Multiplicity and uniqueness of positive solutions for a superlinear-singular $(p,q)$-Laplacian equation on locally finite graphs
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913763630776320 |
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| author | Zhang, Xuechen Zhang, Xingyong |
| author_facet | Zhang, Xuechen Zhang, Xingyong |
| contents | We investigate the multiplicity and uniqueness of positive solutions for the superlinear singular $(p,q)$-Laplacian equation \begin{eqnarray*}
\begin{cases} -Δ_p u-Δ_q u+a(x)u^{p-1}+b(x)u^{q-1}=f(x)u^{-γ}+λg(x)u^α, \;\;\;\;\hfill \mbox{in}\;\; V,\\ u>0,\;\;u\in W_a^{1,p}(V) \cap W_b^{1,q}(V),
\end{cases} \end{eqnarray*} on a weighted locally finite graph $G=(V,E)$, where $0<γ<1<q\leq p<α+1$, $λ$ is a parameter, the potential functions $a(x)$ and $b(x)$ satisfy some suitable conditions, $f>0, g \geq 0$, $f\in L^1(V)\cap L^{\frac{p}{p-1+γ}}(V) \cap L^{\frac{q}{q-1+γ}}(V)$ and $g\in L^1(V)\cap L^\infty(V)$. By making use of the method of Nehari manifold and the Ekeland's variational principle, we prove that there exist two positive solutions for $λ$ belonging to some precise interval. Besides, we also investigate the existence and uniqueness of positive solution for $λ<0$. We overcome some difficulties which are caused by: $(i)$ the singular term; $(ii)$ the definition of gradient $|\nabla u|$ on graph which is different from that on $\mathbb{R}^N$; $(iii)$ the lack of compactness of Sobolev embedding. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22086 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicity and uniqueness of positive solutions for a superlinear-singular $(p,q)$-Laplacian equation on locally finite graphs Zhang, Xuechen Zhang, Xingyong Analysis of PDEs We investigate the multiplicity and uniqueness of positive solutions for the superlinear singular $(p,q)$-Laplacian equation \begin{eqnarray*} \begin{cases} -Δ_p u-Δ_q u+a(x)u^{p-1}+b(x)u^{q-1}=f(x)u^{-γ}+λg(x)u^α, \;\;\;\;\hfill \mbox{in}\;\; V,\\ u>0,\;\;u\in W_a^{1,p}(V) \cap W_b^{1,q}(V), \end{cases} \end{eqnarray*} on a weighted locally finite graph $G=(V,E)$, where $0<γ<1<q\leq p<α+1$, $λ$ is a parameter, the potential functions $a(x)$ and $b(x)$ satisfy some suitable conditions, $f>0, g \geq 0$, $f\in L^1(V)\cap L^{\frac{p}{p-1+γ}}(V) \cap L^{\frac{q}{q-1+γ}}(V)$ and $g\in L^1(V)\cap L^\infty(V)$. By making use of the method of Nehari manifold and the Ekeland's variational principle, we prove that there exist two positive solutions for $λ$ belonging to some precise interval. Besides, we also investigate the existence and uniqueness of positive solution for $λ<0$. We overcome some difficulties which are caused by: $(i)$ the singular term; $(ii)$ the definition of gradient $|\nabla u|$ on graph which is different from that on $\mathbb{R}^N$; $(iii)$ the lack of compactness of Sobolev embedding. |
| title | Multiplicity and uniqueness of positive solutions for a superlinear-singular $(p,q)$-Laplacian equation on locally finite graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.22086 |