Mountain pass solutions for an entire semipositone problem involving the Grushin subelliptic operator - Malanchini, Paolo; Bisci, Giovanni Molica; Secchi, Simone (Topological Methods in Nonlinear Analysis, 2026)
For $N\ge 3$ we study the following semipositone problem\begin{equation*} -Δ_γ u = g(z) f_a(u) \quad \hbox{in $\mathbb{R}^N$},\end{equation*}where $Δ_γ$ is the Grushin operator$$ Δ_ γ u(z) = Δ_x u(z) + \abs{x}^{2γ} Δ_y u (z) \quad (γ\ge 0),$$%$z=(x,y)\in \mathbb{R}^N = \mathbb{R}^m× \mathbb{R}^\ell$, $g\in L^1\big(\mathbb{R}^N\big)\cap L^∞\big(\mathbb{R}^N\big)$ 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.