Mountain pass solutions for an entire semipositone problem involving the Grushin subelliptic operator

Abstract

For $N\ge 3$ we study the following semipositone problem \begin{equation*} -\Delta_\gamma u = g(z) f_a(u) \quad \hbox{in $\mathbb{R}^N$}, \end{equation*} where $\Delta_\gamma$ is the Grushin operator $$ \Delta_ \gamma u(z) = \Delta_x u(z) + \abs{x}^{2\gamma} \Delta_y u (z) \quad (\gamma\ge 0), $$% $z=(x,y)\in \mathbb{R}^N = \mathbb{R}^m\times \mathbb{R}^\ell$, $g\in L^1\big(\mathbb{R}^N\big)\cap L^\infty\big(\mathbb{R}^N\big)$ 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 mountain pass solutions in a suitable Sobolev space associated to the Grushin operator, extending the results found in \cite{alves} to the Grushin operator.