Special Issue in honor of John W. Neuberger

Electronic Journal of Differential Equations, Special Issue 02 (2023), pp. 175–192.

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

LOCAL MIN-ORTHOGONAL PRINCIPLE AND ITS

APPLICATIONS FOR SOLVING MULTIPLE SOLUTION

PROBLEMS

MEIQIN LI, BINGBING JI, JIANXIN ZHOU

In memory of Prof. John W. Neuberger

Abstract. In this article we establish a double-orthogonal principle, and a
local min-orthogonal method with its step size rule, and its convergence un-

der assumptions more general than those in its previous versions. With such

a general framework, we justify mathematically the two new algorithms pro-
posed for solving W-type problems. Numerical examples for finding multiple

solutions to W-type and to mixed M-W-type problems illustrate the flexibility

of this method.

1. Introduction

Consider the semilinear elliptic equation

−∆u(x)− λu(x) + κF (x, u(x)) = 0, x ∈ Ω, (1.1)

satisfying zero Dirichlet or Neumann boundary conditions, where Ω ⊂ Rn is an open
bounded domain, λ and κ are physical parameters, and F : Ω × R → R satisfies
certain growth and regularity conditions [6]. Denoting ∂

∂uf(x, u) = F (x, u), its
variational functional is

J(u) =

∫
Ω

[
1

2
(|∇u(x)|2 − λu2(x)) + κf(x, u(x))]dx. (1.2)

For a function J ∈ C1(H,R) where H is a Hilbert space, a point u∗ ∈ H is called
a critical point of J if its Frechet derivative J ′(u∗) = 0; A critical point u∗ is
called a k-saddle if it is a local maximum point of J in a k-dimensional subspace
and a local minimum point in the corresponding k-co-dimensional subspace. Such
index k can be used to measure the instability of the critical point u∗. Thus a 0-
saddle is a local minimum point of J and corresponds to a stable local equilibrium
state in a physical system; while k-saddles with k ≥ 1 correspond to unstable local
equilibria or excited states. When J is C2 and u∗ is a critical point, we denote the
spectral decomposition of J ′′(u∗) by H = H− ⊕H0 ⊕H+ where H−, H0, H+ are
respectively the maximum negative, the null, and the maximum positive subspaces
of J ′′(u∗) with dim(H0) < ∞, and MI(u∗) = dim(H−) is called the Morse index

2020 Mathematics Subject Classification. 35B38, 58E05, 65N20.

Key words and phrases. Multiple solution; numerical algorithms; convergence analysis.
©2023 This work is licensed under a CC BY 4.0 license.
Published March 27, 2023.

175



176 M. LI, B. JI, J. ZHOU EJDE/SI/02

of u∗. A critical point u∗ is non-degenerate if H0 = {0} and degenerate otherwise.
A non-degenerate critical point u∗ with MI(u∗) = k is a k-saddle.

Let λ1 < λ2 < . . . be the eigenvalues of −∆ with zero boundary condition, and
let v1, v2, . . . be the corresponding eigenfunctions. For simplicity, we first consider
F (x, u(x)) = |u(x)|p−1u(x) with p > 1. Equation (1.1) is called focusing (M-type)
if κ < 0 and defocusing (W-type) if κ > 0. It is known that when κ > 0 and
λk < λ < λk+1, then 0 is the only index k-saddle, all nontrivial saddles have
index greater than k, and for all u ∈ [v1, . . . , vk]⊥, there is tu > 0 such that
tu = arg maxt>0 J(tu); when κ < 0 and λk < λ < λk+1, then 0 is the only index
k-saddle, all nontrivial saddles have index less than k and for all u ∈ [v1, . . . , vk],
there is tu > 0 such that tu = arg mint>0 J(tu). These two types of problems are
very different in physical nature and in mathematical structure as well, see Figure 1.
In the literature, these problems have to be solved by two very different types of
variational methods.

k−saddle

k−saddle

∩-shape in [v1, . . . , vk], M-shape in [v1, . . . , vk]⊥;
∪-shape in [v1, . . . , vk]⊥, W-shape in [v1, . . . , vk].

Figure 1. Typical functional profiles of M-type (left) vs. W-type
(right). Because of the difference in space dimensions, they are not
upside-down to each other.

The case will be much more complex if F (x, u) contains both convex and concave
nonlinear terms and becomes a mixed M-W-type problem, see Figure 2,

F (x, u) = a(x)|u(x)|q−1u(x) + b(x)|u(x)|p−1u(x) . (1.3)

See [1, 2, 11] where 0 < q < 1 < p < 2∗, 2∗ = N+2
N−2 if N ≥ 3 or 2∗ =∞ if N = 1, 2

for some given nonnegative functions a(x) and b(x).

locally M−type

locally W−type

Figure 2. Energy profile consists of locally M and/or W parts.

To solve M-type problems for multiple solutions, a local min-max method (LMM)
was developed in [4]. Theoretically it was extended to a local min-orthogonal
method in [16]. But it has not been used to solve the W-type or other type prob-
lems. In this paper, we generalize the local min-orthogonal method and explore its
flexibilities to solve the M-type, the W-type and even mixed M-W-type problems.



EJDE-2023/SI/02 SHORT TITLE 177

Let L be a closed subset of H. We denote SL⊥ = {v ∈ H : ‖v‖ = 1, v⊥L}, and
for each v ∈ SL⊥ , let [v, L] = Span{v, L}. Let

v∗ = arg min
v∈S

L⊥
J(p(v)),

where for the local min-max method (LMM) [4, 2001], p(v) = arg maxu∈[v,L] J(u)
and for the local min-orthogonal method [16], p(v) ∈ [v, L] such that J ′(p(v))⊥[v, L].
Then u∗ = p(v∗) is a critical point of J . Since when p(v) ∈ [v, L] is a local maximum
point of J in [v, L], we have J ′(p(v))⊥[v, L] and the local min-orthogonal method
becomes LMM which was first developed in [4] and successfully used to solve many
M-type problems.

