A weak solution for a ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian elliptic problem with a singular term

Abstract

Abstract Here, we consider the following elliptic problem with variable components: − a ( x ) Δ p ( x ) u − b ( x ) Δ q ( x ) u + u | u | s − 2 | x | s = λ f ( x , u ) , $$ -a(x)\Delta _{p(x)}u - b(x) \Delta _{q(x)}u+ \frac{u \vert u \vert ^{s-2}}{|x|^{s}}= \lambda f(x,u), $$ with Dirichlet boundary condition in a bounded domain in R N $\mathbb{R}^{N}$ with a smooth boundary. By applying the variational method, we prove the existence of at least one nontrivial weak solution to the problem.

Keywords

• ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian problem
• Singular term
• Variational method