Startseite Mathematik One-dimensional boundary blow up problem with a nonlocal term
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One-dimensional boundary blow up problem with a nonlocal term

  • Taketo Inaba und Futoshi Takahashi EMAIL logo
Veröffentlicht/Copyright: 24. Oktober 2025
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Advances in Nonlinear Analysis
Aus der Zeitschrift Advances in Nonlinear Analysis Band 14 Heft 1

Abstract

In this article, we study a nonlocal boundary blow up problem on an interval and obtain the precise asymptotic formula for solutions when the bifurcation parameter in the problem is large.

Keywords: nonlocal elliptic equations; boundary blow up solutions
MSC 2020: Primary 34C23; Secondary 37G99

1 Introduction

In this article, we study the following one-dimensional nonlocal elliptic problem:

(1) M ( ‖ u ‖ L q ( I ) ) u ″ ( x ) = λ u p ( x ) , x ∈ I = ( − 1 , 1 ) , u ( x ) > 0 , x ∈ I , u ( x ) → + ∞ as x → ∂ I = { − 1 , + 1 } ,

where λ > 0 is a given constant and M : [ 0 , + ∞ ) → R + is a continuous function. We call u a solution of (1) if u ∈ C 2 ( I ) ∩ L q ( I ) and u solves the equation for all x ∈ I . Throughout of this article, we assume that

(2) p > 1 and 0 < q < p − 1 2 .

We consider λ > 0 in (1) as a bifurcation parameter; the number of solutions of (1) is changed according to the value of λ . In most cases, we consider M ( t ) = ( t q + b ) r   ( t ≥ 0 ) for constants r > 0 and b ≥ 0 , i.e., we consider

(3) ( ‖ u ‖ L q ( I ) q + b ) r u ″ ( x ) = λ u p ( x ) , x ∈ I = ( − 1 , 1 ) , u ( x ) > 0 , x ∈ I , u ( x ) → + ∞ as x → ∂ I = { − 1 , + 1 } .

For any p > 1 , it is a classical fact that there exists a unique solution U p ∈ C 2 ( I ) satisfying

(4) U p ″ ( x ) = U p p ( x ) , x ∈ I = ( − 1 , 1 ) , U p ( x ) > 0 , x ∈ I , U p ( x ) → + ∞ as x → ∂ I = { − 1 , + 1 } ,

see Proposition 1.8 and Remark 1.10 in [4]. Note that the nonlinearity f ( s ) = s p   ( p > 1 ) satisfies the famous Keller–Osserman condition [5,11]

∫ a ∞ 1 F ( t ) d t < + ∞ for some a > 0 ,

where F ( t ) = ∫ 0 t f ( s ) d s . By the uniqueness, we further see that U p is even: U p ( − x ) = U p ( x ) for any x ∈ I ; otherwise V p ( x ) = U p ( − x ) will be another solution of (4) different from U p . Later in Lemma 1, we prove that U p ∈ L q ( I ) for 0 < q < p − 1 2 , and we compute the exact value of ‖ U p ‖ L q ( I ) .

In the following, we put

(5) L p = ∫ 1 ∞ d t t p + 1 − 1

for p > 1 . Also B ( x , y ) = ∫ 0 1 t x − 1 ( 1 − t ) y − 1 d t denotes the Beta function.

First, we consider the case b = 0 in (3).

Theorem 1

Assume (2) and let r > 0 and b = 0 in (3). Denote B p , q = B p − 2 q − 1 2 ( p + 1 ) , 1 2 .

  1. Assume q r − p + 1 ≠ 0 . Then for any λ > 0 , there exists a unique solution u λ ∈ C 2 ( I ) ∩ L q ( I ) of (3), which satisfies

    (6) u λ ( x ) = λ 1 q r − p + 1 ‖ U p ‖ q − q r q r − p + 1 U p ( x ) = λ 1 q r − p + 1 p + 1 2 r ( p − 1 − q ) ( p − 1 ) ( q r − p + 1 ) L p − r ( 2 q − p + 1 ) ( p − 1 ) ( q r − p + 1 ) B p , q − r q r − p + 1 U p ( x ) ,

    for x ∈ I , where U p is the unique solution of (4).

  2. Assume q r − p + 1 = 0 . Then (3) admits a solution if and only if λ = ‖ U p ‖ L q ( I ) q r = p + 1 2 − r ( p − 1 − q ) p − 1 L p r ( 2 q − p + 1 ) p − 1 B p , q r . In this case, u λ ∣ λ = ‖ U p ‖ q q r = U p .