Definition 1.1 ([16]). The set-valued mapping P : SL⊥ → 2H is called the L-
orthogonal mapping if P (v) := {u ∈ [L, v] : J ′(u)⊥[L, v]} for each v ∈ SL⊥ . A
function p : SL⊥ → H is called an L-orthogonal selection if p(v) ∈ P (v) ∀v ∈ SL⊥ .
If p is locally defined then p is called a local L-orthogonal selection.

Lemma 1.2 ([16]). If J is C1, then G = {(u, v) : v ∈ SL⊥ , u ∈ P (v)} is closed.

Since p(v) = tv + vL for some vL ∈ L, it is clear that if p is continuous then
p(v∗) is a critical point of J if and only if there is a neighborhood N (v∗) of v∗ in
L⊥ such that

J ′(p(v∗))⊥p(v) ∀v ∈ N (v∗) ∩ SL⊥ . (1.4)

We call (1.4) a double-orthogonal principle for critical points. An important feature
of this principle is that it involves only J ′ not the functional J . It implies that this
principle can treat nonvariational multiple solution problems where J ′(v) = 0 is
replaced by A(v) = 0 for a nonlinear operator A. For example, it is applied to
design algorithms to find nonvariational multiple solutions, see [5, 13]. In this
paper, we focus on solving variational multiple solution problems with different
mathematical structures by exploring variations of this principle. In numerical
implementation, the outer-layer is an infinite-dimensional problem and the inner-
layer consists of finite-dimensional sub-problems. When the outer-layer orthogonal
condition in (1.4) is achieved by a minimization process. It becomes a local min-
orthogonal principle for characterizing variational critical points

min
v∈S

L⊥
J(p(v)). (1.5)

This result is first announced in [4] and then published in [16]. As one of the most
used mathematical frameworks in critical point theory, the Ljusternik-Schnirelman
principle (LSP) [15] characterizes a critical point as a solution to the min-max
problem

min
A∈A

max
u∈A

J(u) (1.6)

where A is a collection of certain compact sets A, e.g., a k-dimensional simplex,
the min and max are all in the global sense, therefore LSP is not for algorithm
implementation. Let us compare our double-orthogonal principle to LSP, one can
see that the former is more general, where (1) the two global min and max in LSP
are replaced by two ⊥-conditions which can be satisfied, e.g., by local min and/or
max process, and (2) compact sets in LSP are generalized to closed subspaces which
can be either finite dimensional or infinite dimensional. This represents a significant
advantage of our double-orthogonal principle over LSP, in particular, in algorithm
implementation.



178 M. LI, B. JI, J. ZHOU EJDE/SI/02

In 1960, the Nehari manifold [7]

N = {tuu 6= 0 : u ∈ H, ‖u‖ = 1, J ′(tuu)⊥u} (1.7)

was introduced and widely applied later on in the literature to prove the existence
of 1-saddle of various nonlinear problems through minu∈N J(u). In 2005 and after
[8, 9, 12]], a generalized Nehari manifold

NL = {u ∈ [v, L] : u 6= 0, v ∈ SL⊥ , J ′(u)⊥[v, L]}

was proposed with L = [v1, . . . , vk] where v1, . . . , vk are the first k eigenfunctions
of −∆ and applied to prove the existence of a k-saddle of semilinear elliptic PDEs
through minu∈NL

J(u). It is also proved that such a manifold NL is C1. It is
clear that the generalized Nehari method is just a special case of our local min-
orthogonal method. Actually for the double-orthogonal principle, there are many
different ways to acheive the inner-layer orthogonal condition. In this paper, we
first prove this principle with a weaker continuity condition, then establish two new
variations to show how this principle can be modified to find multiple solutions to
W-type and even mixed M-W-type problems.

2. A new local min-orthogonal method

For analysis purpose, we first extend the domain of p to S+
L⊥ = {v ∈ L⊥, ‖v‖ ≈ 1}

where by ‖v‖ ≈ 1 we mean 1 − δ < ‖v‖ < 1 + δ for a given δ > 0. While in our
algorithm computation and convergence analysis we will evaluate p only at v or
v+sw
‖v+sw‖ where v, w ∈ L⊥, ‖v‖ = 1 for small s. Thus the closed set SL⊥ will be

enough for us to define LDLC of p.

Definition 2.1. For J ∈ C1(H,R), a set-valued mapping P : S+
L⊥ → H2 is called

the L-orthogonal mapping if for each v ∈ S+
L⊥

P (v) = {u ∈ [v, L] : J ′(u)⊥[v, L]}.

An L-orthogonal selection p : S+
L⊥ → H is a mapping such that

p(v) ∈ P (v),∀v ∈ S+
L⊥ .

If p is locally defined near v ∈ S+
L⊥ , then p is called a local L-orthogonal selection

near v.

Definition 2.2. A local L-orthogonal selection p is said to be locally directional
Lipschitz continuous (LDLC) at v ∈ SL⊥ in w ∈ L⊥, if there is a constant `0
depending on v and w, such that for all s > 0 small, it holds

‖p(v + sw)− p(v)‖ ≤ `0s‖w‖,

where the term ‖w‖ can be removed since `0 depends on w. We say that p is LDLC
at v ∈ SL⊥ if it is LDLC at v ∈ SL⊥ in each w ∈ L⊥. We say that p is LDLC if it
is LDLC at each v ∈ SL⊥ .

