USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 89–99. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

A one-dimensional nonlinear degenerate elliptic equation ^{∗}

Florin Catrina & Zhi-Qiang Wang

Abstract

We study the one-dimensional version of the Euler-Lagrange equation associated to finding the best constant in the Caffarelli-Kohn-Nirenberg inequalities. We give a complete description of all non-negative solutions which exist in a suitable weighted Sobolev space D_{a}^{1,2}(Ω). Using these results we are able to extend the parameter range for the inequalities in higher dimensions when we consider radial functions only, and gain some useful information about the radial solutions in the N-dimensional case.

1 Introduction

The motivation of this paper are the following inequalities due to Caffarelli, Kohn and Nirenberg (see [3]): for some positive constants C_{a,b}

(∫_{ℝ^{N}} |x|^{−bp}|u|^{p} dx)^{2/p} ≤ C_{a,b} ∫_{ℝ^{N}} |x|^{−2a}|∇u|^{2} dx  (1)

holds for all u ∈ C_{0}^{∞}(ℝ^{N}), if and only if

for N ≥ 2 : −∞ < a < \frac{N − 2}{2}, a ≤ b ≤ a + 1, and p = \frac{2N}{N − 2(1 + a − b)},

(2)

(with the case N = 2 excluding a = b), and

for N = 1 : −∞ < a < −\frac{1}{2}, a+ \frac{1}{2} < b ≤ a+1, and p = \frac{2}{−1 + 2(b − a)}. (3)

Let D_{a}^{1,2}(ℝ^{N}) be the completion of C_{0}^{∞}(ℝ^{N}), with respect to the norm || · ||_{a} induced by the inner product

(u, v) = ∫_{ℝ^{N}} |x|^{−2a}∇u · ∇v dx.  (4)

∗Mathematics Subject Classifications: 35J20, 35J70.

Key words: best constant, ground state solutions, wighted Sobolev inequalities.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published January 8, 2001.

Then we see that (1) holds for u ∈ D_{a}^{1,2}(ℝ^{N}). Define the best embedding constants

S(a, b) = inf_{u∈D_{a}^{1,2}(ℝ^{N})\{0}} E_{a,b}(u),  (5)

where

E_{a,b}(u) = \frac{∫_{ℝ^{N}} |x|^{−2a}|∇u|^{2} dx}{(∫_{ℝ^{N}} |x|^{−bp}|u|^{p} dx)^{2/p}} .  (6)

The extremal functions for S(a, b) are ground state solutions of the Euler equation

−div(|x|^{−2a}∇u) = |x|^{−bp}u^{p−1}, u ≥ 0, in ℝ^{N}.  (7)

This equation is a prototype of more general nonlinear degenerate elliptic equations from some physical phenomena related to equilibrium of anisotropic continuous media which possibly are somewhere perfect insulators and somewhere are perfect conductors (e.g., [8]).

Note that the classical Sobolev inequality (a = b = 0) and the Hardy inequality (a = 0, b = 1) are special cases of (1), (see also generalizations in [14] by Lin). These inequalities have been studied by many authors. In [1], [16], the best constant and the minimizers for the Sobolev inequality (a = b = 0) were given by Aubin, and Talenti. In [13], Lieb considered the case a = 0, 0 < b < 1 and gave the best constants and explicit minimizers. In [7], Chou and Chu considered the case a ≥ 0 and gave the best constants and explicit minimizers. Also, Lions in [15] (for a = 0) , and Wang and Willem in [17] (for a > 0), have established the compactness of all minimizing sequences up to dilations. The symmetry of the minimizers has been studied in [13] and [7]. In fact, all nonnegative solutions in D_{a}^{1,2}(ℝ^{N}) for the corresponding Euler equation (7) are radial solutions and explicitly given ([1], [16], [13], [7]). This was established in [7], using a generalization of the moving plane method (e.g., [9], [2], [6]). The case a < 0 has been studied recently in [11], [4], [19]; in [5] we have studied the case a < 0 and discovered some new phenomena including symmetry breaking of the ground state solutions for certain values of the parameters a and b.

