Existence and regularity for a p-Laplacian problem in ℝ^N with singular, convective, and critical reaction

Lemma 2.1

Let 1 < p < N and ( u n ) ⊆ D 0 1 , p ( R N ) , u ∈ D 0 1 , p ( R N ) be such that u n ⇀ u in D 0 1 , p ( R N ) . Then, ( u n ) + ⇀ u + in D 0 1 , p ( R N ) .

Proof

Fix any R > 0 . Since D 0 1 , p ( R N ) ↪ c L p ( B R ) , we have u n → u in L p ( B R ) . Then, up to sub-sequences, u n → u in B R , and there exists U ∈ L p ( B R ) such that ∣ u n ∣ ≤ U in B R for all n ∈ N (see [9, Theorem 4.9]). Thus, Lebesgue’s dominated convergence theorem and the continuity of the real function t ↦ t + imply ( u n ) + → u + in L p ( B R ) .

We observe that ( u n ) is bounded in D 0 1 , p ( R N ) . Stampacchia’s lemma (cf., e.g., [17, Theorem 4.4]) guarantees that ∇ ( v + ) = χ { v > 0 } ∇ v for all v ∈ D 0 1 , p ( R N ) , and this implies that ( ( u n ) + ) is bounded in D 0 1 , p ( R N ) . By reflexivity, there exists v ∈ D 0 1 , p ( R N ) such that ( u n ) + ⇀ v in D 0 1 , p ( R N ) . Reasoning as above, we infer ( u n ) + → v in L p ( B R ) , forcing v = u + . Arbitrariness of R gives ( u n ) + ⇀ u + in D 0 1 , p ( R N ) .□

Lemma 2.2

(Lions, [40, Lemma I.1]) Let 1 ≤ p < N . Suppose ( u n ) ⊆ D 0 1 , p ( R N ) to be such that u n ⇀ u in D 0 1 , p ( R N ) , and both ∣ ∇ u n ∣ p ⇀ μ , ∣ u n ∣ p * ⇀ * ν in the sense of measures, for some u ∈ D 0 1 , p ( R N ) and μ , ν bounded non-negative measures on R N . Then, there exist some at most countable set A , a family of distinct points ( x j ) j ∈ A ⊆ R N , and two families of numbers ( ν j ) j ∈ A , ( μ j ) j ∈ A ⊆ ( 0 , + ∞ ) fulfilling

Lemma 2.3

(Ben-Naoum et al. [7, Lemma 3.3]) Let 1 ≤ p < N . Suppose that ( u n ) ⊆ D 0 1 , p ( R N ) is bounded and define

Then, it holds S ν ∞ p ⁄ p * ≤ μ ∞ and

where ν and μ are as in Lemma 2.2.

Remark 2.4

Although not explicitly stated in [40, Lemma I.1], for all j ∈ A , one has ν j = ν ( { x j } ) and one can assume μ j = μ ( { x j } ) in Lemma 2.2 (see, e.g., [43, Theorem 2.5]). Indeed, for all ε > 0 , (2.2) implies

so letting ε → 0 produces ν ( { x j } ) = ν j for all j ∈ A . On the other hand, it is readily seen that μ ≥ μ ( { x j } ) δ x j for all j ∈ A . Since { ∣ ∇ u ∣ p } ∪ { δ x j : j ∈ A } is a set consisting of pairwise mutually singular measures, as well as (2.2) ensures μ ≥ ∣ ∇ u ∣ p , one has

Using (2.2) again, one has μ ( { x j } ) ≥ μ j ≥ S ν j p ⁄ p * . Thus, replacing μ j with μ ( { x j } ) for all j ∈ A leaves (2.2) unaltered.

Theorem 2.5

(Ambrosetti, Rabinowitz; cf. [44, Theorem 5.40]) Let ( X , ‖ ⋅ ‖ X ) be a Banach space and J ∈ C 1 ( X ) . Let u 0 , u 1 ∈ X , ρ > 0 such that the “mountain pass geometry” is fulfilled, i.e.,

Set

If J satisfies (PS) c M , then c M ≥ b and there exists u ∈ X such that both J ( u ) = c M and J ′ ( u ) = 0 . Moreover, if c M = b , then u can be taken on ∂ B ρ ( u 0 ) .