Next, we consider the case b > 0 in (3). For the following theorems, we define

(7) g ( t ) = ( t q + b ) r t p − 1 for t > 0 .

Theorem 2

Assume (2) and let r > 0 and b > 0 in (3). Assume q r − p + 1 > 0 and put

t 0 = b ( p − 1 ) q r − p + 1 1 q , λ 0 = ‖ U p ‖ q p − 1 g ( t 0 ) ,

where g is in (7). Then

  1. If 0 < λ < λ 0 , there exists no solution in C 2 ( I ) ∩ L q ( I ) for (3).

  2. If λ = λ 0 , there exists a unique solution in C 2 ( I ) ∩ L q ( I ) for (3).

  3. If λ > λ 0 , there exist exactly two solutions u 1 , λ and u 2 , λ in C 2 ( I ) ∩ L q ( I ) for (3). Moreover, as λ → + ∞ ,

    u 1 , λ ( x ) = b r p − 1 λ − 1 p − 1 1 + r p − 1 b q r − p + 1 p − 1 λ − q p − 1 ‖ U p ‖ q q ( 1 + o ( 1 ) ) U p ( x ) , u 2 , λ ( x ) = m p , q λ 1 q r − p + 1 − b r m p , q 1 − q q r − p + 1 λ 1 − q q r − p + 1 ( 1 + o ( 1 ) ) U p ( x ) ‖ U p ‖ q

    for any x ∈ I , where m p , q = ‖ U p ‖ q 1 − p q r − p + 1 .

Theorem 3

Assume (2) and let r > 0 and b > 0 in (3). Assume q r − p + 1 ≤ 0 . Then

  1. If q r − p + 1 < 0 , there exists a unique solution in C 2 ( I ) ∩ L q ( I ) of (3) for any λ > 0 .

  2. If q r − p + 1 = 0 , there exists a unique solution in C 2 ( I ) ∩ L q ( I ) of (3) for any λ > ‖ U p ‖ q p − 1 and no solution for λ ≤ ‖ U p ‖ q p − 1 .

As a special case of Theorem 3, we have the following.

Theorem 4

Let p > 1 , b > 0 , and q = 1 < p − 1 2 = r . Then for any λ > 0 , there exists a unique solution u λ ∈ C 2 ( I ) ∩ L q ( I ) of (3) of the form

u λ ( x ) = ‖ U p ‖ 1 + ‖ U p ‖ 1 2 + 4 b λ 2 p − 1 2 λ − 2 p − 1 U p ( x ) .

The problem (3) is a one-dimensional toy model of much more general boundary blow up problem with a nonlocal term

(8) M ( ‖ u ‖ L q ( Ω ) ) Δ u = λ f ( u ) in Ω , u > 0 in Ω , u ( x ) → + ∞ as d ( x ) ≔ dist ( x , ∂ Ω ) → 0 ,

where Ω is a bounded domain in R N , N ≥ 1 , f is a continuous nonlinearity, λ > 0 , q > 0 , and M : [ 0 , ∞ ) → R + is a continuous function. u is called a solution of (8) if u ∈ C 2 ( Ω ) ∩ L q ( Ω ) and u satisfies the equation for x ∈ Ω . See some discussion for f ( u ) = u p in §6. As far as the authors know, the problem (8), or even (1) and (3), have not been studied so far. Therefore, in this article, we start to study (3) and obtain the precise information of the solution set according to the bifurcation parameter λ .

2 Exact L q -norm of U p

In this section, first, we compute the precise value of the L q -norm of U p , which is the unique solution of (4), when p and q satisfy (2).

Lemma 1

For a given p > 1 , let U p be the unique solution of the problem (4). Then U p ∈ L q ( I ) if and only if 0 < q < p − 1 2 . Also we obtain

(9) μ p ≔ min x ∈ I U p ( x ) = U p ( 0 ) = p + 1 2 L p 2 p − 1 ,

(10) ‖ U p ‖ L q ( I ) q = 2 p + 1 μ p 2 q − p + 1 2 B p − 2 q − 1 2 ( p + 1 ) , 1 2 ,

where L p is defined in (5) and B ( x , y ) denotes the Beta function.