In this paper we concentrate on the case N = 1. Under conditions (3), equation (7) becomes

−(|x|^{−2a}u′)′ = |x|^{−bp}u^{p−1}, u ≥ 0, in Ω ⊂ ℝ,  (8)

with Ω = ℝ, and more generally, we consider this equation in an open interval Ω ⊂ ℝ (possibly unbounded). We are interested in solutions in

D_{a}^{1,2}(Ω) := \overline{C_{0}^{∞}(Ω)}^{||·||_{a}}.

This problem is of critical case in the sense that there is a family of dilations with two parameters that leave the problems invariant (see (11) in Section 2). This feature distinguishes the case N = 1 from the case N ≥ 2, for the latter case a one-parameter family of dilations exists. Nevertheless, we are able to give a complete and detailed solution for the structure of the solutions of (8) for both the ground states and the bound states, as well as some interesting qualitative properties of solutions. First, we give the following which is a corollary of our main results.

Theorem 1.1 Let a, b, p satisfy (3) with b < a+1. Then (8) has a ground state solution in D_{a}^{1,2}(Ω) if and only if 0 ∈ \overline{Ω} and Ω is unbounded, or 0 ∉ \overline{Ω} and Ω is bounded.

Remark 1.2 Due to the degeneracy, the ground state solutions given here may not be continuous at 0 and can be identically zero in a subinterval of Ω. In fact, when Ω = (−c, ∞) with 0 < c ≤ ∞, the ground state solution is positive in [0, ∞) and equal to zero in (−c, 0). This will be clear from the proof (see also Remarks 2.4, 4.2).

On the other hand, as applications of the results for N = 1, especially of the results for Ω = ℝ_{+} (the half real line), we can expand the parameter range of the validity for inequalities (1) for N ≥ 3 when we consider radially symmetric functions only. This generalizes a result due to Lieb ([13]) where he considered the case a = 0. More precisely, for N ≥ 1 define C_{0,R}^{∞}(ℝ^{N}) = {u ∈ C_{0}^{∞}(ℝ^{N}) | u is radial} and

D_{a,R}^{1,2}(ℝ^{N}) = \overline{C_{0,R}^{∞}(ℝ^{N})}^{||·||_{a}}

Theorem 1.3 Inequality (1) holds for any u ∈ D_{a,R}^{1,2}(ℝ^{N}) if and only if

a < \frac{N − 2}{2}, a − \frac{N − 2}{2} < b ≤ a + 1 and p = \frac{2N}{N − 2(1 + a − b)}.

Remark 1.4 (i) The best constants in the above embeddings can be explicitly given (see (22)). (ii) We note that Theorem 1.3 shows that for N ≥ 3, inequalities (1) hold for radial functions in a range substantially larger than (2) for a and b. For the case a = 0, this fact was noted in [10], and it was considered again by Lieb in [13] with a direct proof rather than reduction to the one dimensional case.

The paper is organized as follows. In Section 2, we shall first recall some of our results from [5] and outline the methods there. In Section 3 we indicate how the methods can be used to study (8) on the interval (0, ∞). Theorem 1.3 will also be proved in Section 3, involving studies of equation (8) on the interval (0, ∞). The study can be carried out using ideas from our earlier work [5]. Finally in Section 4, we consider equation (8) in an interval other than the whole real line or the half real line. Theorem 1.1 follows from these results.

2 The whole real line

First, we need the following ODE

−v_{tt} + λ^{2}v = v^{p−1}, v > 0, in ℝ  (9)

with p > 2. The only positive solutions which are in H^{1}(ℝ), are translates of

v(t) = (\frac{λ^{2}p}{2})^{1/(p−2)} (cosh (\frac{p − 2}{2}λt))^{−2/(p−2)} .  (10)

We describe briefly the results we obtained in [5], for the case N = 1. Any nonzero solution of (7), u ∈ D_{a}^{1,2}(ℝ) is a critical point for the energy

E_{a,b}(u) = \frac{∫_{ℝ} |x|^{−2a}|u′|^{2} dx}{(∫_{ℝ} |x|^{−bp}|u|^{p} dx)^{2/p}} .

