Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term

Lemma 2.1 (Comparison Principle).

Let u , w ∈ W loc 2 , n ⁢ ( Ω ) ∩ C ⁢ ( Ω ¯ ) , and assume that L ⁢ u ≥ H ⁢ ( x , u ) in Ω and L ⁢ w ≤ H ⁢ ( x , w ) in Ω. If u ≤ w on ∂ ⁡ Ω , then u ≤ w in Ω.

Proof.

Given ε > 0 ,

Let δ > 0 sufficiently small such that u - ( w + ε ) ≤ 0 on ∂ ⁡ Ω δ . Suppose that the open set

is non-empty. Since L ⁢ ( u - ( w + ε ) ) ≥ H ⁢ ( x , u ) - H ⁢ ( x , w + ε ) ≥ 0 in Ω 0 δ and the coefficients of L are bounded on Ω δ , the maximum principle applies (see [12, Theorem 9.1]) and we conclude that u - ( w + ε ) ≤ 0 on Ω 0 δ . This is an obvious contradiction to the assumption that Ω 0 δ is non-empty. Therefore, we must have u ≤ w + ε in Ω δ . Since δ > 0 is arbitrary, we conclude that u ≤ w + ε in Ω. Since ε > 0 is arbitrary, we find that u ≤ w on Ω, as desired. ∎

Remark 2.2.

1. Any regularly varying function at infinity with index 1 < q < ∞ satisfies conditions (A2) and (A3). We recall that f is said to be regularly varying at infinity of index q ∈ ℝ if f is a measurable function defined on ( a , ∞ ) for some a > 0 and

   lim t → ∞ ⁡ f ⁢ ( ξ ⁢ t ) f ⁢ ( t ) = ξ q   for all  ⁢ ξ > 0 .

2. f ⁢ ( t ) = t satisfies (A3) but not (A2), while f ⁢ ( t ) = e t - 1 satisfies (A2) but not (A3).

3. If f satisfies (A1), then it is clear that F ⁢ ( t ) ≤ t ⁢ f ⁢ ( t ) for all t ≥ 0 . Moreover, if f satisfies both (A1) and (A2), then

   lim inf t → ∞ ⁡ F ⁢ ( t ) t ⁢ f ⁢ ( t ) ≤ 1 2 .

   We refer to Lemma A.1 in Appendix A for a proof of this assertion.

Lemma 2.3.

Suppose that f satisfies (A1), (A2) and (A3). Then the following hold:

1. lim sup t → ∞ ⁡ F ⁢ ( t ) f ⁢ ( t ) ⁢ ∫ t ∞ F ⁢ ( s ) - 1 / 2 ⁢ 𝑑 s < ∞ ,

2. lim sup t → ∞ ⁡ t f ⁢ ( t ) ⁢ ( ∫ t ∞ F ⁢ ( s ) - 1 / 2 ⁢ 𝑑 s ) 2 < ∞ .

Corollary 3.1.

Suppose that f satisfies conditions (A1), (A2) and (A3). Then

Proof.

We prove the first limit and omit the second as the proof is similar. We have

On noting that η ⁢ ( 0 + ) = 0 , the claim follows from Lemma 2.3 (ii).∎

Remark 3.2.

It is clear that (DB) and (DC) hold when the coefficients 𝐛 and c of L are bounded on Ω. Therefore, Corollary 3.1 holds when the coefficients of L are bounded. In fact, in this case, condition (A3) is not needed. One only needs to recall (2.1).

Remark 3.3.

On noting that g * ⁢ ( 1 ) = 0 , we see that Ξ * ≥ 0 . Moreover, we also have Ξ * ≤ 0 for any f that satisfies (A1) and (A3). We refer the reader to Lemma A.3 in Appendix A.

Lemma 3.4.

Given g ∈ C ⁢ ( ∂ ⁡ Ω ) , the following Dirichlet problem admits a solution u ∈ W loc 2 , p ⁢ ( Ω ) ∩ C ⁢ ( Ω ¯ ) for each 1 ≤ p < ∞ :

Remark 3.5.

