On semilinear Grushin--Schrödinger equation in $\mathbb{R}^N$
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908870368034816 |
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| author | Carvalho, Jônison Viana, Arlúcio |
| author_facet | Carvalho, Jônison Viana, Arlúcio |
| contents | We establish the existence of nontrivial nonnegative weak solutions to the following equation
\begin{equation*}
-Δ_γu + V(z)u = Q(z)f(u), \quad z\in \mathbb{R}^N,
\end{equation*}
where $Δ_γ$ denotes the so-called Grushin-type operator in $\mathbb{R}^N$. The potentials $V$ and $Q$ are assumed to be controlled below and above, respectively, by functions of type $(1+|z|)^a$, $a\in\mathbb{R}$. The main result is the embedded of the space $E_V^γ$ into the weighted Lebesgue space $L_Q^p(\mathbb{R}^N)$, under suitable conditions. Finally, we derive regularity results for the obtained weak solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06417 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On semilinear Grushin--Schrödinger equation in $\mathbb{R}^N$ Carvalho, Jônison Viana, Arlúcio Analysis of PDEs Functional Analysis 35J70, 35A15, 35J61, 35H20 We establish the existence of nontrivial nonnegative weak solutions to the following equation \begin{equation*} -Δ_γu + V(z)u = Q(z)f(u), \quad z\in \mathbb{R}^N, \end{equation*} where $Δ_γ$ denotes the so-called Grushin-type operator in $\mathbb{R}^N$. The potentials $V$ and $Q$ are assumed to be controlled below and above, respectively, by functions of type $(1+|z|)^a$, $a\in\mathbb{R}$. The main result is the embedded of the space $E_V^γ$ into the weighted Lebesgue space $L_Q^p(\mathbb{R}^N)$, under suitable conditions. Finally, we derive regularity results for the obtained weak solutions. |
| title | On semilinear Grushin--Schrödinger equation in $\mathbb{R}^N$ |
| topic | Analysis of PDEs Functional Analysis 35J70, 35A15, 35J61, 35H20 |
| url | https://arxiv.org/abs/2603.06417 |