There is a two-sided dilation invariance of (8): for (τ_{+}, τ_{−}) ∈ (0, ∞)^{2}

u(x) → u_{τ_{+},τ_{−}}(x) = \{ \begin{matrix}τ_{+}^{− \frac{1+2a}{2}} u(τ_{+}x) & x > 0 \\ τ_{−}^{− \frac{1+2a}{2}} u(τ_{−}x) & x < 0. \end{matrix}  (11)

That is, if u is a solution of (8) then so is u_{τ_{+},τ_{−}}. More generally, the energy functional E_{a,b}(u) is invariant under these two-sided dilations.

To reveal the relation with equation (9), to a function u ∈ D_{a}^{1,2}(ℝ), we associate a ℝ^{2}-valued function w(t) = (w_{1}(t_{1}), w_{2}(t_{2})) for (t_{1}, t_{2}) ∈ C with C being the union of two real lines ℝ ∪ ℝ, where

u(x) = (−x)^{(1+2a)/2}w_{1}(− ln(−x)), for x < 0,
u(x) = x^{(1+2a)/2}w_{2}(− ln x),  for x > 0, (12)

and t_{1} = − ln(−x) for x < 0, and t_{2} = − ln x for x > 0. Under the transformation (12) we have a Hilbert space isomorphism between D_{a}^{1,2}(ℝ) and H^{1}(C, ℝ^{2}) (see [5] for details) and equation (8) is equivalent to the system of autonomous equations

−w_{tt} + (\frac{1 + 2a}{2})^{2} w = ∇W (w),  (13)

where W (w) = (|w_{1}|^{p} + |w_{2}|^{p})/p. Note that each of the two equations is the same as (9), where λ = \frac{1+2a}{2}.

Critical points of E_{a,b}(u) on D_{a}^{1,2}(ℝ) now correspond to critical points of a new energy functional on H^{1}(C, ℝ^{2})

F_{a,b}(w) = \frac{∫_{ℝ} |w_{t}|^{2} + (\frac{1 + 2a}{2})^{2} |w|^{2} dt}{(∫_{ℝ} pW (w) dt)^{2/p}}, w ∈ H^{1}(C, ℝ^{2}).

Each of the two ODE’s of (13) has the zero solution, and the only (positive) homoclinic solutions are translates of

v(t) = (\frac{(1 + 2a)^{2}}{4(1 − 2(1 + a − b))})^{\frac{1−2(1+a−b)}{4(1+a−b)}} (cosh \frac{(1 + 2a)(1 + a − b)}{1 − 2(1 + a − b)}t)^{− \frac{1−2(1+a−b)}{2(1+a−b)}} .

(14)

The minimizers of F_{a,b}(w) are achieved by w, for which one of the two components w_{1} or w_{2} is identically zero and the other is a translate of v(t) given above. Using v and F_{a,b} we have

S(a, b) = \frac{(−1 − 2a)^{2(b−a)}}{2^{2(1+a−b)}(−1 + 2(b − a))^{−1+2(b−a)}(1 + a − b)^{2(1+a−b)}}

× (\frac{Γ^{2} (\frac{1}{2(1+a−b)})}{Γ (\frac{1}{1+a−b})})^{2(1+a−b)} .  (15)

We observe that as b ↘ a + \frac{1}{2}, we obtain S(a, b) → −1 − 2a. Note that when both w_{1} and w_{2} are nonzero and are (possibly different) translates of v(t) in (14) we get the energy F_{a,b}(w) to be higher

R(a, b) = 2^{2(1+a−b)}S(a, b),

which is the least energy in the radial class. On this energy level, there is a two parameter family of positive solutions, according to the two parameters that control by how much w_{1} and w_{2} are translated from (14).