Note that when v ∈ SL⊥ , w ∈ L⊥ are given, s > 0 is small, we have v+sw ∈ S+
L⊥ ;

For v(s) = v+sw
‖v+sw‖ , we have ‖v(s)‖ = 1 and [v(s), L] = [v + sw,L] and then

p(v(s)) = tsv(s) + uL(s) =
ts

‖v + sw‖
(v + sw) + uL(s) = p(v + sw),



EJDE-2023/SI/02 SHORT TITLE 179

with uL(s) ∈ L. It implies that if p is LDLC at v ∈ SL⊥ in w ∈ L⊥, there is `0 > 0
such that when s > 0 small, we have

‖p(v(s))− p(v)‖ = ‖p(v + sw)− p(v)‖ ≤ `0|s|‖w‖ = O(s). (2.1)

It is clear that if the constant `0 is independent of w, p becomes locally Lipschitz
continuous at v in all directions in L⊥. If the Gateaux derivative δp(v;w) of p exists
at v in w ∈ L⊥ and

lim
s→0

1

s
(p(v + sw)− p(v)) = δp(v;w),

where δp(v;αw) = αδp(v;w) for any scalar α, but not necessarily linear in w.
Denote `0 = max{1, 2

‖w‖ |δp(v;w)|} > 0, then there is s0 > 0, such that when

s0 > |s| > 0, it holds

‖p(v + sw)− p(v)‖ < `0|s|‖w‖,
or p is LDLC at v in w. We conclude that the Gateaux-differential of p at v in
w ∈ L⊥ implies LDLC at v in w, then the directional continuity at v in w but not
the continuity at v. The directional continuity is clearly weaker than the weak-
continuity since the latter implies the continuity in any finite-dimensional space.

Lemma 2.3 (Stepsize Rule). Let p be a local L-orthogonal selection of J in S+
L⊥

such that p is LDLC at v ∈ SL⊥ and dis(p(v), L) > 0. If d = −J ′(p(v)) 6= 0, set
w = d/C where C = max{1, ‖d‖} and v(s) = v+sw

‖v+sw‖ , then there is s0 > 0 such

that when s0 > s > 0, we have a stepsize rule

J(p(v(s)))− J(p(v)) < −1

4
tvs‖d‖2/C. (2.2)

Furthermore, if p(vk) → p(v) as vk → v, then there exists N > 0 such that when
k > N , we have a uniform stepsize rule

J(p(vk))− J(p(v)) <
−tv

4
‖J ′(p(v))‖2/C. (2.3)

Proof. Denote p(v) = tvv+uv, p(v(s)) = tsv(s)+uL(s) for tv, ts > 0 and uv, uL(s) ∈
L. Then there is s0 > 0 such that when s0 > s > 0, we have

J(p(v(s)))− J(p(v)) = 〈J ′(p(v)), p(v(s))− p(v)〉+ o(‖p(v(s))− p(v)‖)

=
tss

‖v + sw‖
〈J ′(p(v)), w〉+ o(‖p(v + sw)− p(v)‖)

= − tss

‖v + sw‖
‖d‖2/C + o(s)

< −1

4
tvs‖d‖2/C,

(2.4)

where we have used the properties J ′(p(v))⊥[v, L] and ‖p(v(s)) − p(v)‖ = O(s) in
(2.1). The uniform stepsize rule follows from the fact that J is C1, p(vk) → p(v)
leads to J ′(p(vk))→ J ′(p(v)) as vk → v. �

Theorem 2.4 (Local min-orthogonal Characterization). Let p be an L-orthogonal
selection of J in S+

L⊥ such that

(1) p is LDLC at v ∈ SL⊥ and dis(p(v), L) > 0,
(2) v = arg local minu∈S

L⊥ J(p(u)).

Then p(v) is a critical point of J .



180 M. LI, B. JI, J. ZHOU EJDE/SI/02

Proof. If d = −J ′(p(v)) 6= 0, set w = d/C where C = max{1, ‖d‖} and v(s) =
v+sw
‖v+sw‖ ∈ SL⊥ , then by the step size rule (2.2), as s > 0 sufficiently small, we have

J(p(v(s)))− J(p(v)) < −1

4
tvs‖d‖2/C.

It violates assumption (2). �

3. A numerical local min-orthogonal algorithm

The local min-orthogonal characterization of a critical point, Theorem 2.4 sug-
gests a local min-orthogonal method with the stepsize rule (2.2) if the min is ap-
proximated by using a steepest decent method. We present the steps below:

Step 1: Given ε > 0, λ > 0 and n previously found critical points w1, w2, . . . , wn of
J , of which wn has the highest critical value. Set L = span{w1, w2, . . . , wn}.
Let v1 ∈ SL⊥ be an ascent direction at wn. Let t00 = 1, v0

L = wn and set
k = 0.

Step 2: Using the initial guess u = tk0v
k + vkL, solve tk0 , t

k
1 , . . . , t

k
n from

〈J ′(t0vk0 + t1w1 + · · ·+ tnwn), vk〉 = 0,

〈J ′(t0vk0 + t1w1 + · · ·+ tvwn), wj〉 = 0,

for j = 1, . . . , n. Denote uk ≡ p(vk) = tk0v
k
0 + vkL = tk0v

k
0 + vk1w1 + · · · tknwn.

Step 3: Compute the steepest descent vector dk = −J ′(uk).
Step 4: If ‖dk‖ ≤ ε then output wn+1 = uk, stop; else goto Step 5.

