Mountain Pass Solutions for an entire semipositone problem involving the Grushin Subelliptic Operator
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909262811234304 |
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| author | Bisci, Giovanni Molica Malanchini, Paolo Secchi, Simone |
| author_facet | Bisci, Giovanni Molica Malanchini, Paolo Secchi, Simone |
| 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 $D_γ(\mathbb{R}^N)$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10742 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mountain Pass Solutions for an entire semipositone problem involving the Grushin Subelliptic Operator Bisci, Giovanni Molica Malanchini, Paolo Secchi, Simone Analysis of PDEs 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 $D_γ(\mathbb{R}^N)$ |
| title | Mountain Pass Solutions for an entire semipositone problem involving the Grushin Subelliptic Operator |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.10742 |