Note that the two-sided dilations (11) correspond to the translations invariance of C for (13). Correspondingly, u(x) defined in (12) is a two parameter family of solutions for (8), which possibly after a dilation given in (11) is radial in ℝ. Explicitly, u_{τ_{+},τ_{−}}(x) is equal to

\{ \begin{matrix}τ+^{\frac{1+2a}{2}} (\frac{(1+2a)^{2}}{1−2(1+a−b)})^{\frac{1−2(1+a−b)}{4(1+a−b)}} _{\frac{x^{1+2a}}{[1+(τ_{+}x)^{\frac{2(1+2a)(1+a−b)}{1−2(1+a−b)}} ]^{\frac{1−2(1+a−b)}{2(1+a−b)}}}} & x > 0 \\ τ−^{\frac{1+2a}{2}} (\frac{(1+2a)^{2}}{1−2(1+a−b)})^{\frac{1−2(1+a−b)}{4(1+a−b)}} \frac{(−x)^{1+2a}}{[1+(−τ_{−}x)^{\frac{2(1+2a)(1+a−b)}{1−2(1+a−b)}} ]^{\frac{1−2(1+a−b)}{2(1+a−b)}}} & x < 0. \end{matrix}

(16)

Summarizing these, we have the following theorems from [5].

Theorem 2.1 (Best constants and existence of ground states) Let a, b, p satisfy (3) with b < a + 1.Then S(a, b) is explicitly given in (15), and up to a dilation of the form (11) it is achieved at a function of the form (12) with either w_{1} = 0 and w_{2} given by (14) or vice versa. Consequently, the ground states for S(a, b) are always nonradial.

Theorem 2.2 (Bound state solutions) Let a, b, p satisfy (3) with b < a + 1. Then the only bound state solutions of (8) besides the ground state solutions are given by (16).

Remark 2.3 In [5] we also proved the nonexistence of extremal functions when b = a + 1 for which S(a, a + 1) = (\frac{1+2a}{2})^{2}, as well as the asymptotic property of S(a, b) as b → (a + \frac{1}{2})^{+}. Note that all solutions of (8), possibly after a dilation

given in (11), satisfy the modified inversion symmetry u(x) = |x|^{1+2a}u (\frac{x}{|x|^{2}} ). This was also discovered in [5] for bound state solutions when N ≥ 2.

Remark 2.4 Standard elliptic regularity arguments break down for problem (8) at x = 0 due to the existence of the weights. In fact, solutions of (8) need not even be continuous at x = 0. We observe that the ground states are never continuous and that the only bound states other than the ground states are continuous if and only if τ_{+} = τ_{−} in (16). By a direct computation we can see easily that the solutions that are continuous belong to C^{1}(ℝ) if and only if b < \frac{(1+2a)^{2}}{4a}.

3 Half line domain

Recall from Section 1, for an open interval Ω ⊂ ℝ (possibly unbounded), we let

D_{a}^{1,2}(Ω) = \overline{C_{0}^{∞}(Ω)}^{||·||_{a}}.

We follow the same idea from [5], as used in Section 2. Under the transformation

u(x) = x^{(1+2a)/2}v(− ln x),  (17)

we still have a Hilbert space isomorphism between D_{a}^{1,2}(Ω) and H_{0}^{1}(Ω̃) where Ω̃ ⊂ C is the image of Ω. Especially when Ω = (0, ∞), Ω̃ is ℝ, one of the two components of C. In this section we look at the problem

−(|x|^{−2a}u′)′ = |x|^{−bp}u^{p−1}, u ≥ 0, u ∈ D_{a}^{1,2}(Ω)  (18)

with Ω = (0, ∞). Under the above transformation, equation (18) becomes (9) with λ = \frac{1+2a}{2}. According to (10), after transformed back to Ω, the only solutions of (18) are

u_{τ} (x) = τ^{\frac{1+2a}{2}} (\frac{(1 + 2a)^{2}}{1 − 2(1 + a − b)})^{\frac{1−2(1+a−b)}{4(1+a−b)}} \frac{x^{1+2a}}{(1 + (τx)^{\frac{2(1+2a)(1+a−b)}{1−2(1+a−b)}} )^{\frac{1−2(1+a−b)}{2(1+a−b)}}} .