Theorem 2.6

(Schauder; cf. [30, Theorem 6.3.2 p. 119]) Let K be a non-empty bounded convex subset of a normed linear space E, and let T : K → K be a compact map. Then, T has a fixed point.

Lemma 2.7

Let ( b k ) k = 0 ∞ ⊆ [ 0 , + ∞ ) satisfy, for some c > 0 and K , α > 1 , the recursion

If

then ( b k ) is bounded.

Proof

By iteration, we obtain

The conclusion then follows by (2.3) observing that, for all k ∈ N ,

since ∑ j = 0 ∞ q α j is convergent for all q ∈ ( 0 , 1 ) .□

Lemma 2.8

Let u , v ∈ D 0 1 , p ( R N ) satisfy

for all φ ∈ D 0 1 , p ( R N ) such that φ ≥ 0 in R N and φ ≡ 0 on { u ≥ v } . Then, u ≤ v in R N .

Proof

Testing (2.4) with ( u − v ) + ∈ D 0 1 , p ( R N ) , besides recalling Stampacchia’s lemma (cf., e.g., [17, Theorem 4.4]), yields

Hence, [14, Lemma 2.1] entails ∇ ( u − v ) + = 0 in R N , which implies ( u − v ) + = 0 in R N , since ( u − v ) + ∈ D 0 1 , p ( R N ) .□

Lemma 2.9

Let w satisfy ( H w ) and let γ ∈ ( 0 , 1 ) . Then, there exists a unique u ∈ C loc 1 , α ( R N ) solution to

Moreover, w u − γ ∈ L 1 ( R N ) ∩ L ∞ ( R N ) .

Proof

For all n ∈ N , consider the regularized problems

Fix any n ∈ N . Direct methods of calculus of variations (see [52, Theorem I.1.2]) ensure that there exists u n ∈ D 0 1 , p ( R N ) solution to ( P ̲ n ). Note that

so u n ∈ C loc 1 , α ( R N ) according to nonlinear regularity theory [15, Corollary p. 830]. Moreover, by the strong maximum principle (cfr. [49, Theorem 1.1.1]), u n > 0 in R N .

Testing ( P ̲ n ) with u n , besides using Hölder’s and Sobolev’s inequalities, yields

being ζ > 1 such that 1 ζ + 1 − γ p * = 1 . We deduce ‖ ∇ u n ‖ p ≤ ( S − 1 − γ p ‖ w ‖ ζ ) 1 p − 1 + γ , so that ( u n ) is bounded in D 0 1 , p ( R N ) . By reflexivity, u n ⇀ u in D 0 1 , p ( R N ) for some u ∈ D 0 1 , p ( R N ) , up to sub-sequences.

Observing that u n + 1 n > u n + 1 + 1 n + 1 on { u n > u n + 1 } , we obtain the weak inequality

According to Lemma 2.8, it turns out that u n ≤ u n + 1 in R N . Thus, we can define a measurable function u ˜ such that u n ↗ u ˜ in R N .

We show that u = u ˜ in R N . For all k ∈ N , D 0 1 , p ( R N ) ↪ c L p ( B k ) , so u n → u in L p ( B k ) and u n → u in B k , up to sub-sequences. A diagonal argument ensures that u n → u in R N , whence u = u ˜ in R N (see, e.g., [21, p. 3044] for details).

Now, we prove that we can pass to the limit in the weak formulation of ( P ̲ n ). Taking any φ ∈ D 0 1 , p ( R N ) , we have

according to [21, Proposition 1] applied to the sequence ( ∣ ∇ u n ∣ p − 2 ∇ u n ) ⊆ L p ′ ( R N ) . On the other hand, splitting φ = φ + − φ − , Beppo Levi’s monotone convergence theorem guarantees