Fractional double phase Robin problem involving variable‐order exponents and logarithm‐type nonlinearity

摘要

The paper deals with the logarithmic fractional equations with variable exponents (−Δ)p1(·)s1(·)(u)+(−Δ)p2(·)s2(·)(u)+|u|p‾1(x)−2u+|u|p‾2(x)−2u=λb(x)|u|α(x)−2u+μa(x)|u|r(x)−2ulog|u|+μc(x)|u|η(x)−2u,x∈Ω,Np1(·)s1(·)(u)+Np2(·)s2(·)(u)+β(x)(|u|p‾1(x)−2u+|u|p‾2(x)−2u)=0,x∈ℝNΩ‾,$$ left{begin{array}{ll}& kern-5pt {left(-Delta right)}_{p_1left(cdotp right)}^{s_1left(cdotp right)}(u)+{left(-Delta right)}_{p_2left(cdotp right)}^{s_2left(cdotp right)}(u)+{left|uright|}^{{overline{p}}_1(x)-2}u+{left|uright|}^{{overline{p}}_2(x)-2}u=lambda b(x){left|uright|}^{alpha (x)-2}u {}& +mu a(x){left|uright|}^{r(x)-2}ulog mid umid +mu c(x){left|uright|}^{eta (x)-2}u, {}& xin Omega, {}& kern-5pt {mathcal{N}}_{p_1left(cdotp right)}^{s_1left(cdotp right)}(u)+{mathcal{N}}_{p_2left(cdotp right)}^{s_2left(cdotp right)}(u)+beta (x)left({left|uright|}^{{overline{p}}_1(x)-2}u+{left|uright|}^{{overline{p}}_2(x)-2}uright)=0, {}& xin {mathbb{R}}^Nbackslash overline{Omega},end{array}right. $$ where (−Δ)pi(·)si(·)$$ {left(-Delta right)}_{p_ileft(cdotp right)}^{s_ileft(cdotp right)} $$ and Npi(·)si(·)$$ {mathcal{N}}_{p_ileft(cdotp right)}^{s_ileft(cdotp right)} $$ denote the variable si(·)$$ {s}_ileft(cdotp right) $$‐order pi(·)$$ {p}_ileft(cdotp right) $$‐fractional Laplace operator and the nonlocal normal pi(·)$$ {p}_ileft(cdotp right) $$‐derivative of si(·)$$ {s}_ileft(cdotp right) $$‐order, respectively, with si(·):ℝ2N→(0,1)$$ {s}_ileft(cdotp right):{mathbb{R}}^{2N}to left(0,1right) $$ and pi(·):ℝ2N→(1,∞)$$ {p}_ileft(cdotp right):{mathbb{R}}^{2N}to left(1,infty right) $$ ( i∈{1,2}$$ iin left{1,2right} $$) being continuous. Here, Ω⊂ℝN$$ Omega subset {mathbb{R}}^N $$ is a bounded smooth domain with N>pi(x,y)si(x,y)$$ N&{p}_ileft(x,yright){s}_ileft(x,yright) $$ ( i∈{1,2}$$ iin left{1,2right} $$) for any (x,y)∈Ω‾×Ω‾,λ$$ left(x,yright)in overline{Omega}times overline{Omega},lambda $$ and μ$$ mu $$ are a positive parameters, r(·)$$ rleft(cdotp right) $$ and η(·)$$ eta left(cdotp right) $$ are two continuous functions, while variable exponent α(x)$$ alpha (x) $$ can be close to the critical exponent p2s2∗(x)=Np‾2(x)/(N−s‾2(x)p‾2(x))$$ {p}_{2{s}_2}^{ast }(x)=N{overline{p}}_2(x)/left(N-{overline{s}}_2(x){overline{p}}_2(x)right) $$, given with p‾2(x)=p2(x,x)$$ {overline{p}}_2(x)={p}_2left(x,xright) $$ and s‾2(x)=s2(x,x)$$ {overline{s}}_2(x)={s}_2left(x,xright) $$ for x∈Ω‾$$ xin overline{Omega} $$. Precisely, we consider two cases. In the first case, we deal with subcritical nonlinearity, that is, α(x)