Multiplicity and uniqueness of positive solutions for a superlinear-singular $(p,q)$-Laplacian equation on locally finite graphs

TL;DR

This work studies the existence, multiplicity, and uniqueness of positive solutions for a singular $(p,q)$-Laplacian equation on a weighted locally finite graph, with $0<\gamma<1<q\le p<\alpha+1$. The authors develop a constrained variational framework on the cone $W_+$, employing the Nehari manifold and Ekeland's variational principle to obtain two distinct positive solutions for $0<\lambda<\Lambda_{*}$, and they prove existence and uniqueness of a positive solution for $\lambda<0$. A key technical challenge is the singular term $f(x)u^{-\\gamma}$, the graph-specific gradient, and the lack of compact Sobolev embedding; these are addressed via a graph-adapted Brezis-Lieb lemma and careful decomposition of the Nehari manifold into $\mathcal{D}^+_\lambda$, $\mathcal{D}^-_\lambda$, and $\mathcal{D}^0_\lambda$. The paper provides explicit parameter thresholds $X(\lambda)$, $S(\lambda)$ and their critical values at $\Lambda_{*}$, along with strong convergence results that extend known results from Euclidean domains to discrete graph settings, enriching the theory of singular nonlinear PDEs on graphs.

Abstract

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.

Figures (3)

• Figure 1: the behaviour of $\varphi_u(t)$.
• Figure 2: the first kind of behaviour of $J_\lambda(tu)$.
• Figure 3: the second kind of behaviour of $J_\lambda(tu)$.