the exponential growth in R2$\mathbb {R}^{2}$ in the corresponding weighted Sobolev space.