Proof

Following the article by Shibata [12], we use a time-map method to compute ‖ U p ‖ L q ( I ) . In [12], the unique solution W p of the problem

− W p ″ ( x ) = W p p ( x ) , x ∈ I = ( − 1 , 1 ) , W p ( x ) > 0 , x ∈ I = ( − 1 , 1 ) , W p ( ± 1 ) = 0

is considered and its maximum value and the L p -norm is computed.

Multiply the equation by U p ′ ( x ) , we have

( U p ″ ( x ) − U p p ( x ) ) U p ′ ( x ) = 0 .

This implies

1 2 ( U p ′ ( x ) ) 2 − 1 p + 1 U p p + 1 ( x ) ′ = 0 ,

thus

1 2 ( U p ′ ( x ) ) 2 − 1 p + 1 U p p + 1 ( x ) = constant = 0 − 1 p + 1 μ p p + 1 .

Therefore, we have

(11) U p ′ ( x ) = 2 p + 1 ( U p p + 1 ( x ) − μ p p + 1 ) .

By (11) and the change of variables s = U p ( x ) and s = μ p t , we compute

1 = ∫ 0 1 1 d x = ∫ 0 1 U p ′ ( x ) d x 2 p + 1 ( U p p + 1 ( x ) − μ p p + 1 ) = p + 1 2 ∫ μ p ∞ d s ( s p + 1 − μ p + 1 ) = p + 1 2 ∫ 1 ∞ μ p d t μ p p + 1 ( t p + 1 − 1 ) = p + 1 2 μ p 1 − p 2 ∫ 1 ∞ d t t p + 1 − 1 .

From this, we obtain

μ p = U p ( 0 ) = p + 1 2 L p 2 p − 1 .

Next, by the formula μ p , we see

‖ U p ‖ L q ( I ) q = 2 ∫ 0 1 ∣ U p ( x ) ∣ q d x = 2 ∫ 0 1 U p q ( x ) U p ′ ( x ) d x 2 p + 1 ( U p p + 1 ( x ) − μ p p + 1 ) = 2 ( p + 1 ) ∫ μ p ∞ s q d s ( s p + 1 − μ p p + 1 ) ( s = U p ( x ) ) = 2 ( p + 1 ) ∫ 1 ∞ μ q + 1 t q d t μ p p + 1 ( t p + 1 − 1 ) t = s μ p = 2 ( p + 1 ) μ p 2 q − p + 1 2 ∫ 1 ∞ t q d t t p + 1 − 1 .

Since

∫ 1 ∞ t q d t ( t p + 1 − 1 ) 1 2 = ∫ 0 1 s − p − 2 q − 3 2 ( p + 1 ) 1 − s d s s = 1 t p + 1 = 1 p + 1 B p − 2 q − 1 2 ( p + 1 ) , 1 2 ,

we obtain the formula of ‖ U p ‖ L q ( I ) .□

Remark 1

Since L p is uniformly bounded with respect to p , we see

log μ p = 2 p − 1 p + 1 2 + 2 p − 1 log L p → 0 ( p → ∞ ) .

This implies lim p → ∞ μ p = 1 .

Remark 2

By the homogeneity of the nonlinearity f ( s ) = s p and the uniqueness of U p , it is obvious that for any λ > 0 , w λ ( x ) = λ − 1 p − 1 U p ( x ) is the unique solution of the problem

(12) w ″ ( x ) = λ w p ( x ) , x ∈ I = ( − 1 , 1 ) , w ( x ) > 0 , x ∈ I , w ( ± 1 ) = + ∞ .

Next lemma is an analogue of Theorem 2 in [1] and is important throughout of this article.

Lemma 2

Let M : [ 0 , + ∞ ) → R + be continuous in (1). Then for a given λ > 0 , problem (1) has the same number of positive solutions as that of the positive solutions (with respect to t > 0 ) of

(13) M ( t ) t p − 1 = λ ‖ U p ‖ q 1 − p ,

where U p is the unique solution of (4). Moreover, any solution u λ of (1) must be of the form

(14) u λ ( x ) = t λ U p ( x ) ‖ U p ‖ q ,

where t λ > 0 is a solution of (13).

Proof

Let u ∈ C 2 ( I ) ∩ L q ( I ) be any solution of (1) and put v = γ u , where

γ = M ( ‖ u ‖ q ) − 1 p − 1 λ 1 p − 1 .

