Existence and multiplicity of solutions for anisotropic elliptic equations with variable exponent

Abstract

In this article we study the following nonlinear anisotropic elliptic equations

$$ (P) \ left \ {\ begin {array} {lr}

- \ sum_ {i = 1} ^ {N} \ partial_ {x_ {i}} a_ {i} (x, \ partial_ {x_ {i}} u) + b (x) | u | ^ {P _ {+ } ^ {+} - 2} u = \ lambda (x) f (x, u) + \ mu (x) g (x, u) \ quad dans \ quad \ Omega, \\

u = 0 \ qquad sur \ qquad \ partial \ Omega.

\ end {array} \ right. $$

We  set up that the problem (P)  admits at least two weak solutions under suitable conditions.