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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.10742 |
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Table of Contents:
- For $N\ge 3$ we study the following semipositone problem $$ -Δ_γu = g(z) f_a(u) \quad \hbox{in $\mathbb{R}^N$}, $$ where $Δ_γ$ is the Grushin operator $$ Δ_ γu(z) = Δ_x u(z) + \vert x \vert^{2γ} Δ_y u (z) \quad (γ\ge 0), $$ $g\in L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ 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 $D_γ(\mathbb{R}^N)$