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Let $Ω$ be an open bounded on class $C^1$. We consider, on $Ω$, the problem with the following limits:

$(P)quad left{ begin{array}{lcc}
-Delta u_1 +alpha u_2=f_1 & text{on } Omega, \
\ -Delta u_2 +beta u_1=f_2 & text{on } Omega, \
\ u_1 =u_2=0 & text{on } partial Omega,
end{array}
right.$

where $alpha,betain L^{infty}(Omega)$ and $f_1,f_2in L^{2}(Omega)$.

We note $H=H_0^1(Omega)times H_0^1(Omega)$. If $uin H$, then $u=(u_1,u_2)$, with $u_1,u_2in H_0^1(Omega)$, and

$||u||^2_{H}=||u_1||^2_{H^1_0(Omega)}+||u_2||^2_{H^1_0(Omega)}$.

1. 
   

   Show, for an element $u=(u_1,u_2)∈ H^2(Ω)×H^2(Ω)$, the equivalence between the boundary problem $(P)$ and the variational problem $(P_λ)$, with $λ> 0$:

   
   
   

   $(P_λ)quad left{ begin{array}{lcc}
   uin H \
   \ forall vin H, quad a_{λ}(u,v)=L_{λ}(v),
   end{array} right.$,

   
   

where

$a_{λ}(u,v)=int_{Omega}(∇u_1 · ∇v_1 + λ∇u_2 · ∇v_2) dx +int_{Omega}(αu_2v_1 + λβu_1v_2) dx$,

$L_{λ}(v)=int_{Omega}(f_1v_1 +lambda f_2v_2)dx$,

$u=(u_1,u_2), v=(v_1,v_2)in H$

2. Suppose $λ = 1$. Show that there exists $C> 0$, depending only on
   $Ω$, such that if $|| α + β ||_{infty} ≤ C$, then there is one and only one
   solution $uin H$ of $(P_{lambda})$
   

I did the variational formulation, but I do not know how I should make the $lambda$ of the $(P)$ problem appear.

And for the part 2. I do not know how to relate the constant $C$ and the $|| α + β ||_{infty} ≤ C$

Multiply the first equation $-Delta u_1 +alpha u_2=f_1$ by $v_1$ and the second equation $-Delta u_2 +beta u_1=f_2$ by $lambda v_2$. Take the sum, integrate over $Omega$ and then integrate by parts.