Special Issue in honor of John W. Neuberger

Electronic Journal of Differential Equations, Special Issue 02 (2023), pp. 87–100.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

REGULAR SOLUTIONS TO ELLIPTIC EQUATIONS

ALFONSO CASTRO, JON JACOBSEN

Dedicated to our admired friend John W. Neuberger

Abstract. A review of results and techniques on the existence of regular ra-

dial solutions to second order elliptic boundary value problems driven by linear
and quasilinear operators is presented. Of particular interest are results where

the solvability of a given elliptic problem can be analyzed by the relationship

between the spectrum of the operator and the behavior of the nonlinearity
near infinity and at zero. Energy arguments and Pohozaev type identities are

used extensively in that analysis. An appendix with a proof of the contraction

mapping principle best suited for using continuous dependence to ordinary
differential equations on initial conditions is presented. Another appendix on

the phase plane analysis as needed to take advantage of initial conditions is

also included. For studies on singular solutions the reader is referred to Ardila
et al., Milan J. Math (2014) and references therein.

1. Introduction

Motivated by the solvability of boundary value problems such as

− u′′(t) = g(t, u(t)) for t ∈ (0, π), u(0) = 0, u(π) = 0, (1.1)

extensive understanding of the solutions of equations of the type

−∆u(x) = g(x, u(x)) for x ∈ Ω, u(x) = 0 for x ∈ ∂Ω, (1.2)

has been achieved when Ω ⊆ RN is either a ball centered at the origin, an annulus
(the region between two concentric balls in RN ) centered at the origin, or RN , and g
is radial in its first variable, i.e., g(x, u) = g(y, u) if ‖x‖ = ‖y‖. In (1.2), ∆ denotes
the Laplace operator ∂2

x1
+ · · · + ∂2

xN
. Here we present a summary of results and

techniques involved in the analysis of such equations.
For a general bounded region W there is a sequence 0 < λ1 < λ2 ≤ · · · → +∞

and corresponding functions ϕ1, ϕ2, . . . such that

−∆ϕi(x) = λiϕi(x) for x ∈W, ϕ(x) = 0 for x ∈ ∂W. (1.3)

The set σ(−∆) = {λi : i = 1, 2, . . .} is known as the spectrum of −∆ subject
to the boundary condition in (1.2). The eigenvalue λ1 is simple, all others have

2020 Mathematics Subject Classification. 3502, 35J25, 34B16, 34B32.
Key words and phrases. Nonlinear elliptic equation; radial solution; regular radial solution;

singular radial solution; bifurcation analysis; Pohozaev identity; shooting method;
superlinear nonlinearity; subcritical nonlinearity; sub-super critical nonlinearity;
jumping nonlinearity.
©2023 This work is licensed under a CC BY 4.0 license.

Published March 27, 2023.

87



88 A. CASTRO, J. JACOBSEN EJDE/SI/02

finite multiplicity. When W is a ball or an annulus, σ(−∆) contains a subsequence
0 < ρ1 = λ1 < ρ2 < · · · → +∞ with all the ρi’s having a radial eigenfunction. The
eigenvalues ρi are simple in the space of radial functions. The general theme is that
the solvability of (1.2) is determined, to a large extent, by the relation between
the range of g(x, u)/u and the spectrum σ(−∆). Similarly, when Ω is a ball or
an annulus and g is radial in x, the existence of radial solutions to (1.2) is largely
determined by the relation between g(x, u)/u and the set σrad(−∆) = {ρi : i =
1, 2, . . .}.

This motivates classifying the nonlinearity g according to the following behaviors:

Definition 1.1 (Classification of nonlinearities). We say that

• g is asymptotically linear if lim|u|→+∞ g(x, u)/u ∈ R.
• g is resonant if lim|u|→+∞ g(x, u)/u ∈ σ(−∆).
• g is a jumping nonlinearity if limu→+∞ g(x, u)/u, limu→−∞ g(x, u)/u ∈ R

with limu→+∞ g(x, u)/u 6= limu→−∞ g(x, u)/u.
• g is superlinear if limu→+∞ g(x, u)/u = +∞.

• g is superlinear with subcritical growth if lim|u|→+∞
g(x,u)
|u|p−1u ∈ R with 1 <

p < N+2
N−2 .

• g is superlinear with critical growth if lim|u|→+∞
g(x,u)
|u|p−1u ∈ R with p = N+2

N−2 .

• g is superlinear with supercritical growth if lim|u|→+∞
g(x,u)
|u|p−1u ∈ R with

p > N+2
N−2 .