Suppose that all coefficients of L are bounded and belong to C α ⁢ ( Ω ) for some 0 < α < 1 . If, in addition to the hypotheses on H, we suppose H ∈ C α ⁢ ( Ω × ℝ ) , then problem (3.2) admits a solution u ∈ C 2 ⁢ ( Ω ) ∩ C ⁢ ( Ω ¯ ) . This is a consequence of the elliptic regularity theory, see [12, Theorem 9.19] with k = 1 .

Lemma 3.6.

Let u * , u * ∈ W loc 2 , p ⁢ ( Ω ) ∩ C ⁢ ( Ω ) for 1 < p < ∞ be such that

Assume that u * ∈ L ∞ ⁢ ( Ω ) or u * = ∞ on ∂ ⁡ Ω . If u * ≤ u * in Ω, then there exists a solution u ∈ W loc 2 , p ⁢ ( Ω ) ∩ C ⁢ ( Ω ) of (3.3) with u * ≤ u ≤ u * a.e. in Ω.

Proof.

Let us first consider the case when u * ∈ L ∞ ⁢ ( Ω ) . For each positive integer j ≥ sup Ω ⁡ u * , by Lemma 3.4, we let u j ∈ W loc 2 , p ∩ C ⁢ ( Ω ¯ ) be a solution of

By the Comparison Principle, Lemma 2.1, we have

By proceeding as in the proof of [18, Lemma 3.2], we conclude that { u j } converges locally uniformly to a solution u ∈ C 2 ⁢ ( Ω ) of problem (3.3) on Ω with u * ≤ u ≤ u * .

Now assume that u * = ∞ on ∂ ⁡ Ω . Let 𝒪 j := { x ∈ Ω : d ⁢ ( x ) > 1 / j } for j = 1 , 2 , … , and let u j ∈ W loc 2 , p ⁢ ( 𝒪 j ) ∩ C ⁢ ( 𝒪 ¯ j ) be a solution of

By the Comparison Principle, we see that u * ≤ u j ≤ u * on 𝒪 ¯ j for each j ≥ 1 . Consequently,^[1] we also note that u j ≤ u j + 1 in 𝒪 j . Thus, we have

Again, by proceeding as in the proof of [18, Lemma 3.2], we conclude that { u j } converges locally uniformly to a solution u ∈ W loc 2 , p ⁢ ( Ω ) ∩ C ⁢ ( Ω ) of problem (3.3) with u * ≤ u ≤ u * on Ω. ∎

Theorem 3.7.

Suppose that f satisfies (A3) and (f4). We also assume that h ∈ C ⁢ ( Ω ) satisfies (h1). Then problem (1.3) has a solution u ∈ W loc 2 , p ⁢ ( Ω ) ∩ C ⁢ ( Ω ) for 1 ≤ p < ∞ . Moreover, u is the maximal solution.

Proof.

There is no loss of generality in supposing that p ≥ n . Let { 𝒪 j } be the sequence of open subsets of Ω introduced in the proof of Lemma 3.6 above. Let us first show that the following has a solution:

To this end, let v j be a solution of L ⁢ v j = f ⁢ ( v j ) in 𝒪 j and v j = ∞ on ∂ ⁡ 𝒪 j . If h ≥ 0 in 𝒪 j , then w j := v j is a super-solution of (3.4). Otherwise, let z j be a solution of L ⁢ z j = min 𝒪 ¯ j ⁡ h in 𝒪 j with z j = 0 on ∂ ⁡ 𝒪 j . Note that z j > 0 . Then w j := v j + z j satisfies

and

Thus, in any case w j is a super-solution of (3.4). We now proceed to construct a sub-solution of (3.4). To this end, we claim that there are positive constants A and α such that

is a sub-solution of (1.3) in Ω. To see this, let ϱ := d in (2.2), and we estimate (2.2) in Ω μ as follows:

In the last inequality, we have used the fact that - α ⁢ c ≥ 0 for any α > 0 . By (h1), since Ξ * > Θ * , we let A such that g * ⁢ ( A ) > Θ * . Let us choose ε > 0 sufficiently small so that

