Supercritical Hénon-type equation with a forcing term

1 Introduction

We are interested in the structure of solutions to the elliptic problem for a Hénon-type equation with a forcing term:

where N ≥ 3 , p > 1 , κ > 0 , and μ is a nontrivial nonnegative Radon measure in R N , and α is a positive continuous function in R N \ { 0 } such that

for some a > − 2 and b ∈ R . Nonlinear elliptic equations with forcing terms in R N arise naturally in the study of stochastic processes. In particular, problem ( P κ ) with α ≡ 1 in R N appeared in establishing some limit theorems for super-Brownian motion (see, e.g., [6,8,10, 20] for a brief history and background of problem ( P κ ) ).

In this article, under suitable assumptions on the exponent p and the forcing term μ , we show the existence of a threshold κ * > 0 with the following properties:

1. problem ( P κ ) possesses a minimal solution if 0 < κ < κ * ;

2. problem ( P κ ) possesses a unique solution if κ = κ * ;

3. problem ( P κ ) possesses no solutions if κ > κ * .

This article is motivated by [16], which treats the elliptic problem for the scalar field equation with a forcing term:

where N ≥ 2 , p > 1 , and μ is a nontrivial nonnegative Radon measure in R N with compact support. In [16], under the following condition on μ :

• G * μ ∈ L q ( R N ) for some q ∈ ( p , ∞ ] with q > N ( p − 1 ) ∕ 2 , where G is the fundamental solution to the elliptic operator − Δ + 1 in R N ,

Unfortunately, the arguments in [16] depend heavily on the exponential decay of the fundamental solution G at the space infinity, and they are not applicable to our problem ( P κ ) (see, e.g., [17,18,22,23] for related results on problem (S)).

We introduce some notations. In what follows, unless otherwise stated, let N ≥ 3 . For any R > 0 , let

We denote by ℳ + the set of all nontrivial nonnegative Radon measures in R N . Let Γ be the fundamental solution to − Δ v = 0 in R N , i.e.,

where ω N is the volume of the ball B ( 0 , 1 ) in R N . We denote by D 1 , 2 the completion of C c ∞ ( R N ) with respect to the norm ‖ ∇ ⋅ ‖ L 2 ( R N ) . It follows from the Sobolev inequality that

For any q ∈ [ 1 , ∞ ] and β , γ ∈ R , we define

where

Note that if f ∈ C ( R N \ { 0 } ) satisfies f ( x ) = O ( ∣ x ∣ β ) as ∣ x ∣ → 0 and f ( x ) = O ( ∣ x ∣ γ ) as ∣ x ∣ → ∞ , then f ∈ L β , γ q for any q ∈ [ 1 , ∞ ] .

We define solutions to problem ( P κ ) .

Now, we are ready to state our results on problem ( P κ ) . Theorem 1.1 concerns properties (A1) and (A3), and it shows the existence of a threshold on the existence/nonexistence of solutions to problem ( P κ ) .

We note that under Condition (1.1), problem ( P κ ) possesses no solutions generally if:

See, e.g., [7, Theorem 3.3]. (See also [13] and [24, Section 8.1] for the case of α ≡ 1 in R N .)

Theorem 1.2 concerns property (A2), and it is the main result of this article. Let

for η ∈ R . Then, p J L = p * ( 0 ) and

We remark that if p ≥ p * ( a − ) , even the existence of solutions to problem ( P κ ) with κ = κ * is open.

As corollaries of Theorems 1.1 and 1.2, we have the following results in the case of α ≡ 1 in R N , i.e., the elliptic problem for the Lane-Emden equation with a forcing term:

where N ≥ 3 , p > 1 , κ > 0 , and μ ∈ ℳ + .