• g is sub-super critical if limu→+∞
g(x,u)
up ∈ R and limu→−∞

g(x,u)
|u|q−1u ∈ R,

with 1 < p < N+2
N−2 < q <∞.

All limits are assumed to be uniform with respect to x.

Writing the Laplace operator in spherical coordinates one sees that the regular
(classical) radial solutions of the partial differential equation (1.2) are solutions to
the ordinary differential equation

− u′′(r)− N − 1

r
u′(r) = g(r, u(r)), r ∈ I, Bu = 0, (1.4)

where I = (0, b) when Ω is the ball of radius b centered at the origin and Bu =
u′(0) = u(b), I = (a, b) with 0 < a < b when Ω is an annulus and B(u) = u(a) =
u(b), and I = (0,+∞) when Ω = RN and B(u) = u′(0) = u(∞). When Ω is an
annulus the differential equation in (1.4) is a regular ordinary differential equation,
when Ω is a ball it is a singular ordinary differential equation with a singularity at
0 and when Ω = RN , by transforming (0,∞) into a bounded interval we can view
the differential equation in (1.4) as a singular ordinary differential equation with
singularities at both ends of the interval. In order to simplify the presentation we
will assume that g(r, u) = g(u) with g differentiable. See [12] for recent results on
the case g(r, u) = g(u)W (r) and W is sign-changing.

Under additional hypothesis on g, see [30], the positive solutions to (1.4) are the
only positive solutions to (1.2). For sign-changing solutions this is not valid. For
example, when Ω is a ball or annulus and g is superlinear and odd-like, (1.2) may
have infinitely many non-radial solutions [1, 2, 9]. See also [3, 11].

Under adequate conditions on N and g, the second order differential equation
in (1.4) has solutions in the sense of distributions that blow up at the origin



EJDE-2023/SI/02 REGULAR SOLUTIONS 89

(limr→0+ u(r) = +∞). The reader is referred to [6] for a study on such solu-
tions. Finally we remark that the results here presented extend to more general
boundary conditions such as ∂u/∂η = 0, αu + β∂u/∂η = 0, and other nonlinear
boundary conditions.

2. The shooting method and phase plane analysis

Much of the understanding of solutions to (1.4) is based on the analysis of the
initial value problem

− u′′(r)− N − 1

r
u′(r) = g(u) r ∈ I, u(0) = d, u′(0) = 0, (2.1)

when Ω is a ball or RN and

− u′′(r)− N − 1

r
u′(r) = g(u) r ∈ I, u(a) = 0, u′(a) = d, (2.2)

when Ω is an annulus. Arguments based on the Contraction Mapping Principle (see
Appendix A) show that if g is locally Lipschitz and g′ satisfies adequate growth
conditions then (2.1) and (2.2) have unique solutions that are defined on [a, b] and
depend continuously on the initial value d. This is not the case if g(r, u) = W (r)g(u)
and W is sign-changing. For such cases, the solutions to (2.1) may blow-up where
W is negative (see [12]).

For any constant d, a solution of (2.1) that satisfies u(b) = 0 is a solution of
(1.4) when Ω is the ball of radius b centered at the origin. Similarly, a solution of
(2.2) that satisfies u(b) = 0 is a solution of (1.4) when Ω is the annulus of inner
radius a and outer radius b. This observation motivates the shooting method based
on studying the behavior of solutions to (2.1) in the (u, u′) plane as d varies. To
indicate the dependence on d we let u(t; d) denote a solution of (2.1), but when the
role of d is not essential we will continue to use u or u(r) to denote the solution.
We outline the key steps and refer to [17, 19] for more details.

Suppose that g is superlinear (see Definition 1.1) and let u be the solution of
(2.1). Applying the integrating factor rN−1 to (2.1) yields

−rN−1u′(r) =

∫ r

0

sN−1g(u(s)) ds.

Let k ∈ (0, 1). For a given d > 0 let r0 > 0 be the first value such that u(r0; d) =
kd and u(r; d) > kd for r ∈ (0, r0). If in addition g has subcritical, critical, or
supercritical growth, for 0 < r ≤ r0 it follows that

− u′(r) = O(rdp). (2.3)

Integrating from 0 to r0 one sees that r0 = O(d
1−p
2 ). Combining this estimate with

the energy

E(r) =
(u′(r))2

2
+G(u(r)), (2.4)

where G(u) =
∫ u

0
g(s) ds, and Pohozaev energy

P (r) = rN−1
(
rE(r) +

N − 2

2
u(r)u′(r)

)
=

∫ r

0

sN−1
(
NG(u(s))− N − 2

2
u(s)g(u(s))

)
ds,

(2.5)



90 A. CASTRO, J. JACOBSEN EJDE/SI/02

we can gain insight into the behavior of the solutions to (2.1). For example, using

r0 = O(d
1−p
2 ) and assuming δup ≤ g(u) ≤ up for some δ ∈ (0, 1], for d large, there

exists positive constants K,M such that

rNE(r) ≥ KrN0 E(r0) ≥ KrN0 (G(kd)) ≥MdN( 1−p
2 )dp+1.