Therefore, there exists 0 < δ < μ such that for x ∈ Ω δ , we have

Recalling (2.1) and Corollary 3.1, we can take δ > 0 sufficiently small so that for all x ∈ Ω δ ,

Likewise, by the definition of Θ * := Θ * ⁢ ( h ) , we have (by shrinking δ further, if necessary)

Putting (3.5), (3.6) and (3.7) together, we find that w ⁢ ( x ) = A ⁢ ϕ ⁢ ( d ⁢ ( x ) ) - α satisfies the following for all x ∈ Ω δ :

Next we choose α so that w is a sub-solution of L ⁢ w = f ⁢ ( w ) + h ⁢ ( x ) on Ω δ . For this, let

We should point out that η is independent of α. The hypothesis on f shows that

Thus, we can choose α > 0 sufficiently large so that

Having fixed such α, we see that in Ω δ the following holds:

Thus, we have shown that w is a sub-solution of (1.3) in Ω. Moreover, for each positive integer j, the Comparison Principle shows that w ≤ w j in 𝒪 j . Therefore, by Lemma 3.6, problem (3.4) has a solution u j such that w ≤ u j ≤ w j in 𝒪 j . Since u j is bounded on 𝒪 j - 1 , the Comparison Principle shows that u j ≤ u j - 1 on 𝒪 j - 1 , that is, { u k } k = j - 1 ∞ is a decreasing sequence in 𝒪 j - 1 . Let us set

Since { u j } is locally uniformly bounded, by standard Schauder estimates, we see that u ∈ C 2 ⁢ ( Ω ) satisfies L ⁢ u = f ⁢ ( u ) + h in Ω. From this and the inequality w ≤ u in Ω, we see that u is a solution of (1.3), as desired.

If v is any solution of (1.3), then Comparison Principle shows that v ≤ u j on 𝒪 j for all j ≥ 1 . Consequently, we have v ≤ u in Ω, and therefore u is a maximal solution of (1.3), as claimed. ∎

Remark 3.8.

If the coefficients of L are bounded in Ω, we note that condition (A3) is not needed in the above theorem, see Remark 3.2.

Remark 4.1.

If h is bounded from above on Ω, then Θ * ≤ Θ * ≤ 0 .

Lemma 4.2.

Let h ∈ C ⁢ ( Ω ) .

1. If h satisfies (h2) , then there exists a positive constant A * such that for any solution u ∈ W loc 2 , n ⁢ ( Ω ) of ( 1.3 ), we have

   (4.1) lim sup d ⁢ ( x ) → 0 ⁡ u ⁢ ( x ) ϕ ⁢ ( d ⁢ ( x ) ) ≤ A * .

2. If h satisfies (h1) and is bounded from above, then there exists a positive constant A * such that for any solution u ∈ W loc 2 , n ⁢ ( Ω ) of ( 1.3 ), we have

   (4.2) A * ≤ lim inf d ⁢ ( x ) → 0 ⁡ u ⁢ ( x ) ϕ ⁢ ( d ⁢ ( x ) ) .

Proof.

For any 0 < ρ < μ , let us consider the sets

Proof of (4.1). For an appropriate choice of a positive constant A * , we will show that

is a super-solution of (1.3) on Ω ρ - for all 0 < ρ < μ and sufficiently small μ. To this end, we estimate (2.2) with ϱ ⁢ ( x ) := d ⁢ ( x ) - ρ as follows. Recalling that c ≤ 0 in Ω, we estimate

Let ε > 0 , to be specified later. By (2.1), we can take μ > 0 sufficiently small so that for all x ∈ Ω ρ - ,

Therefore, from (4.3) and (4.4), we conclude that for x ∈ Ω μ - ,

Now, let ε be chosen so that Θ * - 2 ⁢ ε > Ξ * . We pick a number A * := A * ⁢ ( Λ , f , Θ * ⁢ ( h ) ) > 0 such that

