Critical $(p,q)$-fractional problems involving a sandwich type nonlinearity

TL;DR

This work studies weak solutions to a nonlocal sandwich-type problem driven by a pair of fractional operators $(−Δ)^{s_1}_{p}$ and $(−Δ)^{s_2}_{q}$ with critical growth, formulated on a general open set $Ω$. Using a variational framework on $E=Z^{s_1,p}(Ω)\cap Z^{s_2,q}(Ω)$ and a careful truncation plus genus theory, the authors establish finite multiplicity results: for each fixed $j$ there exist parameters $λ_*$ and $λ^*$ and a decreasing sequence $\{θ_j\}$ such that for small $θ$ and $λ$ in $(λ_*,λ^*)$ the problem has at least $j$ distinct weak solutions with negative energy. They also prove the existence of a nonnegative weak solution with negative energy when $λ$ is large and $θ$ is small, via a direct minimization and Ekeland principle on a constrained energy, extended to general open domains. The results address compactness issues created by the critical Sobolev term and contribute a framework for sandwich-type nonlinearity in fractional $(p,q)$-Laplacian problems on nonstandard domains. Collectively, the paper advances multiplicity and sign-information results for nonlocal problems with double-growth, providing tools that may inform stability and qualitative behavior in related nonlocal PDEs.

Abstract

In this paper, we deal with the following $(p,q)$-fractional problem $$ (-Δ)^{s_{1}}_{p}u +(-Δ)^{s_{2}}_{q}u=λP(x)|u|^{k-2}u+θ|u|^{p_{s_{1}}^{*}-2}u \, \mbox{ in }\, Ω,\qquad u=0\, \mbox{ in }\, \mathbb{R}^{N} \setminus Ω, $$ where $Ω\subseteq\mathbb{R}^{N}$ is a general open set, $0<s_{2}<s_{1}<1$, $1<q<k<p<N/s_{1}$, parameter $λ,\ θ>0$, $P$ is a nontrivial nonnegative weight, while $p_{s_{1}}^{*}=Np/(N-ps_{1})$ is the critical exponent. We prove that there exists a decreasing sequence $\{θ_j\}_j$ such that for any $j\in\mathbb N$ and with $θ\in(0,θ_j)$, there exist $λ_*$, $λ^*>0$ such that above problem admits at least $j$ distinct weak solutions with negative energy for any $λ\in (λ_*,λ^*)$. On the other hand, we show there exists $\overlineλ>0$ such that for any $λ>\overlineλ$, there exists $θ^*=θ^*(λ)>0$ such that the above problem admits a nonnegative weak solution with negative energy for any $θ\in(0,θ^*)$.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Definition 1
• Proposition 3.2
• Lemma 3.3
• proof