In the subcritical growth case, 1 < p < N+2
N−2 and so the exponent on d will be

positive in which case E(r) → ∞ as d → ∞. From these estimates if we consider
phase plane dynamics for (u(r; d), u′(r; d)) and define

ρ2(r; d) = (u(r; d))2 + (u′(r; d))2

then ρ2(r; d) → ∞ as d → ∞, uniformly for r ∈ [0, R]. Therefore, there exists a
unique continuous argument function θ(r; d) such that

u(r; d) = ρ(r; d) cos θ(r; d) (2.6)

u′(r; d) = −ρ(r; d) sin θ(r; d) (2.7)

with θ(0; d) = 0 (see Appendix B). A direct calculation shows

θ′(r; d) = sin2 θ(r; d) +
cos θ(r; d)

ρ(r; d)

(N − 1

r
u′ + g(u)

)
.

Using this one can show that for any fixed R > 0 the argument θ(R; d) → ∞
as d → ∞. Since θ is continuous one can use the intermediate value theorem
to construct an infinite sequence of values d1 < · · · < dk < · · · → ∞ such that
θ(R; dk) = kπ + π

2 (hence u(R; dk) = 0), which will be a radial solution of (1.4)
on the ball of radius R, establishing infinitely many solutions for the ball [17]. A
similar analysis can be used to establish infinitely many radial solutions on the ball
for sub-super critical nonlinearities, see [15].

The Pohozaev energy (2.5) utilized above is obtained by multiplying (2.1) by
µ(r) = φ(r)u + ψ(r)u′ for some to be determined coefficients φ, ψ and integrating
by parts:

− φuu′ − ψ

2
(u′)2

∣∣β
α

+

∫ β

α

(
φ′ − N − 1

r
φ
)
uu′

+
(
φ+

ψ′

2
− N − 1

r
ψ
)

(u′)2 dr

=

∫ β

α

µ(r)g(u) dr.

(2.8)

Choosing φ = rN−1 will eliminate the uu′ integral. Similarly, for this choice of φ
the function ψ = 2

N−2r
N will eliminate the (u′)2 integral, reducing (2.8) to

− rN−1uu′ − rN

N − 2
(u′)2

∣∣β
α

=

∫ β

α

rN−1ug(u) +
2

N − 2
rNu′g(u) dr. (2.9)

Integrating the u′g(u) = (G(u))′ term and multiplying by −N−2
2 this simplifies to

rN−1
(
rE(u) +

N − 2

2
uu′
)∣∣∣β
α

=

∫ β

α

(
NG(u)− N − 2

2
ug(u)

)
rN−1 dr. (2.10)

In particular, with α = 0 and β = r0 we have

P (r0) =

∫ r0

0

(
NG(u)− N − 2

2
ug(u)

)
rN−1 dr. (2.11)



EJDE-2023/SI/02 REGULAR SOLUTIONS 91

For example, if f(u) = |u|p−1u then the integrand is negative for p > N+2
N−2 hence

the integral is negative, but we showed earlier that P (r0) > 0, a contradiction, so it
follows from the Pohozaev energy that there are no radial solutions of (2.1) when
p > N+2

N−2 .

3. Bifurcation analysis

Suppose now that ug(u) > 0 for |u| 6= 0, and g′(0) > 0. For λ > 0 consider the
equation

− u′′(r)− N − 1

r
u′(r) = λg(u) r ∈ [0, 1), u′(0) = 0, u(1) = 0. (3.1)

For g(u) = u and u(0) 6= 0, the solutions to (3.1) are the radial eigenfunctions and
corresponding eigenvalues to (1.3) when Ω is the unit ball centered at the origin.
We will denote such eigenvalues by ρ1 < ρ2 < · · · < ρk < · · · → +∞. It turns out
that such eigenvalues are simple in the space of radial functions. That is, for such
eigenvalues, all the radial eigenfunctions are linearly independent. Also, ρ1 = λ1.

