On Removable Singularities of Solutions of Higher-Order Differential Inequalities

Definition 1.1.

A function u is called a weak solution of (1.1) if u∈L1⁢(B1∖Bε) and f⁢(x)⁢g⁢(|u|)∈L1⁢(B1∖Bε) for all ε∈(0,1) and, moreover,

for any non-negative function φ∈C0∞⁢(B1∖{0}).

Definition 1.2.

A weak solution of (1.1) has a removable singularity at zero if u∈L1⁢(B1), f⁢(x)⁢g⁢(|u|)∈L1⁢(B1), and (1.2) is valid for any non-negative function φ∈C0∞⁢(B1), i.e.

Theorem 2.1.

Suppose that

and

where

for some real number σ>1. If

then any weak solution of (1.1) has a removable singularity at zero.

Theorem 2.2.

Let λ>1 and

where

for some real number σ>1. If

then any weak solution of (1.3) has a removable singularity at zero.

Corollary 2.3 (Brezis and Véron [3]).

Suppose that u∈C2⁢(B1∖{0}) is a solution of (1.4), where

Then u∈C2⁢(B1) and, moreover,

Proof.

By the Kato theorem [5], the function u+⁢(x)=max⁡{u⁢(x),0} is a weak solution of the inequality

Applying Theorem 2.2, we obtain that u+ has a removable singularity at zero in the weak sense, i.e. u+∈Lλ⁢(B1) and

Since the right-hand side of the last expression is non-negative, we have

therefore, u+∈L∞⁢(B1/2). To verify the validity of (2.7), it suffices to take

where

are Steklov–Schwartz averaging kernels for some non-negative function ω∈C0∞⁢(B1) such that

It is obvious that u+ε is an infinitely smooth function and, moreover,

Taking into account (2.6), we obtain

for all x∈B1/2 and ε∈(0,12). Hence,

for all ε∈(0,12). By the maximum principle, we have

for all ε∈(0,12). In the limit as ε→+0, this yields (2.7).

Analogously, one can show that u-⁢(x)=max⁡{-u⁢(x),0}∈L∞⁢(B1/2); therefore, u∈L∞⁢(B1/2). Further, putting

where |S1| is an (n-1)-dimensional volume of the unit sphere in ℝn and φ∈C0∞⁢(B1) is some function equal to one on B1/2, we obtain

Thus, u-v∈C∞⁢(B1/2) as u-v is a bounded harmonic function in B1/2∖{0}. The condition

implies that v∈C1⁢(B1/2). Therefore, u belongs to

This, in turn, implies that v∈C2⁢(B1/2) and, accordingly, u∈C2⁢(B1). Consequently, u satisfies equation (2.5) in the classical sense. ∎

Remark 2.4.

Condition (2.4) also guarantees the removability of a singularity at zero for weak non-negative solutions of (1.5) since, in Theorem 2.2, it does not matter what sign the Laplace operator faces. However, unlike (1.4), we can not argue that these solutions belong to C2⁢(B1) even if (1.5) is understood in the classical sense. In fact, if

then (1.5) has a solution of the form

where c>0 is some constant [4, Theorem 1.3]. The function u is a weak solution of the equation

in the sense of distributions, but it does not satisfy (2.8) in the classical sense.

Example 2.5.

Consider the inequality

where λ and s are real numbers and m is a positive even integer. By Theorem 2.2, if

then any weak solution of (2.9) has a removable singularity at zero. For m=2, condition (2.10) coincides with the analogous condition given in [10, Example 6.1.1]. In turn, if m=2, s=0, and n≥3, then (2.10) coincides with (2.4).

Let us examine the critical exponent s=λ⁢(n-m)-n in the right-hand side of (2.9). Namely, assume that u is a weak solution of the inequality

where λ and ν are real numbers and m is a positive even integer. By Theorem 2.2, if

then u has a removable singularity at zero. For m=2, condition (2.12) coincides with the analogous condition obtained in [10, Example 6.1.2]

It can be seen that, in the case of λ≤1, for any c, s, and ν there exist real numbers k>0 and l>0 such that

is a weak solution of both (2.9) and (2.11) with an unremovable singularity at zero. Therefore, the first inequality in (2.10) and (2.12) is exact.

Assume now that λ>1 and ν<-1. Let us put

and

It is obvious that

In the case of m<n, for any 1≤i≤m2 we also have

or in other words,

with some constants c1>0 and c2>0 for all r>0 in a neighborhood of zero. Hence, for any c>0 there are real numbers ε>0 and δ>0 such that the function

is a weak solution of (2.11) with an unremovable singularity at zero. Thus, the second inequality in (2.12) is exact for all m<n. Since solutions of (2.11) are also solutions of (2.9) for any s>λ⁢(n-m)-n, we have simultaneously showed the exactness of the second inequality in (2.10).