Then

v ″ ( x ) = γ u ″ ( x ) = ( 1 ) γ λ M ( ‖ u ‖ q ) u p ( x ) = γ λ M ( ‖ u ‖ q ) v ( x ) γ p = γ 1 − p λ M ( ‖ u ‖ q ) v p ( x ) = v p ( x ) .

Thus, v is a solution of

v ″ ( x ) = v p ( x ) , x ∈ I = ( − 1 , 1 ) , v ( x ) > 0 , x ∈ I , v ( ± 1 ) = + ∞ .

By the uniqueness, v ≡ U p . This leads to

(15) u = γ − 1 U p = M ( ‖ u ‖ q ) 1 p − 1 λ − 1 p − 1 U p .

Now, we put t = ‖ u ‖ q > 0 . Then by taking a L q -norm of the both sides of (15), we have

t = ‖ u ‖ q = M ( t ) 1 p − 1 λ − 1 p − 1 ‖ U p ‖ q ,

which is equivalent to (13). This shows that

♯ { u ∈ C 2 ( I ) ∩ L q ( I ) : solutions of (1) } ≤ ♯ { t > 0 : solutions of (13) } ,

where ♯ A denotes the cardinality of the set A .

On the other hand, let t > 0 be any solution of (13) and put

u ( x ) = t U p ( x ) ‖ U p ‖ q ∈ C 2 ( I ) ∩ L q ( I ) .

Then we have u ( x ) > 0 , u ( ± 1 ) = + ∞ , t = ‖ u ‖ q and

M ( ‖ u ‖ q ) u ″ ( x ) = M ( t ) t ‖ U p ‖ q U p ″ ( x ) = ( 4 ) M ( t ) t ‖ U p ‖ q U p p ( x ) = M ( t ) t ‖ U p ‖ q ‖ U p ‖ q t u ( x ) p = M ( t ) ‖ U p ‖ q p − 1 t p − 1 u p ( x ) = λ u p ( x ) .

This shows that

♯ { t > 0 : solutions of (13) } ≤ ♯ { u ∈ C 2 ( I ) ∩ L q ( I ) : solutions of (1) } . □

3 Proof of Theorem 1.

In this section, we prove Theorem 1.

Proof

Let q r + 1 − p ≠ 0 . By virtue of Lemma 2, we only need to prove that (13) has the unique solution for any λ > 0 . Since b = 0 in (3), M ( t ) = t q r and equation (13) reads

(16) t q r − p + 1 = λ ‖ U p ‖ q 1 − p .

The map t ↦ t q r − p + 1 is strictly monotone increasing (resp. decreasing) for t > 0 if q r − p + 1 > 0 (resp. if q r − p + 1 < 0 ) and maps ( 0 , + ∞ ) onto ( 0 , + ∞ ) . Therefore, equation (16) admits the unique solution

t λ = λ 1 q r − p + 1 ‖ U p ‖ q 1 − p q r − p + 1

for any λ > 0 . By (14), this corresponds to the unique solution

u λ ( x ) = t λ U p ( x ) ‖ U p ‖ q = λ 1 q r − p + 1 ‖ U p ‖ q − q r q r − p + 1 U p ( x )

of (3). Inserting the formula (10) in ‖ U p ‖ q , we obtain (6). This proves (i).

If q r − p + 1 = 0 , (16) is satisfied if and only if λ = ‖ U p ‖ q p − 1 = ‖ U p ‖ q q r . Thus, in this case, u λ = U p follows. This proves (ii).□

4 Proof of Theorem 2

In this section, we prove Theorem 2.

Proof

According to Lemma 2, the number of solutions of (3) is the same as the number of solutions

(17) g ( t ) = ( t q + b ) r t p − 1 = λ ‖ U p ‖ q 1 − p .

Since

g ′ ( t ) = ( t q + b ) r − 1 t p { ( q r − p + 1 ) t q − b ( p − 1 ) } ,

and q r − p + 1 > 0 , equation g ′ ( t ) = 0 admits the unique solution

t 0 = b ( p − 1 ) q r − p + 1 1 ⁄ q

and g ′ ( t ) < 0 if 0 < t < t 0 , g ′ ( t ) > 0 if t > t 0 and g ( t 0 ) = min t ∈ R + g ( t ) . Also lim t → + 0 g ( t ) = + ∞ since b > 0 , and lim t + ∞ g ( t ) = + ∞ since q r > p − 1 . Thus, equation (in t ) (17) admits

  1. no solution if λ ‖ U p ‖ q 1 − p < g ( t 0 ) .