That is,

We also suppose that μ is sufficiently small, so that for ( x , ρ ) ∈ Ω ρ - × ( 0 , μ ) ,

Let us now suppose that Θ * ⁢ ( h ) ≤ 0 . In this case, for ( x , ρ ) ∈ Ω ρ - × ( 0 , μ ) , we have

from the definition of Θ * . Let B * := max ⁡ { u ⁢ ( x ) : d ⁢ ( x ) ≥ μ } . Then u ≤ w * + B * on ∂ ⁡ Ω ρ - and

By the Comparison Principle, Lemma 2.1, we conclude that u ≤ w * + B * in Ω ρ - . Therefore,

On letting ρ → 0 + , we see that the following holds on Ω μ :

Now, let d ⁢ ( x ) → 0 to obtain (4.1) when Θ * ⁢ ( h ) ≤ 0 .

We now consider the case Θ * ⁢ ( h ) > 0 . Then h is non-negative near ∂ ⁡ Ω . Let { 𝒪 j } be the sequence of open subsets of Ω defined as in the proof of Lemma 3.6 and m := min ⁡ { h ⁢ ( x ) : x ∈ Ω ¯ } . We note that m > - ∞ . Let v j be a solution of (3.4) with h ≡ m . If u is any solution of (1.3), then L ⁢ u = f ⁢ ( u ) + h ≥ f ⁢ ( u ) + m . By the Comparison Principle, u ≤ v j in 𝒪 j for all j. Proceeding as in the proof of Theorem 3.7, one can show that v = lim j → ∞ ⁡ v j is a solution of (1.3) on Ω with h ≡ m on Ω. Clearly, u ≤ v in Ω, and since v is a large solution of L ⁢ v = f ⁢ ( v ) + m on Ω and Θ * ⁢ ( m ) = 0 , the case considered above applied to v shows that

where A * = A * ⁢ ( Λ , f , Θ * ⁢ ( h ) ) , and therefore (4.1) holds.

Proof of (4.2).

We consider the function

For an appropriate choice of A * , we show that w * is a sub-solution in Ω ρ + for all ρ < μ , assuming that μ is sufficiently small. We assume first that Ξ * > 0 . Let us fix ε > 0 such that 3 ⁢ ε < Ξ * . Then, by definition, there exists a positive real number A * such that Ξ * / 2 + 3 ⁢ ε / 2 < g * ⁢ ( A * ) . That is,

Therefore, assuming that μ is sufficiently small, the following holds in Ω ρ + for all 0 < ρ < μ :

By shrinking μ if necessary, we can invoke (2.1) to estimate

Thus, for any 0 < ρ < μ , the following chain of inequalities hold on Ω ρ + :

Recall that Ξ * > 0 . Since h is bounded from above on Ω, we can assume μ is small enough so that

Thus, for Ξ * > 0 and sufficiently small μ > 0 , we have shown that

Now let us suppose that Ξ * = 0 . Then there exists A * > 0 such that for ε > 0 and d + ρ small, we have

which can be rewritten as

Using this in estimate (4.5), we find

By (h1), we recall that - ∞ ≤ Θ * < Ξ * = 0 . There is no loss in generality if we assume that Θ * > - ∞ . At this point we use condition (h1), that is,

On noting that f ⁢ ( ϕ ⁢ ( d + ρ ) ) ≤ f ⁢ ( ϕ ⁢ ( d ) ) , and using (4.7), we find

Choosing ε = - Θ * / 4 in the above inequality, we find

Inserting the latter estimate into (4.6), for sufficiently small ρ > 0 , yields

In conclusion, we have shown that in either of the cases, the following holds:

Let B * := A * ⁢ ϕ ⁢ ( μ ) . Note that w * - B * ≤ u on ∂ ⁡ Ω ρ + and

Therefore, w * - B * ≤ u in Ω ρ + and

On letting ρ → 0 + , we see that

On recalling that ϕ ⁢ ( d ⁢ ( x ) ) → ∞ as d ⁢ ( x ) → 0 , we get

and the proof is complete. ∎