Since g(0) = 0, u0 ≡ 0 is a solution to (3.1) for any λ > 0 (the trivial branch).
We are interested in nontrivial branches of solutions that bifurcate from the trivial
branch at some value of λ. If λ̂g′(0) 6∈ σ(−∆), arguments based on the implicit
function theorem show that there exists ε > 0 such that u0 is the only solution to

(3.1) for |λ− λ̂| < ε and |u| < ε. This begs the question: What if λ̂g′(0) ∈ σ(−∆)?
The answer is given by the Crandall-Rabinowitz Theorem [23, Theorem 1.7] which
captures the nature of the bifurcation that occurs. Let

Y =
{
y : [0, 1]→ R : y is continuous and

∫ 1

0

y(r)ϕk(r)rN−1dr = 0
}
.

The Crandall-Rabinowitz theorem says that since the eigenvalue ρk = λ̂g′(0) is
simple there exists ε > 0 and continuous functions v : (−ε, ε)→ Y and L : (−ε, ε)→
R such that v(0) = L(0) = v′(0) = L′(0) = 0 and all nontrivial solutions to (3.1) in

a neighborhood of (λ̂, 0) = ( ρk
g′(0) , 0) are of the form

λ(t) =
ρk
g′(0)

+ L(t) and u(t) = tϕk + v(t). (3.2)

General global bifurcation results (e.g., [24]) imply that (3.1) has two connected
sets of solutions Sk,1 and Sk,2 where Sk,1 contains {(λ(t), u(t)) : t ∈ (0, ε)} and
Sk,2 contains {(λ(t), u(t)) : t ∈ (−ε, 0)}. Moreover, such branches of solutions are
either unbounded or contain two points of the form (ρk, 0) in their closure. In our
case, since g(0) = 0, by the uniqueness of solutions to initial value problems, if
(λ, u) is a solution to (3.1) and u(x) = 0 then u′(x) 6= 0. This implies that if S is a
connected set of solutions to (3.1) the number of zeros for nontrivial solutions along
the branch is constant and u(0) does not change sign. Thus we may assume that
u(0) > 0 for (λ, u) ∈ Sk,1 and u(0) < 0 for (λ, u) ∈ Sk,2. In particular, it follows
that Sk,j ∩ Sn,m = ∅ if (k, j) 6= (n,m). Hence no branch can have two points of
the form (ρk, 0) in its closure and so all continua Sk,j are unbounded. With more
precise information about the nonlinearity g one can refine the behavior of these
unbounded continua to establish existence and multiplicity results. For example,
Figures 1 and 2 illustrate the solution branches for the superlinear subcritical case
and the asymptotically linear case, which we discuss in the next section.



92 A. CASTRO, J. JACOBSEN EJDE/SI/02

6

-

‖u‖∞

λρ1
g′(0)

ρ2
g′(0)

ρ3
g′(0)

. . .

. . . ρk
g′(0)

|
. . .

. . .
|

Figure 1. Bifurcation curves for superlinear g with subcritical growth.

4. Asymptotically linear nonlinearities

For the sake of simplicity in the notations we write:

g′(+∞) = lim
u→+∞

g(u)

u
, g′(−∞) = lim

u→−∞

g(u)

u
, g′(∞) = lim

|u|→+∞

g(x, u)

u
, (4.1)

when they exist.
Let us illustrate how to combine the above bifurcation arguments to prove the

following theorem.

Theorem 4.1. If 0 < g′(0) ∈ (ρj , ρj+1), 0 < g′(∞) ∈ (ρk, ρk+1), ug(u) > 0 for
u 6= 0, then (1.4) has 2|k − j|+ 1 solutions.

Proof. Without loss of generality we may assume that j ≤ k. If j = k, taking u = 0
we have the conclusion of the theorem. Thus we assume j < k. As stated in Section
3, for each i = 1, 2, . . ., there exist two unbounded connected sets of solutions
Si,1 and Si,2 bifurcating from (ρi/g

′(0), 0). Taking a sequence (ρn, un) ∈ Si,τ , for
τ ∈ {1, 2}, with limn→+∞ ‖un‖ = +∞ one sees that {ρn} converges to ρi/g

′(∞).
Since ρi/g

′(0) > 1 > ρi/g
′(∞) for i = j + 1, . . . , k, the bifurcation curve Si,1

contains elements of the form (λ, u) with λ close to λi/g
′(0) > 1 and u close to

0. Since Si,1 is unbounded it contains elements of the form (λ, u) with λ close to
ρi/g

′(∞) < 1. By the connectedness of Si,1 there must exist (λ, ui,1) ∈ Si,1 with
λ = 1, in which case ui,1 is a solution to (1.4). Similarly, one has a solution ui,2



EJDE-2023/SI/02 REGULAR SOLUTIONS 93

6

-

‖u‖∞

λρ1
g′(0)

ρ2
g′(0)

ρ3
g′(0). . .

