In this paper, we are concerned in establishing properties about the function \(\vartheta \) and versions of the classical Keller–Osserman condition to prove existence of solutions to the Schrödinger quasilinear elliptic problem

where \(\Omega \subset {\mathbb {R}}^N\), with \(N\ge 3\), is a bounded domain, \(a:{\bar{\Omega }} \rightarrow [0,\infty )\) and \(g:[0,\infty ) \rightarrow [0,\infty )\) are suitable nonnegative continuous functions, \(\vartheta :{\mathbb {R}}\rightarrow (0,\infty )\) is a \(C^1\)-function satisfying appropriated hypotheses, and \(d(x)=\mathrm{dist}(x,\partial \Omega )\) stands for the distance function to the boundary of \(\Omega \). By exploring a dual approach and the relationship among the properties of \(\vartheta \) with its corresponding Keller–Osserman condition, we were able to show existence of solutions for this problem.

在本文中，我们关注建立函数\（\ vartheta \）和经典Keller-Osserman条件的版本的性质，以证明Schrödinger拟线性椭圆问题的解的存在

其中\（\ Omega \ subset {\ mathbb {R}} ^ N \）与\（N \ ge 3 \）是一个有界域，\（a：{\ bar {\ Omega}} \ rightarrow [0 ，\ infty）\）和\（g：[0，\ infty）\ rightarrow [0，\ infty} \）是合适的非负连续函数\（\ vartheta：{\ mathbb {R}} \ rightarrow（0， \ infty）\）是满足适当假设的\（C ^ 1 \） -函数，并且\（d（x）= \ mathrm {dist}（x，\ partial \ Omega）\）代表到\（\ Omega \）的边界。通过探索对偶方法以及\（\ vartheta \）属性之间的关系 借助其相应的Keller-Osserman条件，我们能够证明存在该问题的解决方案。