Mountain Pass Solutions for an entire semipositone problem involving the Grushin Subelliptic Operator

Abstract

For $N\ge 3$ we study the following semipositone problem $$-{\Delta }_{\gamma }u=g(z)f_a(u)in{\mathbb{R}}^N,$$ where ${\Delta }_{\gamma }$ is the Grushin operator $${\Delta }_{\gamma }u(z)={\Delta }_{x}u(z)+|x{|}^{2\gamma }{\Delta }_{y}u(z)(\gamma \ge 0),$$ $g\in L^1({\mathbb{R}}^N)\cap L^{\infty }({\mathbb{R}}^N)$ is a positive function, $a>0$ is a parameter and $f_a$ is a continuous function on $\mathbb{R}$ that coincides with $f(t)-a$ for $t\in {\mathbb{R}}^{+}$, where $f$ is a continuous function with subcritical and Ambrosetti-Rabinowitz type growth and which satisfies $f(0)=0$. Depending on the range of $a$, we obtain the existence of positive mountain pass solutions in $D_{\gamma }({\mathbb{R}}^N)$