Blow-Up Solutions for a Class of Schrödinger Quasilinear Operators with a Local Sublinear Term

Abstract

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 Schrodinger quasilinear elliptic problem $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \mathrm{div}\left( \vartheta (u)\nabla u\right) -\frac{1}{2}\vartheta '(u)|\nabla u|^2=a(x)g(u)~ \text{ in }~ \Omega ,\\ u\ge 0\ \text{ in }~\Omega ,\ u(x){\mathop {\longrightarrow }\limits ^{d(x)\rightarrow 0}} \infty , \end{array} \right. \end{aligned}$$ 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.