  2. exactly one solution if λ ‖ U p ‖ q 1 − p = g ( t 0 ) .

  3. exactly two solutions if λ ‖ U p ‖ q 1 − p > g ( t 0 ) .

Thus, putting λ 0 = ‖ U p ‖ q p − 1 g ( t 0 ) and recalling Lemma 2, we conclude the former part of Theorem 2.

Now, we prove the asymptotic formulae in the case (iii) and λ > > 1 . By a simple consideration using the graph of g ( t ) , we see that two solutions 0 < t 1 < t 2 of (17) satisfy t 1 < t 0 < t 2 and

t 1 → + 0 and t 2 → + ∞

as λ → + ∞ . Since

λ ‖ U p ‖ q 1 − p = g ( t 2 ) = ( t 2 + b ) r t 2 p − 1 ∼ t 2 q r t 2 p − 1

as λ → + ∞ , t 2 can be expressed as follows:

t 2 = λ 1 q r − p + 1 ‖ U p ‖ q 1 − p q r − p + 1 + R ,

where R = o ( λ 1 q r − p + 1 ) as λ → ∞ . In the following, we put

m p , q = ‖ U p ‖ q 1 − p q r − p + 1

for simplicity. Then we write

(18) t 2 = m p , q λ 1 q r − p + 1 + R .

We insert this expression into g ( t 2 ) = λ ‖ U p ‖ q 1 − p , which is equivalent to

(19) ( t 2 q + b ) r = t 2 p − 1 λ ‖ U p ‖ q 1 − p .

Then Taylor expansion implies

(LHS) of (19) = t 2 q r 1 + b t 2 q r = t 2 q r 1 + r b t 2 q ( 1 + o ( 1 ) ) = ( 18 ) m p , q λ 1 q r − p + 1 + R q r 1 + r b m p , q λ 1 q r − p + 1 + R q ( 1 + o ( 1 ) ) = m p , q λ 1 q r − p + 1 q r 1 + R m p , q λ 1 q r − p + 1 q r 1 + r b m p , q λ 1 q r − p + 1 q ( 1 + o ( 1 ) ) = m p , q λ 1 q r − p + 1 q r 1 + q r R m p , q λ 1 q r − p + 1 ( 1 + o ( 1 ) ) 1 + r b m p , q λ 1 q r − p + 1 q ( 1 + o ( 1 ) ) = m p , q λ 1 q r − p + 1 q r 1 + q r R m p , q λ 1 q r − p + 1 + r b m p , q λ 1 q r − p + 1 q ( 1 + o ( 1 ) )

as λ → ∞ . On the other hand,

(RHS) of (19) = t 2 p − 1 λ ‖ U p ‖ q 1 − p = ( 18 ) λ ‖ U p ‖ q 1 − p m p , q λ 1 q r − p + 1 + R p − 1 = λ ‖ U p ‖ q 1 − p m p , q p − 1 λ p − 1 q r − p + 1 1 + ( p − 1 ) R m p , q λ 1 q r − p + 1 ( 1 + o ( 1 ) ) = m p , q λ 1 q r − p + 1 q r 1 + ( p − 1 ) R m p , q λ 1 q r − p + 1 ( 1 + o ( 1 ) )

as λ → ∞ . We compare these two equations and obtain

q r R m p , q λ 1 q r − p + 1 + b r ( m p , q λ 1 q r − p + 1 ) q = ( p − 1 ) R m p , q λ 1 q r − p + 1 ( 1 + o ( 1 ) ) .

From this, we obtain

R = − b r m p , q 1 − q q r − p + 1 λ 1 − q q r − p + 1 ( 1 + o ( 1 ) )

as λ → ∞ . Inserting this in (18), we see

t 2 = m p , q λ 1 q r − p + 1 − b r m p , q 1 − q q r − p + 1 λ 1 − q q r − p + 1 ( 1 + o ( 1 ) ) .

Then u 2 , λ = t 2 U p ‖ U p ‖ q by (14), we obtain the asymptotic formula for u 2 , λ ( x ) .