ρk
g′(0) . . .

. . . . . .

ρ1
g′(∞)

ρk
g′(∞)

. . . ρk
g′(0)

1

Sk,1

Figure 2. Bifurcation curves for asymptotically linear g, g′(0) <
λ1 < ρk < g′(∞).

to (1.4) in Si,2. Hence 0, uk+1,1, uk+1,2, . . . , uj,1, uj,2 are 2(j − k) + 1 solutions to
(1.2), which proves the theorem. �

Figure 2 provides a graphical description of the foregoing proof.
In the absence of the hypothesis ug′(u) > 0 for u 6= 0 additional solutions may

arise. For example, the following result is established in [10]:

Theorem 4.2. If g′(0) = g′(∞) ∈ (ρj ,+∞), and g(z) = 0 for some z > 0 then
(1.4) has 4j − 1 solutions.

Figure 3 suggests the proof of Theorem 4.2.

5. Jumping nonlinearities

In [5], for any bounded region in RN , the solvability of the boundary value
problem

−∆u(x) = g(u(x)) + v(x) + cϕ1(x) for x ∈ Ω, u(x) = 0 for x ∈ ∂Ω, (5.1)

was considered assuming that g is a jumping nonlinearity with g convex and

g′(−∞) < λ1 < g′(+∞) < λ2, (5.2)

where λ1 < λ2 are the smallest eigenvalues of the Laplacian in Ω subject to the
boundary condition in (5.2), ϕ1 a positive eigenfunction corresponding to the eigen-
value λ1, v ∈ L2(Ω) with

∫
Ω
v(x)ϕ1(x)dx = 0, and c ∈ R. The following was proven



94 A. CASTRO, J. JACOBSEN EJDE/SI/02

in [5]:
For each v ∈ L2(Ω) there exists a real number D(v) such
that:

(a) if c > D(v) then (5.1) has no solution,
(b) if c = D(v) then (5.1) has exactly one solution; and
(c) if c < D(v) then (5.1) has exactly two solutions.

(5.3)

Such a precise result attracted a great deal of interest motivating questions such as:
What if the interval (g′(−∞), g′(+∞)) contains eigenvalues other than λ1? Among
other applications, equation (5.1) arises in the design of suspension bridges, see [35].
A reasonable conjecture is that if (g′(−∞), g′(+∞)) contains k eigenvalues then for
c sufficiently large and negative the problem (5.1) has 2k solutions. Moreover, if
g′(−∞) ∈ R and g′(+∞) = +∞ then for large negative values of c the equation has
a large number of solutions. These conjectures are known as the Lazer-McKenna
conjecture and the literature is rich in contributions to this ample field. See [35, 36].

When we restrict ourselves to the case where Ω is the unit ball in RN and the
solutions are radial, using bifurcation arguments one can show the following [15]:

Theorem 5.1. Suppose

g′(−∞) < ρ1([j/2] + 1)2 < ρk < g′(+∞) < ρk+1. (5.4)

Then for t negative and of sufficiently large magnitude the problem (1.2) has at
least 2(k − j) radial solutions, of which k − j satisfy u(0) > 0.

6

-

.

1

λ

β

b1

d

. G+
1

G+
2 G+

j

Γ+
1 Γ+

2 Γ+
j

Γ−2 Γ−j

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

ρ1
g′(∞)

ρ2
g′(∞)

ρj
g′(∞)

ρ1
g′(0)

ρ2
g′(0)

ρj
g′(0)

Figure 3. Bifurcation curves for g asymptotically linear having a
zero at β > 0.



EJDE-2023/SI/02 REGULAR SOLUTIONS 95

6. Sub-super critical problems

Motivated by the results in [17], the existence of radial solutions to (1.2) on the
unit ball for problems with

lim
u→+∞

g(x, u)

up
∈ R, lim

u→−∞

g(x, u)

−|u|q
∈ R with 1 < p <

N + 2

N − 2
< q <∞ (6.1)

was considered in [19] and the following result was established.

Theorem 6.1. If (6.1) is satisfied, then equation (1.4) has infinitely many solu-
tions.

