An existence result for nonlinear elliptic equations on \(\mathbb R^d\) without sign condition (Q1413608)

The author proves the existence of a solution of the following nonlinear elliptic equation \[ - {{1}\over{2}} \Delta u(x) + \nabla w(x) \cdot \nabla u(x) + (\lambda + H(x, u(x))) u(x) = f(x),\quad x \in \mathbb{R}^d, \tag{1} \] where \(f, H \) and \(w\) are are given functions and \(\lambda \in \mathbb{R}^+ .\) We point out that we do not need sign condition \[ \lambda + H(x,p) \geq 0 , \quad(x,p) \in \mathbb{R}^d \times \mathbb{R}. \] The proof is organized in two steps. In the first one the author proves a sufficient condition on a Lebesgue-measurable function \(q(x)\) on \(\mathbb{R}^d\) such that the equation \[ - {{1}\over{2}} \Delta u(x) + \nabla w(x) \cdot \nabla u(x) + (\lambda + q(x))) u(x) = f(x) \] have a unique solution for a given function \(f.\) We observe that the function \(q(x)\) have not continuity assumption. In the second step the author constructs a sequence \(\{ u_n\}_n \) of solutions to some linear elliptic equations on \(\mathbb{R}^d\) which converges in a suitable topology to a solution of the above equation \((1).\) In this second step it is useful to obtain an a priori estimate independent on \(n .\) These results are due to the existence of the term \(\nabla w \cdot \nabla u.\)