Assumption (1.6) is more general than that of (1.4) with c = 0 . Indeed, if μ satisfies (1.4) with c = 0 , then ‖ Γ * μ ‖ L r ( A ( 0 , R ) ) = O ( R N r ) as R → + 0 . On the other hand, assumption (1.6) does not require any decay estimates of ‖ Γ * μ ‖ L r ( A ( 0 , R ) ) as R → + 0 .

In the proofs of our theorems, for any κ > 0 , we set U − 1 κ ≡ 0 in R N and define approximate solutions { U j κ } to problem ( P κ ) and their differences { V j κ } inductively by:

for a.a. x ∈ R N . By induction, we easily see that

for a.a. x ∈ R N and j = 0 , 1 , 2 , … . Assume the same conditions as in Theorem 1.1. Define

We find j * ∈ { 0 , 1 , 2 , … } such that

(see Lemma 2.4), and set w ≔ u − V j * κ for κ ∈ K * . Then, u is a solution to problem ( P κ ) if and only if w satisfies an integral equation:

Applying the same arguments as in [16], we see that

if K * ≠ ∅ (see Lemma 3.1). Furthermore, we apply the contraction mapping theorem in L 0 , − N + 2 ∞ to find a function w ∈ L 0 , − N + 2 ∞ satisfying (1.11) for all small enough κ > 0 . This means that

(see Lemma 3.2). In addition, for any κ ∈ K * , we find the first eigenvalue λ κ and its corresponding positive eigenfunction φ κ ∈ D 1 , 2 to the linearized eigenvalue problem ( E κ ) to problem ( P κ ) at u κ (see Lemma 5.1). Then, thanks to elliptic regularity theorems and the Kelvin transformation, we see that φ κ ∈ L 0 , − N + 2 ∞ for κ ∈ K * (see Lemma 5.2) and λ κ > 1 for κ ∈ ( 0 , κ * ) (see Lemma 5.3). We also approximate α ( u κ ) p − 1 by functions in L c ∞ to obtain

for all κ ∈ ( 0 , κ * ) (see Lemma 5.4). This leads to that κ * < ∞ and completes the proof of Theorem 1.1.

For the proof of Theorem 1.2, we obtain a uniform energy estimate of { ( w κ ) ν } κ ∈ ( 0 , κ * ) in B ( 0 , 2 ) for all ν ≥ 1 with ν 2 ∕ ( 2 ν − 1 ) < p . Then, under a suitable condition on p , ν , and a , we apply elliptic regularity theorems to obtain a uniform L ∞ ( B ( 0 , 1 ) ) -estimate of { w κ } κ ∈ ( 0 , κ * ) . We also apply the same arguments to the Kelvin transformation of w κ . Then, we obtain a uniform L 0 , − N + 2 ∞ -estimate of { w κ } κ ∈ ( 0 , κ * ) , and show that κ * ∈ K * . Furthermore, we prove that λ κ * = 1 and the uniqueness of solutions to problem ( P κ ) with κ = κ * . Finally, thanks to (1.14), we obtain κ * = κ * and complete the proof of Theorem 1.2.

The rest of this article is organized as follows. In Section 2, we obtain preliminary results on the function space L β , γ q (see Section 2.1) and the functions { U j κ } and { V j κ } (see Section 2.2). In Section 3, we prove (1.12) and (1.13). In Section 4, we study the relation between problem ( P κ ) and the Kelvin transformation. In Section 5, we study the linearized eigenvalue problem to problem ( P κ ) at u κ and prove Theorem 1.1. Section 6 is devoted to uniform L ∞ ( B ( 0 , 1 ) ) -estimates of { w κ } κ ∈ ( 0 , κ * ) and their Kelvin transformations. In Section 7, we complete the proof of Theorem 1.2. Furthermore, we obtain Corollaries 1.1 and 1.2.

2 Preliminary

In this section, we collect some properties of the function space L β , γ q . Furthermore, we obtain some estimates of the functions { U j κ } and { V j κ } . In what follows, we use C to denote generic positive constants and point out that C may take different values within a calculation.