The proof of Theorem 6.1 relies on proving that the energy E satisfies

lim
d→+∞

E(r, d) = +∞, (6.2)

uniformly for r ∈ [0, 1]. This is achieved by proving that the Pohozaev energy
P satisfies limd→+∞ P (r, d) = +∞ uniformly on compact subsets of (0, 1]. The
main idea behind this proof is that for d > 0 large, if xi < xi+1 < xi+2 are three
consecutive zeros of u with u > 0 on (xi, xi+1) and u < 0 on (xi+1, xi+2) then∫ xi+1

xi

sN−1Γ(u(s))ds > −
∫ xi+2

xi+1

sN−1Γ(u(s))ds, (6.3)

where

Γ(u) = NG(u)− N − 2

2
ug(u). (6.4)

Note that 1 < p < N+2
N−2 < q < ∞ implies that Γ(u) > 0 for u > 0 large and

Γ(u) < 0 for u < 0 large.

7. Shooting from singularity to singularity and Laplace-Beltrami
problems

In this section we consider an elliptic equation on a differentiable manifold that
leads to a singular ordinary differential equation like (1.4) on a bounded interval
where the coefficients of u′(r) are singular at both ends of the interval.

Let M ⊂ RN , N ≥ 3, be a compact connected hypersurface of revolution of
class C2 without boundary that intersects its axis of revolution. Without loss of
generality we may assume that {(0, . . . , 0, z); z ∈ R} is the axis of revolution and
that P± = (0, . . . , 0,±1) ∈M . Let d : M×M → [0,∞) denote the geodesic distance
in M and a = max{d(P−1, x);x ∈ M} = d(P−, P+), see [20]. Hence there exist
differentiable functions G, z : [0, a]→ [0,∞) such that:

G(t) = 0 if and only if t ∈ {0, a},
M = {(θ, z(r)) : θ = G(r), r ∈ [0, a]},

G′(0) = −G′(a) = 1, z(0) = −1, and z(a) = 1.

(7.1)

Since M is compact (see [31]) there exists ε2 > 0 such that

U := {x ∈ RN : min{|x− y|; y ∈M} < ε2} = {x+ sη(x) : x ∈M, s ∈ (−ε2, ε2)},
(7.2)

where η : M → RN is a differentiable function such that η(x) is normal to M at
x and |η(x)| = 1 for all x ∈ M (see [31]). Moreover, x1 + s1η(x1) = x2 + s2η(x2)
if and only if x1 = x2 and s1 = s2. For u ∈ C2(M,R), let ũ : U → R be defined



96 A. CASTRO, J. JACOBSEN EJDE/SI/02

by ũ(x + s η(x)) = u(x), for x ∈ M . The Laplace-Beltrami operator of M at u,
denoted ∆Mu, is defined as the restriction to M of the RN -Laplacian of ũ.

A function u : M → R is invariant under rotations of M about its axis of
symmetry if and only if u(x1, . . . , xN−1, xN ) = u(y1, . . . , yN−1, xN ) for x2

1 + · · · +
x2
N−1 = y2

1 + · · · + y2
N−1. In the Appendix of [20] it is shown that a rotationally

symmetric function u : M → R is a solution of

∆Mu+ f(u) = q(r) on M (7.3)

if and only if

v(r) = u(0, G(r), z(r)) = u(x1, . . . , xN−2, xN−1, z(r)) (7.4)

with x2
1 + . . . x2

N−2 = G2(r) satisfies

v′′(r) +
( (N − 2)G′(r)

G(r)

)
v′(r) + f(v(r)) = q(r), (7.5)

subject to

v(0) ∈ R, v(a) ∈ R, v′(0) = v′(a) = 0. (7.6)

Using this equivalence, in [20] the following result is proven.

Theorem 7.1. If f is differentiable and there exist p1, p2 ∈ (1, (N + 1)/(N − 3))
such that

lim
u→−∞

f(u)

|u|p1−1u
∈ (0,∞) and lim

u→+∞

f(u)

|u|p2−1u
∈ (0,∞), (7.7)

then equation (7.3) has infinitely many rotationally symmetric solutions.

The version of this result for the unit sphere in RN and solutions symmetric about
the equator was proven in [14]. The additional symmetry reduces the problem to
an equation in [0, a/2] which is treated shooting from the singular point 0 to a/2,
which is a regular point. Equation (7.5) is singular at both 0 and a, so one is
shooting from singularity to singularity.

The proof of Theorem 7.1 is carried out shooting from the singular point 0 to
the singular point a by considering (7.5) subject to the initial conditions

v(0) = d, v′(0) = 0 (7.8)

and aiming for d such that

lim
r→a

v(r) ∈ R and lim
r→a

v′(r) = 0. (7.9)

In summary, the proof of Theorem 7.1 follows from the following steps, see [20].