Step 5: Set vk(s) = vk+sdk

‖vk+sdk‖ and find

sk = max
{ λ

2m
: m ∈ N, 2m > ‖dk‖,

J(p(vk(
λ

2m
)))− J(wk) ≤ − t

k
0

2
‖dk‖ ‖vk(

λ

2m
)− vk‖

}
.

Initial guess u = tk0v
k( λ

2m )+vkL is used to find p(vk( λ
2m )) in {L, vk( λ

2m )}\L
as similar in Step 2 and where tk0 and vkL are found in Step 2.

Step 6: Set vk+1 = vk(sk) and update k = k + 1 then goto Step 2.

Remark 3.1. (1) The algorithm starts with n = 0, L = {0} to find w1, then n = 1,
L = {w1} to find w2, etc. To stay away from previously found solutions contained
in L, we should choose an initial guess v0 which is at least nearly orthogonal to L.

(2) The finite-dimensional sub-problem for finding p(vk) in Step 2 or p(vk( λ
2m ))

in Step 5 can be solved, for example, by Matlab subroutine fsolve, where tk0 > 0 is
determined by the problem structure. When tk0 = 0, the algorithm fails to find a
new critical point.

(3) In Step 5, the step size rule has been equivalently modified due to the in-
equalities

s‖d‖
‖v + sd‖

≤ ‖v(s)− v‖ ≤
√

2s‖d‖
‖v + sd‖

.

Also using the designated initial guess as in Step 5 is very important to continuously
trace a solution branch and to keep the algorithm stable;

(4) For an initial guess u0, when the Nehari manifold N defined in [7] has only
one branch, we can simply find t0 > 0 such that t0u0 ∈ N and use t0u0 as an
initial guess; when N contains multiple branches, with extra information on the



EJDE-2023/SI/02 SHORT TITLE 181

branches such as the sign of J-values, we can find a proper value t0 > 0 such that
t0u0 belongs to a designated branch.

4. Algorithm convergence analysis

We assume that J satisfies the Palais-Smale (PS) condition, i.e., any sequence
{uk} ⊂ H with {J(uk)} bounded and J ′(uk)→ 0 has a convergent subsequence.

Let {wk} = {p(vk)} be the sequence generated by the local min-orthogonal
method with ε = 0.

Next we present an improved convergence result [16, 18] by modifying the proof
of [18, Theorem 2.4].

Theorem 4.1. If p is defined in S+
L⊥ , and continuous and LDLC on SL⊥ , d(L,wk)>

α>0 and infv∈S
L⊥ J(p(v))>−∞, then

(a) skdk → 0;
(b) there is {vki} ⊂ {vk} such that vki → v∗ with w∗ = p(v∗), J ′(w∗) = 0;
(c) if w∗ is isolated then vk → v∗.

Proof. By the step size rule (2.2),

√
1+(sk‖dk‖)2√

2
> 1/2 and the inequality

sk‖dk‖√
1 + (sk‖dk‖)2

≤ ‖vk+1 − vk‖ ≤
√

2sk‖dk‖√
1 + (sk‖dk‖)2

, (4.1)

we obtain

J(wk+1)− J(wk) ≤ −1

4
|tk0 |sk‖dk‖2Ck ≤

−1

8
|tk0 |‖J ′(wk)‖‖vk+1 − vk‖.

Adding it up for all k = 1, 2, . . . and noting |tk0 | > α > 0, J(wk) > −M > −∞, we
obtain

−∞ < lim
k→∞

J(wk)− J(w1) ≤ −1

4

∞∑
k=1

|tk0 |sk‖dk‖2Ck

≤ −α
4

∞∑
k=1

sk‖dk‖2Ck < −
α

8

∞∑
k=1

‖J ′(wk)‖‖vk+1 − vk‖.
(4.2)

Thus sk‖dk‖2Ck = sk‖J ′(wk)‖2/Ck → 0. Then Ck = max{1, ‖J ′(wk)‖} and 0 <
sk < λ lead to sk‖dk‖ → 0 as in (a).

Next to prove (b), there are totally two cases, either (1) ‖dk‖ > η > 0 for
k = 1, 2, . . . for some 1

2 > η > 0, or (2) there is a subsequence {dki} ⊂ {dk} such

that dki → 0.
In case (1), η < ‖dk‖ ≤ 1, then (4.2) becomes

−∞ < lim
k→∞

J(wk)− J(w1) ≤ −α
8

∞∑
k=1

‖J ′(wk)‖‖vk+1 − vk‖

≤ −αη
8

∞∑
k=1

‖vk+1 − vk‖,
(4.3)

i.e., {vk} ⊂ SL⊥ is a Cauchy sequence in the Hilbert space H. Thus there exists
v∗ ∈ SL⊥ such that vk → v∗ as k →∞. Since p is continuous and J is C1, we have
w∗ = p(v∗) = limk→∞ p(vk) and ‖J ′(w∗)‖ ≥ η. Hence w∗ is not a critical point.
Then skdk → 0 and ‖dk‖ > η > 0 imply sk → 0, which contradicts the uniform
step size rule, Lemma 2.3. Thus case (2) must hold, i.e., there is a subsequence



182 M. LI, B. JI, J. ZHOU EJDE/SI/02

{dki} ⊂ {dk} such that dki → 0. Since {J(wki)} is bounded, by the PS condition,

there is a subsequence denoted by wki = p(vki) = tki0 v
ki + wkiL again for some

wkiL ∈ L, such that wki → w∗ = w∗⊥ + w∗L with w∗⊥ ∈ L⊥, w∗L ∈ L. By the

condition |tk0 | > α > 0, we must have tki0 v
ki → w∗⊥, w

ki
L → w∗L and vki → v∗

for some v∗ ∈ SL⊥ . Thus tki0 → t∗0 for some |t∗0| ≥ α > 0. The continuity of
p then leads to w∗ = p(v∗). We have proved that p is a homeomorphism and
J ′(w∗) = limi→∞ J ′(wki) = limi→∞ dki = 0. So (b) is proved.

