On the existence of solutions to a (p,q)‐Laplacian system on the Sierpiński gasket on ℝN−1

摘要

We study the existence of solutions for the following boundary value problem involving the (p,q)$$ left(p,qright) $$‐Laplacian −Δpu=λk(x)|u|p−2u+αα+βa(x)|u|α−2u|v|βinS∖S0,−Δqv=νl(x)|v|q−2v+βα+βa(x)|u|α|v|β−2vinS∖S0,v=u=0inS0.$$ left{begin{array}{llll}-{Delta}_pu& =lambda k(x){left|uright|}^{p-2}u+frac{alpha }{alpha +beta }a(x){left|uright|}^{alpha -2}u{left|vright|}^{beta }& in& mathcal{S}setminus {mathcal{S}}_0, {}-{Delta}_qv& =nu l(x){left|vright|}^{q-2}v+frac{beta }{alpha +beta }a(x){left|uright|}^{alpha }{left|vright|}^{beta -2}v& in& mathcal{S}setminus {mathcal{S}}_0, {}v=u& =0& in& {mathcal{S}}_0.end{array}right. $$ where S$$ mathcal{S} $$ is the Sierpiński g on ℝN−1$$ {mathbb{R}}^{N-1} $$ for N≥3,S0$$ Nge 3,{mathcal{S}}_0 $$ is its boundary, a,k,l:S→ℝ$$ a,k,l:mathcal{S}to mathbb{R} $$ are appropriate functions and α,β,p$$ alpha, beta, p $$ and q$$ q $$ are reals satisfying an adequate hypothesis.