1. For any d ∈ R the solution v := vd to (7.5) is defined in [0, a).
2. lim|d|→∞(v2(r) + (v′(r))2) = +∞ uniformly for r ∈ [0, d).
3. For each d the solution to (7.5)-(7.8) has finitely many zeros.
4. The number of zeros of v tends to +∞ as |d| tends to +∞.
5. If v is a bounded solution to (7.5)-(7.8) then v satisfies (7.9), that is, u

defined by (7.4) is a solution to (7.3).
6. There exists D > 0 such that {d : |d| ≥ D and limr→a vd(r) = ±∞} is

open.



EJDE-2023/SI/02 REGULAR SOLUTIONS 97

8. Quasilinear equations

Many of the above results extend to equations of the form

− div(ψ(|∇u|)∇u)(x) = g(x, u(x)) for x ∈ Ω, u(x) = 0 for x ∈ ∂Ω. (8.1)

Prominent among such equations are p-Laplace equations and k-Hessian equations.
The p-Laplace operator, denoted ∆p, is given by ψ(y) = yp−2, i.e., ∆pu(x) =
div(|∇u|p−2∇u). Note that ∆2 = ∆. When u is radial this becomes

∆pu(x) = (|up−2
r |ur)r(r). (8.2)

To describe k-Hessian equation, recall that the Hessian of a function u at x is given
by the matrix

D2(u)(x) =
( ∂2u

∂xi∂xj
(x)
)
. (8.3)

The k-Hessian of u at x, denoted Sku(x), is given by the kth-symmetric function
of the eigenvalues of D2u(x). Since the eigenvalues of D2u(x) define the sectional
curvatures of the graph of u at x and the graph of u is invariant under rotations when
u is a radial function, we have Sku(x) = Sku(‖x‖, 0, . . . , 0). A direct calculation
shows that

D2(u)(r, 0, . . . , 0) =


urr 0 0 . . . 0
0 ur

r 0 . . . 0
...

...
...

...
...

0 . . . 0 0 ur

r

 (r, 0, . . . , 0). (8.4)

Hence, for a radial function u,

S1u(x) = urr(r) +
N − 1

r
ur(r) = ∆u(x)

Sku(x) =

(
N − 1

k − 1

)(ur(r)
r

)k−1

urr(r) +

(
N − 1

k

)(ur(r)
r

)k
k = 2, . . . , N − 1

SNu(x) =
(ur(r)

r

)N−1

urr(r),

(8.5)
where ‖x‖ = r. The operator SN is known as the Monge-Ampere operator. As
pointed out in [33], for radial functions the p-Laplacian operator and the k-Hessian
operators are particular cases of operators of the form

Q(u)(r) = r−γ(rα|ur(r)|βur(r))r. (8.6)

For a recent result on the existence of radial solutions for equations such as (8.1)
we refer the reader to [13] where the following extension of Theorem 6.1 is proven.

Theorem 8.1. If

g(s) :=

{
|s|q1 , s ≥ 0

−|s|q2 , s < 0,
(8.7)

with

p− 1 < q1 < p∗ − 1 < q2 < +∞, (8.8)

where p∗ := Np
N−p is the critical Sobolev exponent, then (8.1) has infinitely many

radial solutions.



98 A. CASTRO, J. JACOBSEN EJDE/SI/02

9. Appendix A: The contraction mapping principle

A fundamental tool in proving the existence of solutions to differential equations
with continuous dependence on parameters such as initial conditions, boundary
conditions, or forcing terms, is the following theorem known as the Contraction
Mapping Principle. The Inverse Function Theorem and Implicit Function Theorem
are important consequences of the Contraction Mapping Principle.

Theorem 9.1 (Contraction Mapping Principle). Let (X, d) be a complete metric
space and (Y, δ) a metric space. If f : X × Y → X is continuous and there exists
a real number a ∈ [0, 1) such that

d(f(x1, y), f(x2, y)) ≤ a d(x1, x2) for all x1, x2 ∈ X, y ∈ Y, (9.1)

then there exists a continuous function ϕ : Y → X such that f(ϕ(y), y) = ϕ(y).

Proof. Let x̄ ∈ X and y ∈ Y . Consider the sequence in X defined by x1 = f(x̄, y),
x2 = f(x1, y), . . ., xk+1 = f(xk, y). We show that {xn} is a Cauchy sequence. Let
k be a positive integer. By (9.1), we have

d(xk+1, xk) = d(f(xk, y), f(xk−1, y))

≤ ad(xk−1, xk−2) ≤ . . .