Finally to prove (c) vk → v∗ if w∗ = p(v∗) is an isolated saddle, since p is
a homeomorphism, this means that there is δ > 0 such that there is no point
v′ ∈ SL⊥ satisfying ‖v∗ − v′‖ < δ and J ′(p(v′)) = 0. Let I ⊂ N = {1, 2, . . . }. We
call

∑
i∈I ‖vi+1 − vi‖ the total distance traveled by the subsequence {vi}i∈I . For

any η > 0, let i ∈ I ⊂ N denote the whole index set in N with ‖di‖ > η. Then
(4.2) leads to

−∞ < lim
i→∞

J(wi)− J(w1) ≤ −α
8

∑
i∈I
‖J ′(wi)‖‖vi+1 − vi‖

< −αη
8

∑
i∈I
‖vi+1 − vi‖,

(4.4)

i.e., the total distance traveled by {vi}i∈I is finite. Note that by (a), we have

‖vk+1 − vk‖ ≤
√

2sk‖dk‖√
1 + (sk‖dk‖)2

→ 0. (4.5)

Suppose there is δ3 > 0 such that there are infinitely many points v in {vk} with
‖v − v∗‖ > δ3. By the inequality (4.5), for any 0 < δ1 < δ2 < δ3, there is K > 0
such that when k > K, ‖vk+1− vk‖ < 1

2 (δ2− δ1). Since v∗ is a limit point of {vk},
there are infinitely many points v ∈ {vk} such that 0 < ‖v − v∗‖ < δ1 and there
are also infinitely many points {vki} ⊂ {vk} such that 0 < δ1 < ‖vki − v∗‖ < δ2,
i.e., the sequence {vk} enters the three ring regions centered at v∗ and defined
by 0 < δ1 < δ2 < δ3 infinitely many times. Thus the total distance traveled by
such {vki} has to be infinite. However by (4.4), for any η > 0, the total distance

traveled by all the points vk
′
i ∈ {vk} with ‖J ′(p(vk′i))‖ > η is finite. Thus there

must be infinitely many points {vki′} ⊂ {vki} such that J ′(p(vki′ )) → 0. By the
PS condition, there is a subsequence, denoted by {wki′} = {p(vki′ )} again, such
that wki′ → w′ = t′v′ + w′L for some t′ > 0, v′ ∈ SL⊥ , w′L ∈ L with J ′(w′) = 0
and δ1 ≤ ‖v∗ − v′‖ ≤ δ2. Since 0 < δ1 < δ2 can be any numbers less than δ3,
this contradicts to the assumption that w∗ = p(v∗) is an isolated critical point.
Thus for any δ3 > 0, there can be at most a finite number of points v in {vk}
with ‖v∗ − v‖ > δ3, i.e., vk → v∗. (c) is proved. This completes the proof of the
theorem. �

5. Two new variations to the min-orthogonal method

The local min-orthogonal method is first introduced in [16] as a theoretical ex-
tension of LMM for solving M-type multiple solution problems. Next we show that
this method turns out to be quite flexible. It can be actually modified to solve
W-type or even mixed M-W type problems for multiple solutions. First we observe
closely the profile of a typical W-type functional J in Figure 1, we found that (1) 0
is the only k-saddle and all other critical points must have saddle index less than k;



EJDE-2023/SI/02 SHORT TITLE 183

(2) For any nontrivial solution u∗, it must be a local minimum point of J along its
direction with J(u∗) < J(0) = 0. Based on the variational structures we observed,
we propose two new variations of the min-orthogonal method:

(A1) A min-min-max algorithm. (This method was proposed by Dr. Zhi-Qiang
Wang in 2012 in a discussion.)

min
v∈S

L⊥
min
r>0

max
u∈[v,L],‖u‖=r

J(u).

(A2) A min-max-min algorithm.

min
v∈S

L⊥
max

u∈[v,L],‖u‖≈1
min
t>0

J(tu).

Note that those two algorithms start with L = {0}. In this case, the max-
operation in the algorithms is void and the algorithms find a local minimum point
u∗0 of J . Then by setting L = {u∗0}, the algorithms find a 1-saddle,. . . ,etc. For each
v ∈ SL⊥ , we denote, in the min-min-max algorithm (A1),

p(v) = arg min
r>0

max
u∈[v,L],‖u‖=r

J(u) (5.1)

and in the min-max-min algorithm (A2),

p(v) = arg max
u∈[v,L],‖u‖≈1

min
t>0

J(tu), (5.2)

then the two algorithms can be expressed as minv∈S
L⊥ J(p(v)). Thus to establish

their mathematical justifications by the local min-orthogonal principle, we only
have to show J ′(p(v))⊥[v, L].

5.1. Justification of the min-min-max algorithm. In the min-min-max algo-
rithm (A1), when L is k-dimensional and v ∈ SL⊥ is given, choose an orthonormal
basis in [v, L]. For each u ∈ [v, L], let (r, θ) = (r, θ1, . . . , θk) be the k+1- dimen-
sional spherical coordinates of u in [v, L]. Denote χ : (r, θ) → u = χ(r, θ) the
coordinate transformation from the spherical coordinates to [v, L] with respect to
the orthonormal basis. We denote u(r, θ) the spherical coordinates of u. Since
p(v) ∈ [v, L], to prove J ′(p(v))⊥[v, L], we consider

JS(r, θ) = J(χ(r, θ)) = J(u).