Example 2.6.

We examine the critical exponent λ=1 in the right-hand side of (2.9). Consider the inequality

where ν and s are real numbers and m≤n is a positive even integer.

By Theorem 2.1, if

then any weak solution of (2.15) has a removable singularity at zero. In fact, for any δ∈(0,1) there exists κ∈(0,∞) such that

for all ε∈(0,1) and τ∈(0,∞). To establish the validity of the last inequality, we assume the converse. Then there are a sequences or real numbers εi∈(0,1) and τi∈(0,∞) such that

It is clear that εi⁢τi→∞ as i→∞; otherwise, there are subsequences εij and τij such that τij≤β/εij with some constant β>0 for all j=1,2,…. Hence, taking into account (2.18), we arrive at a contradiction. In particular, one can assert that τi→∞ as i→∞; therefore, (2.18) implies the inequality

for all sufficiently large i, whence it follows that

for all sufficiently large i. Thus,

and we again arrive at a condition.

From (2.17) it follows that

for all ε∈(0,1), r∈(0,1), and t∈(0,∞). Therefore, taking

we obtain (1.6) with

To complete our arguments, it is sufficient to note that (2.16) guarantees the validity of conditions (2.1), (2.2), and (2.3) if δ is small enough.

We also note that both inequalities in (2.16) are exact. In fact, if ν≤m, then for any c and s there exist real numbers k>0 and l>0 such that

is a weak solution of (2.15) with an unremovable singularity at zero. In turn, if ν>m and s>-m, then for any c>0 there exists a real number k>0 such that the function

is a weak solution of (2.15) with an unremovable singularity at zero.

Example 2.7.

Consider the inequality

where λ and ν are real numbers and m<n is a positive even integer.

We are interested in the case of λ>1. By Theorem 2.1, if

then any weak solution of (2.19) has a removable singularity at zero. Indeed, let condition (2.20) be valid. Without loss of generality, it can be assumed that ν<0; otherwise, we replace ν by -1. After this replacement, inequality (2.19) obviously remains valid.

We have

for all real numbers a>0 and b>0. This allows us to assert that

for all ε∈(0,1), r∈(0,1), and t∈(0,∞). Therefore, taking

we obtain (1.6) with

In so doing, it can be verified that (2.1), (2.2), and (2.3) hold.

Inequality (2.20) is exact for all m<n. In fact, if ν<-1, then for any λ>1 and c>0 there exist ε>0 and δ>0 such that the function u defined by (2.14) is a weak solution of (2.19) with an unremovable singularity at zero.

Example 2.8.

Consider the first-order differential inequality

where λ and s are real numbers. By Theorem 2.2, if

then any weak solution of (2.21) has a removable singularity at zero. In can easily be seen that condition (2.22) is exact. Indeed, if λ≤1, then for any real numbers c and s there exist a weak solution of (2.21) with an unremovable singularity at zero. As such a solution, we can take the function u defined by (2.13), where k>0 and l>0 are sufficiently large real numbers. At the same time, if λ>1 and s>λ⁢(n-1)-n, then for any c>0 there exists ε>0 such that

is a weak solution of (2.21) with an unremovable singularity at zero.

Lemma 3.1.

For any integer i≥0 the estimate

is valid.

Proof.

It is sufficient to take

as a test function in (1.2), where ψ∈C∞⁢(ℝ) is a non-negative function such that

Lemma 3.2.

Suppose that

for some sequence of positive integers {ij}j=1∞. Then u has a removable singularity at zero.

Proof.

Applying the Fenchel–Young inequality, we obtain

In so doing,

according to Lemma 3.1, and

according to condition (2.3). Therefore, one can assert that

Let ψ be the function defined in the proof of Lemma 3.1 and let φ∈C0∞⁢(B1) be an arbitrary non-negative function. Put

where

We obviously have

From Lebesgue dominated convergence theorem, it follows that

and

Since

we also obtain

Thus, (3.2) implies (1.2). The proof is completed. ∎

Lemma 3.3.

Suppose that

Then

for all sufficiently large i.

Proof.

In view of (3.3), Lemma 3.1 allows us to assert that

for all sufficiently large i. In so doing, we admit that the constant C>0 in the last expression can depend on the limit in the left-hand side of (3.3) and on the first summand in the left-hand side of (3.1). For us, it is only important that this constant does not depend on i.

Since

inequality (3.5) implies the estimate

whence it follows that