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By continuity on the compact closure, $v$ has a maximum and minimum. If its maximum were greater than one, the boundary hypothesis would force that maximum to occur at an interior point $x_0$. There $\Delta v(x_0)\le 0$, whereasa contradiction. If its minimum were less than minus one, it would again be interior, where $\Delta v\ge 0$ but $v^3-v<0$. ThereforeThis is a direct maximum/minimum argument for a nonlinear elliptic boundary value problem; treating its nonlinearity as a zeroth-order coefficient of fixed nonpositive sign would not be justified.

The Fredholm alternative for an elliptic Dirichlet problem and part (b) give an inverse $S:Y\to X$ of norm at most $K\ge 1$. ChooseOn the closed radius-$\epsilon$ ball $\mathcal{B}\subset X$, set $T(v)=S(\mathcal{Q}(v)+f)$. The stated inequalities giveandThus $T$ is a contraction mapping of the complete metric space $\mathcal{B}$ into itself. The Banach fixed-point theorem gives $u=T(u)$, solving the required nonlinear elliptic boundary value problem. This small-data existence for a nonlinear elliptic Dirichlet problem also gives uniqueness within this small ball; it does not assert global uniqueness.