We consider the existence of solutions of the system (*) $P(D)u^l = F(x,(∂^α u))$, l = 1,...,k, $x ∈ ℝ^n$ $(u=(u¹,...,u^k))$ in Sobolev spaces, where P is a positive elliptic polynomial and F is nonlinear.
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This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F(x): = F̃x^{(2n+1)}$ where $F̃: X^{2n+1} → Y$ is a continuous (2n+1)-linear operator.
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The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form $-∑_{i,j=1}^{n} D_i(a_{ij}D_ju) = f(u,∫_{Ω}g(u^p))$, in a bounded domain Ω in ℝⁿ with some growth assumptions on the nonlinear terms f and g is proved. The method based on the Krasnosel'skiĭ Fixed Point Theorem enables us to find many solutions as well.