LAUSR

This paper deals with the existence and multiplicity of weak solutions to non-uniformly elliptic problem $$\begin{aligned} -\text {div}(a(x,\nabla u))+V(x)\vert u\vert ^{N-2} u=\lambda \Bigg (\exp \Bigg (\alpha \vert u\vert ^{\frac{N}{N-1}}\Bigg )+f(x,u)\Bigg ), \end{aligned}$$-div(a(x,∇u))+V(x)|u|N-2u=λ(exp(α|u|NN-1)+f(x,u)),in $$\mathbb {R}^N$$RN, where $$N\ge 2$$N≥2, $$\alpha $$α is some positive constant and $$\lambda $$λ is some positive parameter. Our method is mainly based on variational arguments.