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

TL;DR

This paper establishes the existence of mountain pass solutions for a semipositone problem involving the Grushin subelliptic operator, with solutions depending on a parameter and growth conditions on the nonlinear term.

Contribution

It introduces new existence results for positive solutions to a semipositone problem with the Grushin operator, extending variational methods to this subelliptic setting.

Findings

Existence of positive mountain pass solutions for certain parameter ranges.

Solutions depend on the growth conditions of the nonlinear function.

Application of variational methods to subelliptic operators.

Abstract

For $N\ge 3$ we study the following semipositone problem $$ -\Delta_\gamma u = g(z) f_a(u) \quad \hbox{in $\mathbb{R}^N$}, $$ where $\Delta_\gamma$ is the Grushin operator $$ \Delta_ \gamma u(z) = \Delta_x u(z) + \vert x \vert^{2\gamma} \Delta_y u (z) \quad (\gamma\ge 0), $$ $g\in L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ 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 $D_\gamma(\mathbb{R}^N)$