(19)

Let E_{a,b}(u, Ω) be the restriction of E_{a,b}(u) on D_{a}^{1,2}(Ω). We have the following result.

Theorem 3.1 Let a, b, p satisfy (3) with b < a + 1. Then the best constant inf_{D_{a}^{1,2}(Ω)\{0}} E_{a,b}(u, Ω) is achieved by a family of functions given by (19). Moreover, all nontrivial bound state solutions of (18) are given by (19).

As a consequence of the above result we can prove the inequality in Theorem 1.3 for radial functions in any space dimension.

Proof of Theorem 1.3: We work in the class of radial functions u(|x|) = u(r) defined on ℝ^{N}. Denote by ω_{N−1} the area of the unit N − 1 dimensional sphere in ℝ^{N}. Then

∫_{ℝ^{N}} |x|^{−2a}|∇u(x)|^{2} dx = ω_{N−1} ∫_{0}^{∞} r^{N−1−2a}u′(r)^{2} dr,  (20)

and

∫_{ℝ^{N}} |x|^{−bp}|u(x)|^{p} dx = ω_{N−1} ∫_{0}^{∞} r^{N−1−bp}|u(r)|^{p} dr.  (21)

If we denote

ā = a − \frac{N − 1}{2}, and b̄ = b − \frac{N − 1}{2} + \frac{(1 + a − b)(N − 1)}{N},

then

p = \frac{2}{−1 + 2(b̄ − ā)}.

With a proof similar to Lemma 2.1 in [5], we have

D_{a,R}^{1,2}(ℝ^{N}) = \overline{C_{0,R}^{∞}(ℝ^{N} \ \{0\})}^{||·||_{a}},

where C_{0,R}^{∞}(ℝ^{N} \ {0}) is the space of radial, smooth functions with compact support in ℝ^{N} \ \{0\}. Since inequalities (1) hold for ā + \frac{1}{2} < b̄ ≤ ā + 1 for functions with compact support in ℝ, they also hold for functions with compact support in (0, ∞). Hence, for b in the interval (a − \frac{N−2}{2}, a + 1], inequalities (1) still hold in the class of radial functions. In fact, with a, b, and p satisfying (2), if we denote by R(a, b) the best constant for the embedding D_{a,R}^{1,2}(ℝ^{N}) into the weighted L_{b}^{p}(ℝ^{N}), we have

R(a, b) = ωN−1^{\frac{p−2}{p}}S(ā, b̄),  (22)

where S(ā, b̄) is given in (15).

When looking for solutions of (7) in D_{a,R}^{1,2}(ℝ^{N}), the results in (0, ∞) solve the problem by symmetry reduction. Equation (7) becomes

−r^{−2a}u′′ − (N − 1 − 2a)r^{−2a−1}u′ = r^{−bp}u^{p−1}.  (23)

Multiplying the reduced equation by r^{N−1}, we obtain

−(r^{N−1−2a}u′)′ = r^{N−1−bp}u^{p−1}.  (24)

Therefore, if −\frac{N−2}{2} < b − a < 1, the only positive radial solutions are those in (19) corresponding to ā and b̄. □

4 Domains other than ℝ or ℝ_{+}

In this section we look at problem (18), with Ω different from ℝ or ℝ_{+}. We divide the section in three cases according to the position of zero, relative to the interval Ω.

A. We consider 0 ∉ \overline{Ω}.

(i) The first subcase is Ω is bounded. From (12), Ω̃ is a bounded interval in one of the two components of C. It is well-known that (9) on such an interval with Dirichlet boundary condition, has a unique positive solution (see e.g. [18], [12]).

(ii) In the second subcase, Ω is unbounded which corresponds to Ω̃ being of the form (−∞, m) or (m, ∞) for a finite m. We use a simple Pohazaev type argument to show that problem (18) has no solution in D_{a}^{1,2}(Ω). For convenience, assume Ω ⊂ (0, ∞). Performing the transformation

u(x) = x^{(1+2a)/2}v(− ln x),  (25)

(denote t = − ln x), equation (18) is equivalent to

−v_{tt} + (\frac{1 + 2a}{2})^{2} v = v^{p−1}, v(t) > 0 on (−∞, m), and v(m) = 0,

where m is some real number. Assuming there is a solution v(t), we multiply by v_{t}(t) and integrate from s ∈ (−∞, m) to m. We get,

v_{t}^{2}(m) = v_{t}^{2}(s) − (\frac{1 + 2a}{2})^{2} v^{2}(s) + \frac{2}{p}v^{p}(s).