Denote, for each r > 0,

θ(r) = arg max
θ
JS(r, θ) (5.3)

and u = p(v) in the spherical coordinates

u(rv, θ(rv)) = arg min
r>0

JS(r, θ(r)). (5.4)

Assume θ(r) is continuous, we show that

dJS(rv, θ(rv)) = (JSr (rv, θ(rv)), J
S
θ (rv, θ(rv))) = (0, 0, . . . , 0).

By (5.3) we have

JSθ (r, θ(r)) = (JSθ1(r, θ(r)), . . . , JSθk(r, θ(r))) = (0, . . . , 0)

or

dJS(r, θ(r)) = (JSr (r, θ(r)), JSθ (r, θ(r))) = (JSr (r, θ(r)), 0, . . . , 0).

Suppose JSr (rv, θ(rv)) 6= 0. Denote

rv(s) = rv − sJSr (rv, θ(rv)).



184 M. LI, B. JI, J. ZHOU EJDE/SI/02

Let (rv(s), θ(s)) be defined in (5.3) where θ(s) = θv(rv(s)). For each fixed s > 0,
we define

g(λ) = JS(rv − λsJSr (rv, θ(rv)), θ(s)).

By the mean-value theorem, there is 0 < λs < 1 such that

g′(λs) = g(1)−g(0) = JS(rv(s), θ(s))−JS(rv, θ(s)) ≥ JS(rv(s), θ(s))−JS(rv, θ(rv))

since by (5.3), JS(rv, θ(rv)) ≥ JS(rv, θ(s)) when s > 0 is small. On the other hand

g′(λs) = 〈JSr (rv − λssJSr (rv, θ(rv)), θ(s)),−sJSr (rv, θ(rv))〉 ≤ −
s

4
‖JSr (rv, θ(rv))‖2

since rv − λssJSr (rv, θ(rv))→ rv implies θ(s)→ θ(rv) as s→ 0. Then we have

JS(rv(s), θ(s))− JS(rv, θ(rv)) < −
s

4
‖JSr (rv, θ(rv))‖2

when s > 0 is small. This contradicts (5.4) if JSr (rv, θ(rv)) 6= 0. Thus we have
shown dJS(rv, θ(rv)) = (JSr (rv, θ(rv)), J

S
θ (rv, θ(rv))) = (0, 0, . . . , 0).

To show J ′(p(v))⊥[v, L], we use the same notation as before. For u = p(v), we
have u = χ(rv, θ(rv)). We note that χ′r(rv, θ(rv)) is the ray-direction of the point u
and χ′θ(rv, θ(rv)) forms a basis for the tangent space of the k+1-dimensional sphere
at the point u. By the chain rule, we have

0 =
∂

∂r
JS(rv, θ(rv)) =

∂

∂r
J(χ(r, θ))|(r,θ)=(rv,θ(rv)) = J ′(p(v))χ′r(rv, θ(rv)),

(0, . . . , 0) =
∂

∂θ
JS(rv, θ(rv)) =

∂

∂θ
J(χ(r, θ))|(r,θ)=(rv,θ(rv))

= J ′(p(v))χ′θ(rv, θ(rv)),

i.e., in the space [v, L], J ′(p(v)) is orthogonal to the vector (ray) u = p(v) and the
tangent space of the sphere at u = p(v). Consequently J ′(p(v))⊥[v, L].

We have theoretically verified that the min-min-max algorithm (A1) fits into
the local min-orthogonal principle framework by introducing a k+1-dimensional
spherical coordinates. In numerical implementation, this is not necessary, we may
conveniently work with the original coordinates system while matching the above
analysis.

Also numerically the two mins can be put into one to form a two-level algorithm.
A similar method has been proposed independently in [14] in the framework of
LSP which actually uses global min and max in its mathematical formulation while
our local min-orthogonal principle uses local min and max in its mathematical
formulation as showed in the previous sections. Since some numerical examples
on W-type problems are already carried out in [14], we will not go further on this
algorithm.

5.2. Justification of the min-max-min algorithm. In the min-max-min algo-
rithm (A2), for each v ∈ SL⊥ , u ∈ [v, L], ‖u‖ ≈ 1, denote

p̄(u) = tuu where tu = arg min
t>0

J(tu) (5.5)

and

p(v) = p̄(uv) = tvuv where uv = arg max
u∈[v,L],‖u‖≈1

J(p̄(u)).



EJDE-2023/SI/02 SHORT TITLE 185

Assume p̄ is continuous and suppose the projection w of J ′(p(v)) onto the subspace
[v, L] satisfies w = J ′(p(v))[v,L] 6= 0. For s > 0 small, we let

uv(s) =
uv + sw

‖uv + sw‖
∈ S[v,L] (5.6)

and p̄(uv(s)) = tsuv(s) = ts
uv+sw
‖uv+sw‖ for some ts > 0. Thus p̄(uv(s)) → p̄(uv) as

s→ 0. For each s > 0, we define

g(λ) = J(
tsuv

‖uv + sw‖
+ λ

tssw

‖uv + sw‖
).

