Advanced Search

We consider elliptic nonlinear equations in a separable Hilbert space and their solutions in spaces of Sobolev type.

We consider the existence of solutions of the system (*) $P\left(D\right)u^l=F(x,\left({\partial }^{\alpha }u\right))$, l = 1,...,k, $x\in {ℝ}^n$ $(u=(u¹,...,u^k))$ in Sobolev spaces, where P is a positive elliptic polynomial and F is nonlinear.

This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F\left(x\right):=F̃x^{(2n+1)}$ where $F̃:X^{2n+1}\to Y$ is a continuous (2n+1)-linear operator.

The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form $-{\sum }_{i,j=1}^n{D}_i\left(a_{ij}{D}_{j}u\right)=f(u,{\int }_{\Omega }g\left(u^p\right))$, 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.