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

TL;DR

This paper investigates the existence and multiplicity of solutions for a fractional $(p,q)$-problem with a nonlinear sandwich-type term, establishing conditions for multiple solutions with negative energy and a nonnegative solution under parameter constraints.

Contribution

It introduces new existence and multiplicity results for a fractional $(p,q)$-problem involving critical and nonlinear terms, with parameter-dependent solutions.

Findings

Existence of multiple solutions with negative energy for small $ heta$ and certain $ ext{lambda}$ ranges.

Existence of a nonnegative solution with negative energy for large $ ext{lambda}$ and small $ heta$.

Identification of parameter thresholds $ heta_j$, $ heta^*$, $ar{ ext{lambda}}$ for solution existence.

Abstract

In this paper, we deal with the following $(p,q)$-fractional problem $$ (-\Delta)^{s_{1}}_{p}u +(-\Delta)^{s_{2}}_{q}u=\lambda P(x)|u|^{k-2}u+\theta|u|^{p_{s_{1}}^{*}-2}u \, \mbox{ in }\, \Omega,\qquad u=0\, \mbox{ in }\, \mathbb{R}^{N} \setminus \Omega, $$ where $\Omega\subseteq\mathbb{R}^{N}$ is a general open set, $0<s_{2}<s_{1}<1$, $1<q<k<p<N/s_{1}$, parameter $\lambda,\ \theta>0$, $P$ 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 $\{\theta_j\}_j$ such that for any $j\in\mathbb N$ and with $\theta\in(0,\theta_j)$, there exist $\lambda_*$, $\lambda^*>0$ such that above problem admits at least $j$ distinct weak solutions with negative energy for any $\lambda\in (\lambda_*,\lambda^*)$. On the other hand, we show there exists $\overline{\lambda}>0$ such that for any $\lambda>\overline{\lambda}$, there exists $\theta^*=\theta^*(\lambda)>0$ such that the above problem admits a nonnegative weak solution with negative energy for any $\theta\in(0,\theta^*)$.