Existence of positive eigenfunctions to an anisotropic elliptic operator via the sub-supersolution method

Abstract

Using the sub-supersolution method, we study the existence of positive solutions for the anisotropic problem 0.1 $$\begin{aligned} -\sum _{i=1}^N\frac{\partial }{\partial x_i}\left( \left| \frac{\partial u}{\partial x_i}\right| ^{p_i-2}\frac{\partial u}{\partial x_i}\right) =\lambda u^{q-1} \end{aligned}$$ where $$\Omega $$ is a bounded and regular domain of $${\mathbb {R}}^N$$ , $$q>1$$ , and $$\lambda >0$$ .