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

• Published in: Mathematical methods in the applied sciences 2022-11, Vol.45 (17), p.11272-11296
• Main Authors: Biswas, Reshmi, Bahrouni, Anouar, Fiscella, Alessio
• Format: Article
• Language: English
• Subjects: Arrays; Continuity (mathematics); critical nonlinearity; double phase problem; Exponents; logarithmic nonlinearity; Logarithms; Nonlinearity; Operators (mathematics); Robin boundary condition; variable‐order fractional p(·)$$ p\left(\cdotp \right) $$‐Laplacian; Variational methods
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.8449

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∈ℝN\Ω‾,$$ \left\{\begin{array}{ll}& \kern-5pt {\left(-\Delta \right)}_{p_1\left(\cdotp \right)}^{s_1\left(\cdotp \right)}(u)+{\left(-\Delta \right)}_{p_2\left(\cdotp \right)}^{s_2\left(\cdotp \right)}(u)+{\left|u\right|}^{{\overline{p}}_1(x)-2}u+{\left|u\right|}^{{\overline{p}}_2(x)-2}u=\lambda b(x){\left|u\right|}^{\alpha (x)-2}u\\ {}& +\mu a(x){\left|u\right|}^{r(x)-2}u\log \mid u\mid +\mu c(x){\left|u\right|}^{\eta (x)-2}u,\\ {}& x\in \Omega, \\ {}& \kern-5pt {\mathcal{N}}_{p_1\left(\cdotp \right)}^{s_1\left(\cdotp \right)}(u)+{\mathcal{N}}_{p_2\left(\cdotp \right)}^{s_2\left(\cdotp \right)}(u)+\beta (x)\left({\left|u\right|}^{{\overline{p}}_1(x)-2}u+{\left|u\right|}^{{\overline{p}}_2(x)-2}u\right)=0,\\ {}& x\in {\mathbb{R}}^N\backslash \overline{\Omega},\end{array}\right. $$ where (−Δ)pi(·)si(·)$$ {\left(-\Delta \right)}_{p_i\left(\cdotp \right)}^{s_i\left(\cdotp \right)} $$ and Npi(·)si(·)$$ {\mathcal{N}}_{p_i\left(\cdotp \right)}^{s_i\left(\cdotp \right)} $$ denote the variable si(·)$$ {s}_i\left(\cdotp \right) $$‐order pi(·)$$ {p}_i\left(\cdotp \right) $$‐fractional Laplace operator and the nonlocal normal pi(·)$$ {p}_i\left(\cdotp \right) $$‐derivative of si(·)$$ {s}_i\left(\cdotp \right) $$‐order, respectively, with si(·):ℝ2N→(0,1)$$ {s}_i\left(\cdotp \right):{\mathbb{R}}^{2N}\to \left(0,1\right) $$ and pi(·):ℝ2N→(1,∞)$$ {p}_i\left(\cdotp \right):{\mathbb{R}}^{2N}\to \left(1,\infty \right) $$ ( i∈{1,2}$$ i\in \left\{1,2\right\} $$) being continuous. Here, Ω⊂ℝN$$ \Omega \subset {\mathbb{R}}^N $$ is a bounded smooth domain with N>pi(x,y)si(x,y)$$ N>{p}_i\left(x,y\right){s}_i\left(x,y\right) $$ ( i∈{1,2}$$ i\in \left\{1,2\right\} $$) for any (x,y)∈Ω‾×Ω‾,λ$$ \left(x,y\right)\in \overline{\Omega}\times \overline{\Omega},\lambda $$ and μ$$ \mu $$ are a positive parameters, r(·)$$ r\left(\cdotp \right) $$ and η(·)$$ \eta \left(\cdotp \right) $$ are