Integrate the equality above again, from t to m, to obtain

(m − t)v_{t}^{2}(m) = ∫_{t}^{m} v_{t}^{2}(s) − (\frac{1 + 2a}{2})^{2} v^{2}(s) + \frac{2}{p}v^{p}(s) ds.  (26)

Since u ∈ D_{a}^{1,2}(Ω) we have that v ∈ H_{0}^{1}(−∞, m) and also in L^{p}(−∞, m). In equality (26), let t → −∞. The right hand side is bounded and this implies v_{t}(m) = 0, hence v is identically zero, which provides the necessary contradiction.

B. The second case is when zero is one of the endpoints of Ω.

(i) If Ω is bounded, then after the transformation (25) we have the same situation as in A(ii), hence there is no solution.

(ii) If Ω is unbounded, we are in the case of the half line treated in Section 3.

C. Finally, the case when 0 ∈ Ω.

(i) If Ω is bounded, we have no solution by the argument in A(ii), carried out on each of the two components of Ω \ {0}.

(ii) If Ω is unbounded, and not equal to ℝ, then Ω\{0} has one component equal to the half line and the other one being a bounded segment. This translates into Ω̃ being the union of a copy of ℝ and an unbounded interval of ℝ. By using a combination of arguments above we conclude that the only nontrivial solution is (19) on the half line, and zero on the bounded segment.

Summarizing the above we have the following complete solution for (8) as we have treated the cases of the whole real line and the half real line in Sections 2 and 3 respectively.

Theorem 4.1 Let a, b, p satisfy (3) with b < a + 1.

(i) Assume Ω is not equal to ℝ or ℝ_{+} and 0 ∈ \overline{Ω}. If Ω is bounded, (18) has no solutions at all. If Ω is unbounded (18) has a unique solution (up to a dilation) which is the ground state, ann it is zero on the bounded component of Ω \ {0}.

(ii). Assume 0 ∉ \overline{Ω}. If Ω is bounded, (18) has a unique solution which is the ground state. If Ω is unbounded (18) has no solution at all.

Now, Theorem 1.1 follows from Theorems 2.1, 3.1, and 4.1.

Remark 4.2 We comment here that a result in the spirit of Theorem 1.1 was given for the case b = 0, −1 < a < −\frac{1}{2} in [4] (p.386, Theorem 4.1) in which it was claimed that (18) has a ground state solution in D_{a}^{1,2}(Ω) if and only if Ω = ℝ, ℝ_{±}, or Ω is bounded with 0 ∉ \overline{Ω}. Here we have shown that (18) also has ground state solutions in D_{a}^{1,2}(Ω) when Ω = (−c, ∞) and (−∞, c) for any c ≥ 0. The reason is that due to the degeneracy the maximum principle does not hold in intervals that contain 0. Therefore there are nonzero nonnegative solutions which are identically zero in a subinterval.

Remark 4.3 For completeness, we remark that for b = a + 1 we have p = 2, and (18) becomes a linear problem:

−(|x|^{−2a}u′)′ = |x|^{−2(a+1)}u, u ≥ 0, in Ω = (α, β) ⊂ ℝ,  (27)

Using transformation (12) and analyzing the resulting equation we can easily conclude that (27) has a nonzero solution if and only if −\frac{3}{2} < a < −\frac{1}{2}, 0 ∉ Ω and \frac{β}{α} = exp(±\frac{2π}{\sqrt{4−(1+2a)^{2}}} ) where ± depends on the sign of β. We leave the

details to reader.

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Florin Catrina & Zhi-Qiang Wang Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA. e-mail: wang@sunfs.math.usu.edu