By the mean-value theorem, there is 0 < λs < 1 such that

g′(λs) = g(1)− g(0) = J(p̄(uv(s)))− J(
tsuv

‖uv + sw‖
) ≤ J(p̄(uv(s)))− J(p̄(uv)),

since ts → tv‖uv‖ as s→ 0 and when s > 0 is small, we have J( tsuv

‖uv+sw‖ ) ≥ J(p̄(uv))

by (5.5). On the other hand when s > 0 is small, we have ts → tv‖uv‖ and then

g′(λs) = 〈J ′( tsuv
‖uv + sw‖

+ λs
tssw

‖uv + sw‖
),

tssw

‖uv + sw‖
〉 > 1

4
tss‖w‖2,

where the last inequality is due to the facts that as s→ 0, J ′( tsuv

‖uv+sw‖+λs
tssw

‖uv+sw‖ )→
J ′(p̄(uv)) and 〈J ′(p̄(uv)), w〉 = ‖w‖2 > 0. Consequently as s > 0 small, we have

J(p̄(uv(s)))− J(p̄(uv) = J(p̄(uv(s)))− J(p(v) >
1

4
tss‖w‖2 > 0, (5.7)

which violates (5.2) unless w = J ′(p(v))[v,L] = 0 or J ′(p(v))⊥[v, L].
Thus we have proved that the min-max-min algorithm (A2) fits into the lo-

cal min-orthogonal principle framework. On the other hand, denote F (t, u) =
〈J ′(tu), u〉 = 0, when

F ′t = 〈J ′′(tuu)u, u〉 6= 0 (> 0 for W-type), (5.8)

by the implicit function theorem, t′(u) = p̄′(u) is actually locally C1 at u. For
many problems in application, p̄(u) = tuu has an explicit expression, then the
algorithm becomes to solve minv∈S

L⊥ maxu∈[v,L],‖u‖≈1 J(p̄(u)), a two-level local
min-max algorithm.

6. Numerical examples

Since the M-type problems have been successfully solved before, in this section,
we present numerical multiple solutions to a W-type problem by the local min-max-
min algorithm (A2) and to a mixed M-W-type problem by the local min-orthogonal
algorithm.

6.1. Numerical examples by the mini-max-min method. A W-type prob-
lem. In (1.1), we set κ = 1, F (x, u(x)) = |u(x)|p−1u(x) with H = H1

0 (Ω),Ω =
(0, 1)2 ⊂ R2 and p = 3.

The operator −∆ subject to zero Dirichlet boundary condition. has eigenvalues
µi = 4.9348, 12.3370, 12.3370, 19.7392, 24.6740, 24.6740, 32.0762, 32.0762, . . . . It
is known that when µn < λ < µn+1, the problem has at least n pairs of solutions.

The local min-max-min algorithm (A2) has numerical solutions u1–u10 shown in
Figs. 3-4, and their numerical data documented in Table 1.



186 M. LI, B. JI, J. ZHOU EJDE/SI/02

Table 1. W-type solutions

nth figure MI support J(·) ‖J ′(·)‖ errmax Nit
1 (u1) 0 NA -313.4176 0.0002 0.0044 8
2 (u2) 1 1 -126.9724 0.0007 0.0052 10
3 (u3) 1 1 -126.9722 0.0004 0.0053 8
4 (u4) 1 1 -109.1923 0.0052 0.0296 9
5 (u5) 1 1 -109.1914 0.006 0.0719 6
6 (u6) 2 [1,2] -31.7034 0.001 0.0075 7
7 (u7) 3 [1,2,6] -5.1074 0.0029 0.0457 5
8 (u8) 3 [1,2,6] -5.1073 0.0014 0.0304 6
9 (u9) 3 [1,2,4] -4.2985 0.001 0.0215 4
10 (u10) 3 [1,2,4] -4.2607 0.0011 0.0709 5

(u1) (u2)

(u3) (u4)

Figure 3. Critical points u1–u4 when λ = 28

As stated in the last paragraph of Section 5, for this problem, p̄(u) has an
explicit expression, the local min-max method can actually be used to find multiple
solutions. Much more numerical details can be found in [3].

6.2. Numerical examples by the min-orthogonal method. A mixed M-W-
type problem. Consider numerically solutions to the problem mixed with concave
and convex nonlinearities [1, 2, 11]

−∆u+ λ(x)u− a|u(x)|q−1u(x)− b|u(x)|p−1u(x) = 0, (6.1)



EJDE-2023/SI/02 SHORT TITLE 187

(u5) (u6)

(u7) (u8)

(u9) (u10)

Figure 4. Critical points u5–u10 when λ = 28

where u ∈ H = H1
0 (Ω), Ω ⊂ RN is open bounded, 0 < q < 1 < p < 2∗, 2∗ = N+2

N−2

if N ≥ 3 and 2∗ =∞ if N ≤ 2, a and b are nonnegative functions on Ω. Its energy
functional is

J(u) =

∫
Ω

[1
2
|∇u(x)|2 +

1

2
λ(x)u2(x)− a(x)

q + 1
|u(x)|q+1 − b(x)

p+ 1
|u(x)|p+1

]
dx. (6.2)

We note that its function profile shows a mixed locally M-type and locally W-type
feature, see Figure 2, and its Nehari manifold N defined in (1.7) consists of two
branches, one is a local M-type where J-values are positive and the other is a local
W-type where J-values are negative. Usually two different variational methods,
e.g., the local min-max method for the locally M-type branch and the min-min-
max algorithm (A1) for the locally W-type branch have to be used separately. We
use the min-orthogonal method described in Section 3, since the ⊥-operation does
not differentiate and can treat both types. The difficulty is how the algorithm



188 M. LI, B. JI, J. ZHOU EJDE/SI/02

search will consistently stay on the desired branch. To resolve this difficulty, in our
numerical computation we add a constraint J > 0 or J < 0 to the ⊥-operation.