Next, we prove the asymptotic formula for u 1 , λ when λ ≫ 1 . In this case, t 1 → 0 as λ → ∞ and q r > p − 1 , thus,

λ ‖ U p ‖ q 1 − p = g ( t 1 ) = ( t 2 q + b ) r t 1 p − 1 = b r t 1 p − 1 ( 1 + o ( 1 ) ) .

From this, we have

t 1 = b r p − 1 λ − 1 p − 1 ‖ U p ‖ q ( 1 + o ( 1 ) ) ,

and we can write

(20) t 1 = b r p − 1 λ − 1 p − 1 ‖ U p ‖ q ( 1 + η ) , η = o ( λ − 1 p − 1 ) .

As before, we insert (20) into

(21) ( t 1 q + b ) r = t 1 p − 1 λ ‖ U p ‖ q 1 − p .

Then Taylor expansion implies

(LHS) of (21) = ( t 1 q + b ) r = b r 1 + r b t 1 q + o ( t 1 q ) . (RHS) of (21) = t 1 p − 1 λ ‖ U p ‖ q 1 − p = ( 20 ) λ ‖ U p ‖ q 1 − p b r p − 1 λ − 1 p − 1 ‖ U p ‖ q ( 1 + η ) p − 1 = b r ( 1 + η ) p − 1 = b r ( 1 + ( p − 1 ) η + o ( η ) ) .

By comparing these equations, we have

η = r b ( p − 1 ) t 1 q ( 1 + o ( 1 ) ) = ( 20 ) r b ( p − 1 ) b r p − 1 λ − 1 p − 1 ‖ U p ‖ q q ( 1 + o ( 1 ) ) = r p − 1 b q r − p + 1 p − 1 λ − q p − 1 ‖ U p ‖ q q ( 1 + o ( 1 ) )

as λ → ∞ . Returning to (20) with this, we see

t 1 = b r p − 1 λ − 1 p − 1 ‖ U p ‖ q 1 + r p − 1 b q r − p + 1 p − 1 λ − q p − 1 ‖ U p ‖ q q ( 1 + o ( 1 ) ) .

Since u 1 , λ = t 1 U p ‖ U p ‖ q , we have the asymptotic formula for u 1 , λ .□

5 Proof of Theorems 3 and 4

In this section, first we prove Theorem 3.

Proof

Since q r − p + 1 ≤ 0 and b > 0 , g ( t ) in (7) is strictly decreasing on ( 0 , + ∞ ) and

lim t → + 0 g ( t ) = + ∞ , lim t + ∞ g ( t ) = 0 , q r − p + 1 < 0 , 1 , q r − p + 1 = 0 .

Thus, equation (17) has the unique solution t λ for any λ > 0 when q r − p + 1 < 0 , and for any λ such that λ ‖ U p ‖ 1 1 − p > 1 when q r − p + 1 = 0 . By Lemma 2 (14), u λ = t λ U p ‖ U p ‖ q is the unique solution of (3). This proves Theorem 3.□

In some special cases, we can obtain the exact value of t λ in Theorem 3. This is the content of Theorem 4, which we prove here.

Proof

In this case, since q = 1 < p − 1 2 = r , equation (17) reduces to

t + b t 2 = λ 2 p − 1 ‖ U p ‖ 1 − 2 .

This is the quadratic equation in t > 0 , and its positive solution is given by

t λ = 1 + 1 + 4 b λ 2 p − 1 ‖ U p ‖ 1 − 2 2 λ 2 p − 1 ‖ U p ‖ 1 − 2 = ‖ U p ‖ 1 ‖ U p ‖ 1 + ‖ U p ‖ 1 2 + 4 b λ 2 p − 1 2 λ − 2 p − 1

Thus, by Lemma 2 (14), we obtain the unique solution u λ = t λ U p ‖ U p ‖ 1 of (3).□

6 Higher dimensional case

In this section, we consider the higher dimensional analogue of the problem (3), namely,

(22) M ( ‖ u ‖ L q ( Ω ) ) Δ u = u p in Ω , u > 0 in Ω , u ( x ) → + ∞ as d ( x ) ≔ dist ( x , ∂ Ω ) → 0 ,

where Ω is a bounded domain in R N , N ≥ 1 , p > 1 , q > 0 , and M : [ 0 , ∞ ) → R + is a continuous function. We call u a solution of (22) if u ∈ C 2 ( Ω ) ∩ L q ( Ω ) and u satisfies (22) for x ∈ Ω .

