Existence, nonexistence, and multiplicity of solutions for the fractional p & q $p&q$ -Laplacian equation in R N $mathbb{R}^{N}$

摘要

In this paper, we study the existence, nonexistence, and multiplicity of solutions to the following fractionalp&q$p&q$-Laplacian equation:0.1(−Δ)psu+a(x)|u|p−2u+(−Δ)qsu+b(x)|u|q−2u+μ(x)|u|r−2u=λh(x)|u|m−2u,x∈RN,$$egin{aligned} &(-Delta)^{s}_{p}u + a(x)ert uert ^{p-2}u +(-Delta)^{s}_{q}u + b(x)ert uert ^{q-2}u +mu(x)ert uert ^{r-2}u &quad= lambda h(x)ert u ert ^{m-2}u,quad xin {mathbb {R}}^{N}, end{aligned}$$where λ is a real parameter,(−Δ)ps$(-Delta)_{p}^{s} $and(−Δ)qs$(-Delta )_{q}^{s} $are the fractionalp&q$p&q$-Laplacian operators with0s1qp,r1$0 s1qp, r1$andspN$sp N$, and the functionsa(x),b(x),μ(x)$a(x), b(x),mu(x)$, andh(x)$h(x)$are nonnegative inRN${mathbb {R}}^{N}$. Three cases onp,q,r,m$p,q,r,m$are considered:pmrps∗$p m r p_{s}^{*}$,max{p,r}mps∗$max{p,r} m p_{s}^{*}$, and1mqrps∗$1 m q r p_{s}^{*}$. Using variational methods, we prove the existence, nonexistence, and multiplicity of solutions to Eq. (0.1) depending onλ,p,q,r,m$lambda, p,q,r,m$and the integrability properties of the ratiohr−p/μm−p$h^{r-p}/mu^{m-p}$. Our results extend the previous work in Bartolo et al. (J. Math. Anal. Appl. 438:29-41, 2016) and Chaves et al. (Nonlinear Anal. 114:133-141, 2015) to the fractionalp&q$p&q$-Laplacian equation (0.1).