≤ akd(x1, x̄).

(9.2)

Hence, for m > n,

d(xm, xn) ≤ d(xm, xm−1) + · · ·+ d(xn+1, xn)

≤ an d(x1, x̄)
(
am−n + · · ·+ 1

)
≤ an

1− a
d(x1, x̄),

(9.3)

which implies that {xn} is a Cauchy sequence. Since (X, d) is complete, there exists
ϕ(y) ∈ X such that limn→∞ xn = ϕ(y). By the continuity of f we have

f(ϕ(y), y) = lim
n→∞

f(xn, y) = lim
n→∞

xn+1 = ϕ(y), (9.4)

hence φ(y) is a fixed point of f(·, y).
Suppose now that f(z, y) = z and f(w, y) = w. From (9.1), we have that

d(z, w) = d(f(z, y), f(w, y)) ≤ a d(z, w). Hence (1 − a) d(z, w) ≤ 0 which proves
that z = w. Thus ϕ(y) is unique and so the function ϕ : Y → X is well-defined.

To show that ϕ : Y → X is continuous, suppose limn→∞ yn = ŷ. Then

d(ϕ(yn), ϕ(y)) = d(f(ϕ(yn), yn), f(ϕ(y), y))

≤ d(f(ϕ(yn), yn), f(ϕ(y), yn)) + d(f(ϕ(y), yn), f(ϕ(y), y))

≤ ad(ϕ(yn), ϕ(y)) + d(f(ϕ(y), yn), f(ϕ(y), y)).

(9.5)

Hence d(ϕ(yn), ϕ(y)) ≤ d(f(ϕ(y), yn), f(ϕ(y), y))/(1−a). Since f is continuous, the
right hand side in the last inequality converges to 0. Hence limn→+∞ ϕ(yn) = ϕ(y)
proving the continuity of ϕ and completing the proof. �

10. Appendix B: Existence of the argument function

Let x : [0, T ]→ R and y : [0, T ]→ R be continuous functions such that

ρ(t) =
√

(x(t))2 + (y(t))2 > 0 for all t ∈ [0, T ].



EJDE-2023/SI/02 REGULAR SOLUTIONS 99

Lemma 10.1. There exists a continuous function θ : [0, T ]→ R such that

x(t) = ρ(t) cos(θ(t)) and y(t) = ρ(t) sin(θ(t)) (10.1)

for all t ∈ [0, T ].

Proof. Without loss of generality we assume x(0) > 0. Let ε > 0 such that x(t) > 0
for all t ∈ [0, ε]. For t ∈ [0, ε], let θ0(t) = tan−1(y(t)/x(t)). Then cos θ0(t) =
x(t)/ρ(t) and sin θ0(t) = y(t)/ρ(t) and so θ0 satisfies (10.1) on [0, ε]. Let S =

{
t ∈

[ε, T ] : there exists continuous θ : [0, t]→ R such that θ satisfies (10.1) and coincides
with θ0 on [0, ε]

}
, and let τ = supS. By definition we know τ ≥ ε. Let us see that

τ = T and hence θ0 is defined on [0, T ]. Suppose τ < T . If x(τ) 6= 0, there exists
δ > 0 such that if |t − τ | < δ then |y(t)/ρ(t)| ≤ (1 + |y(τ)/ρ(τ)|)/2. Let k be an
integer such that θ(τ) ∈ (π2 + kπ, π2 + (k + 1)π) (recall x(τ) 6= 0). Now we define
for t ∈ (τ, τ + δ),

θ(t) = sin−1
k (y(t)/ρ(t))

where sin−1
k is the inverse of sinx on the interval (π2 + kπ, π2 + (k + 1)π). Since

sin−1
k is continuous, we have a contradiction to the definition of τ , and so τ = T ,

proving the lemma. �

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Alfonso Castro

Department of Mathematics, Harvey Mudd College, Claremont CA 91711, USA
Email address: castro@hmc.edu

Jon Jacobsen
Department of Mathematics, Harvey Mudd College, Claremont CA 91711, USA

Email address: jacobsen@hmc.edu


	1. Introduction
	2. The shooting method and phase plane analysis
	3. Bifurcation analysis
	4. Asymptotically linear nonlinearities
	5. Jumping nonlinearities
	6. Sub-super critical problems
	7. Shooting from singularity to singularity and Laplace-Beltrami problems
	8. Quasilinear equations
	9. Appendix A: The contraction mapping principle
	10. Appendix B: Existence of the argument function
	References