As in the 1D case, the important fact is the existence and uniqueness of solutions of the model problem

(23) Δ u = u p in Ω , u > 0 in Ω , u ( x ) → + ∞ as d ( x ) → 0 .

It is well known that there always exists a solution of (23) when p > 1 , if Ω satisfies some regularity assumption (exterior cone condition is enough) [5,11]. A necessary and sufficient condition involving capacity for the existence of solutions of (23) is given in [7]. Also the solution is unique and satisfies the estimate

(24) C 1 d ( x ) − 2 p − 1 ≤ u ( x ) ≤ C 2 d ( x ) − 2 p − 1

for some 0 < C 1 ≤ C 2 , if Ω is Lipschitz [2,3,8]. If p ∈ ( 1 , N N − 2 ) when N ≥ 3 , or p ∈ ( 1 , + ∞ ) if N = 1, 2, no smoothness assumption is needed for the existence of solutions to (23) and the solution is unique if ∂ Ω = ∂ Ω ¯ is satisfied, see [6,9,10,13]. For any p > 1 , (23) admits at most one solution if ∂ Ω is represented as a graph of a continuous function locally [9].

In the following, we assume that Ω is sufficiently smooth, say C 2 . In this case, the unique solution u p of the problem (23) is in L q ( Ω ) for q < p − 1 2 . Indeed, let ε > 0 small and put Ω ε = { x ∈ Ω ∣ d ( x ) > ε } . Then estimate (24) implies

∫ Ω ∣ u p ∣ q d x = ∫ Ω ε ∣ u p ∣ q d x + ∫ Ω \ Ω ε ∣ u p ∣ q d x ≤ C + C ∫ Ω \ Ω ε d ( x ) − 2 q p − 1 d x ≤ C + C ′ ∫ 0 ε t − 2 q p − 1 d t < ∞

if 2 q p − 1 < 1 , i.e., q < p − 1 2 .

As in Lemma 2, we can prove the following.

Theorem 5

Let Ω ⊂ R N be a smooth bounded domain and let M : [ 0 , + ∞ ) → R + be continuous in (22). Then for a given λ > 0 , problem (22) has the same number of positive solutions as that of the positive solutions (with respect to t > 0 ) of

(25) M ( t ) t p − 1 = λ ‖ u p ‖ q 1 − p

where u p is the unique solution of (23). Moreover, any solution u λ of (22) must be of the form

u λ ( x ) = t λ u p ( x ) ‖ u p ‖ q ,

where t λ > 0 is a solution of (25).

The proof of Theorem 5 is completely similar to that of Lemma 2, so we omit it here.

Acknowledgments

The authors are thankful to Osaka Central University Advanced Mathematical Institute (OCAMI), MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics for its kind support.

  1. Funding information: F.T. was supported by JSPS Grant-in-Aid for Scientific Research (B), No. 23K25781.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. All authors have made the equal contribution to the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2024-09-14
Accepted: 2025-08-20
Published Online: 2025-10-24

