... ∗ ,p x ,q x Thus, J u, υ where W −1 ,p 1 ,q x W0 x D1 J u, υ ∗ ,p x ,q x resp ., W −1 ,q RN x ∗ ,p x D2 J u, υ ∗ ,q x , 2.15 1 ,p x RN is the dual space of W0 RN resp ., R , and · ∗ ,p x resp ., · ∗ ,q x ... ≤ , for a. e x ∈ RN Let u ∈ Lq x RN , u / Then, p p |u |p x q x ≤ ⇒ |u |p x q x ≤ |u |p x |u |p x q x ≥ ⇒ In particular, if p x p |u |p x q x ≤ |u| ≤ |u |p x q x , q x p |u |p x q x p x ≤ 2.10 p ... b1 x | u, υ | b2 x | u, υ | , where < p , ∈ Lδ δ x q x x q < p − , q − , q ≤ p , RN ∩ Lβ x RN , p x , p x −1 bi ∈ Lγ q x , q x −1 γ x q x q x , x q x β x q RN ∩ Lβ x RN , x p x p x q ...