Anisotropic obstacle Neumann problems in weighted Sobolev spaces and variable exponent

Proof of Lemma 5.

Firstly, we will show that the operator B n is coercive. For all v , w ∈ W 1 , p → ⁢ ( z ) ⁢ ( Ω , ϱ → ) and by the generalized Hölder-type inequality, we have

which implies that

Let v 0 ∈ K Δ . Due to Hölder’s inequality and (2.5), by the continuous embeddings W 1 , p i ⁢ ( z ) ⁢ ( Ω , ϱ i ) ↪ L p i ⁢ ( z ) ⁢ ( Ω , ϱ i ) , we have

where

Therefore

Then

Using to Jensen’s inequality, we obtain

where

Then

as ∥ v ∥ → + ∞ . From (A.1), we have

Since 〈 Φ n ⁢ v , v 〉 ∥ v ∥ and 〈 Φ n ⁢ v , v 0 〉 ∥ v ∥ are bounded, we get

We conclude that the operator B n = A + Φ n is coercive.

Secondly, it remains to prove that the operator B n is pseudo-monotone. To this end, let ( w k ) k be a sequence in W 1 , p → ⁢ ( z ) ⁢ ( Ω , ϱ → ) such that

We will show that χ = B n ⁢ w and 〈 B n ⁢ w k , w k 〉 → 〈 χ , w 〉 as k → + ∞ . Since W 1 , p → ⁢ ( z ) ( Ω , ϱ → ) ↪ ↪ L p ¯ ( Ω ) , it follows that w k → w strongly in L p ¯ ⁢ ( Ω ) and a.e. in Ω for a subsequence denoted again ( w k ) k . Since ( w k ) k is bounded in W 1 , p → ⁢ ( z ) ⁢ ( Ω , ϱ → ) by (2.5), we have ( Φ i ⁢ ( z , w k , ∇ ⁡ w k ) ) k is bounded in L p i ′ ⁢ ( z ) ⁢ ( Ω , ϱ i * ) . Then there exists a function φ i ∈ L p i ′ ⁢ ( z ) ⁢ ( Ω , ϱ i * ) such that

Moreover, since ( b i n ⁢ ( w k ) ) k is bounded in L p i ′ ⁢ ( z ) ⁢ ( Ω , ϱ i * ) and b i n ⁢ ( w k ) → b i n ⁢ ( w ) a.e. in Ω, we have

For all v ∈ W 1 , p → ⁢ ( z ) ⁢ ( Ω , ϱ → ) using (A.2) and (A.3), we obtain

Hence, we get

as a result

Thanks to (2.6), we have ∑ i = 1 N ∫ Ω ( Φ i ( z , w k , ∇ w k ) - Φ i ( z , w k , ∇ w ) ) ( ∂ i w k - ∂ i w ) d z 〉 ≥ 0 . Then

By (A.2), we obtain

Using (A.4) and (A.5), we have

which implies that

Moreover, since a i ⁢ ( z , w k , ∇ ⁡ w ) converges to Φ i ⁢ ( z , w , ∇ ⁡ w ) strongly in L p i ′ ⁢ ( z ) ⁢ ( Ω , ϱ i ) by (A.6), we get

By Lemma 3, we have w k converges to w strongly in W 1 , p → ⁢ ( z ) ⁢ ( Ω , ϱ → ) and a.e. in Ω, then Φ i ⁢ ( z , w k , ∇ ⁡ w ) converges to Φ i ⁢ ( z , w , ∇ ⁡ w ) weakly in L p i ′ ⁢ ( z ) ⁢ ( Ω , ϱ i ) and b i n ⁢ ( w ) converges to b i n ⁢ ( w ) strongly in L p i ′ ⁢ ( z ) ⁢ ( Ω , ϱ i ) . Then for all v ∈ W 1 , p → ⁢ ( z ) ⁢ ( Ω , ϱ → ) we obtain

which implies that B n ⁢ w = χ . ∎