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

  1. Research Articles
  2. Incompressible limit for the compressible viscoelastic fluids in critical space
  3. Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
  4. Intervals of bifurcation points for semilinear elliptic problems
  5. On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
  6. Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
  7. Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
  8. Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
  9. Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
  10. Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
  11. Existence results for non-coercive problems
  12. Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
  13. Geometric characterization of generalized Hajłasz-Sobolev embedding domains
  14. Subharmonic solutions of first-order Hamiltonian systems
  15. Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
  16. Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
  17. Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
  18. Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
  19. Homoclinic solutions in periodic partial difference equations
  20. Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
  21. Properties of minimizers for L2-subcritical Kirchhoff energy functionals
  22. Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
  23. Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
  24. Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
  25. Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
  26. Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
  27. Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
  28. Existence of positive radial solutions of general quasilinear elliptic systems
  29. Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
  30. Sharp viscous shock waves for relaxation model with degeneracy
  31. Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
  32. Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
  33. Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
  34. Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
  35. Singularity for the macroscopic production model with Chaplygin gas
  36. Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
  37. Global dynamics of population-toxicant models with nonlocal dispersals
  38. α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
  39. High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness
  40. On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
  41. Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
  42. On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
  43. Remark on the analyticity of the fractional Fokker-Planck equation
  44. Continuous dependence on initial data for damped fourth-order wave equation with strain term
  45. Unilateral problems for quasilinear operators with fractional Riesz gradients
  46. Boundedness of solutions to quasilinear elliptic systems
  47. Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
  48. Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
  49. Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals
  50. Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
  51. Shape of extremal functions for weighted Sobolev-type inequalities
  52. One-dimensional boundary blow up problem with a nonlocal term
  53. Doubling measure and regularity to K-quasiminimizers of double-phase energy
  54. General solutions of weakly delayed discrete systems in 3D
  55. Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data
  56. Optimal large time behavior of the 3D rate type viscoelastic fluids
  57. Local well-posedness for the two-component Benjamin-Ono equation
  58. Self-similar blow-up solutions of the four-dimensional Schrödinger-Wave system
  59. Existence and stability of traveling waves in a Keller-Segel system with nonlinear stimulation
  60. Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations
  61. On the global large regular solutions of the 1D degenerate compressible Navier-Stokes equations
  62. Normal forms of piecewise-smooth monodromic systems
  63. Fractional Dirichlet problems with singular and non-locally convective reaction
  64. Sharp forced waves of degenerate diffusion equations in shifting environments
  65. Global boundedness and stability of a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities
  66. Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p-Laplacian
  67. Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p-Laplacian
  68. Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics
  69. Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model
  70. Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system
  71. Generalized quasi-linear fractional Wentzell problems
  72. Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space
  73. Normalized solutions for NLS equations with general nonlinearity on compact metric graphs
  74. Review Article
  75. Existence and stability of contact discontinuities to piston problems
https://doi.org/10.1515/anona-2025-0114
Schlagwörter für diesen Artikel
nonlocal elliptic equations; boundary blow up solutions
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Research Articles
  2. Incompressible limit for the compressible viscoelastic fluids in critical space
  3. Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
  4. Intervals of bifurcation points for semilinear elliptic problems
  5. On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
  6. Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
  7. Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
  8. Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
  9. Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
  10. Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
  11. Existence results for non-coercive problems
  12. Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
  13. Geometric characterization of generalized Hajłasz-Sobolev embedding domains
  14. Subharmonic solutions of first-order Hamiltonian systems
  15. Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
  16. Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
  17. Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
  18. Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
  19. Homoclinic solutions in periodic partial difference equations
  20. Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
  21. Properties of minimizers for L2-subcritical Kirchhoff energy functionals
  22. Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
  23. Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
  24. Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
  25. Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
  26. Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
  27. Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
  28. Existence of positive radial solutions of general quasilinear elliptic systems
  29. Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
  30. Sharp viscous shock waves for relaxation model with degeneracy
  31. Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
  32. Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
  33. Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
  34. Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
  35. Singularity for the macroscopic production model with Chaplygin gas
  36. Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
  37. Global dynamics of population-toxicant models with nonlocal dispersals
  38. α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
  39. High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness
  40. On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
  41. Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
  42. On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
  43. Remark on the analyticity of the fractional Fokker-Planck equation
  44. Continuous dependence on initial data for damped fourth-order wave equation with strain term
  45. Unilateral problems for quasilinear operators with fractional Riesz gradients
  46. Boundedness of solutions to quasilinear elliptic systems
  47. Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
  48. Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
  49. Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals
  50. Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
  51. Shape of extremal functions for weighted Sobolev-type inequalities
  52. One-dimensional boundary blow up problem with a nonlocal term
  53. Doubling measure and regularity to K-quasiminimizers of double-phase energy
  54. General solutions of weakly delayed discrete systems in 3D
  55. Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data
  56. Optimal large time behavior of the 3D rate type viscoelastic fluids
  57. Local well-posedness for the two-component Benjamin-Ono equation
  58. Self-similar blow-up solutions of the four-dimensional Schrödinger-Wave system
  59. Existence and stability of traveling waves in a Keller-Segel system with nonlinear stimulation
  60. Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations
  61. On the global large regular solutions of the 1D degenerate compressible Navier-Stokes equations
  62. Normal forms of piecewise-smooth monodromic systems
  63. Fractional Dirichlet problems with singular and non-locally convective reaction
  64. Sharp forced waves of degenerate diffusion equations in shifting environments
  65. Global boundedness and stability of a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities
  66. Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p-Laplacian
  67. Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p-Laplacian
  68. Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics
  69. Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model
  70. Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system
  71. Generalized quasi-linear fractional Wentzell problems
  72. Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space
  73. Normalized solutions for NLS equations with general nonlinearity on compact metric graphs
  74. Review Article
  75. Existence and stability of contact discontinuities to piston problems
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