Case 1. We set λ(x) = 0, a(x) = 1.4, b(x) = 1, p = 4, q = 0.05, r = 0, and
Ω = (−1, 1)2.
Locally W-type saddles with J < 0. The algorithm finds the solutions in
Figure 5 and generates the numerical data in Table 2.

Table 2. Numerical data for W-type saddles

‖d‖ ε ‖J ′(·)‖∞ J(·) NIt
A Local Min 0.0007 1e-3 0.0031 -0.4314 9

1-saddle 1 0.0008 1e-3 1.7330 -0.1589 8
1-saddle 2 0.0064 7e-3 1.9031 -0.1445 6
2-saddle 1 0.0009 1e-3 1.6276 -0.0930 7
2-saddle 2 0.0008 1e-3 1.6924 -0.0671 8
3-saddle 0.0007 1e-3 1.7164 -0.0756 8

J = −0.4314,
‖u‖∞ = 0.3898 at

(−0.0005,−0.0032)

J = −0.1589,
‖u‖∞ = 0.1428 at
(0.5003, 0.0031)

J = −0.1445,
‖u‖∞ = 0.1497 at

(−0.3853,−0.3897)

(a) A Local Min (b) 1-saddle 1 (c) 1-saddle 2

J = −0.0930,
‖u‖∞ = 0.0901 at

(−0.5012, 0.4955), sid
= a, b

J = −0.0671,
‖u‖∞ = 0.0714 at
(0.0018,−0.6103),

sid= a, d

J = −0.0756,
‖u‖∞ = 0.0662 at

(0.6662, 0.0038), sid
= a, b, d

(d) 2-saddle 1 (e) 2-saddle 2 (f) 3-saddle

Figure 5. Locally W-type saddles with with J < 0

Locally M-type saddles with J > 0 The algorithm finds the solutions in Figure 6
and generates the numerical data in Table 3.
Case 2. To investigate possible bifurcation phenomenon, we set λ(x) = 0, a(x) =
1.4, b(x) = 1, p = 4, q = 0.05, r = 4, and Ω = (−1, 1)2.



EJDE-2023/SI/02 SHORT TITLE 189

Table 3. Numerical data for locally M-type saddles

‖d‖ ε ‖J ′(·)‖∞ J(·) NIt
1-saddle 0.0008 1e-3 0.0281 2.1680 18

2-saddle 1 0.0009 1e-3 1.7601 18.0417 45
2-saddle 2 0.0009 1e-3 0.0630 19.5637 31
2-saddle 3 0.0009 1e-3 0.0556 19.5638 31
3-saddle 0.0029 3e-3 1.7163 56.6547 44

4-saddle 1 0.0007 1e-3 0.9094 53.2727 36
4-saddle 2 0.0036 4e-3 1.5387 65.1722 60

Locally M-type saddles with J > 0 The algorithm finds the solutions in Figure 7
and generates the numerical data in Table 4.

Table 4. Numerical data for locally M-type saddles

‖d‖ ε ‖J ′(·)‖∞ J(·) NIt
1-saddle 0.0009 1e-3 0.0916 17.6073 98

2-saddle 1 0.0009 1e-3 1.6902 38.7233 98
2-saddle 2 0.0009 1e-3 1.6926 39.8931 93
2-saddle 3 0.0010 1e-3 1.6629 39.8933 93
3-saddle 0.0009 1e-3 0.9003 56.6546 171

4-saddle 1 0.0010 1e-3 1.7205 82.5416 154

From Figure 7 (a), a symmetry breaking phenomenon can be clearly observed.
To further explore such a bifurcation process, we use the same initial guess u0, the
eigenfunction of −∆ corresponding to the first eigenvalue µ1. Thus u0 is symmetric
about the x-axis, the y-axis and the lines x = y and x = −y. The min-orthogonal
method is applied with L = {0} for the first NMO times and then it is followed by a
Newton method to speedup local convergence. Since the Newton method is symme-
try invariant and insensitive to numerical errors [10], we obtain the following three
solutions with different symmetries as in Figure 8, a typical symmetry-breaking
phenomenon due to a bifurcation process along the parameter r.

Since in this case, the locally W-type saddles did not show any bifurcation phe-
nomenon and their solution profiles are similar to those in Figure 5, they are omitted
here. It is interesting to compare the three solutions in Figure 8 with the three so-
lutions in [18, Figure 9], where a = 0, r = 2 and NMO = 4, 14, 20. They have
exactly the same symmetries in order. Since a = 0, the latter is an M-type prob-
lem. The differences in NMO values imply that the latter is much easier to bifurcate
to an asymmetric solution. This may indicate the influence by the concave term
−a|u(x)|q−1u(x) in the problem.

Final remarks. From a double-orthogonal principle, which does not have to have
a variational structure, to a local min-orthogonal principle, which needs only a
general variational structure, to its numerical algorithm, we established its solution
characterization, step size rule and convergence. Then we used the local min-
orthogonal principle as a general mathematical framework to justify two algorithms,
a min-max-min algorithm and a min-min-max algorithm, used to solve W-type
problems for multiple solutions. In the final section, to illustrate the flexibilities of



190 M. LI, B. JI, J. ZHOU EJDE/SI/02

J = 2.1680,
‖u‖∞ = 2.1990 at

(−0.0005,−0.0032)

J = 18.0417,
‖u‖∞ = 3.3582 at
(0.3972, 0.3990)

J = 19.5637,
‖u‖∞ = 3.4559 at
(0.5003, 0.0031)

(a) 1-saddle (b) 2-saddle 1 (c) 2-saddle 2

J = 19.5638,
‖u‖∞ = 3.4559 at

(−0.0014,−0.5010)

J = 56.6547,
‖u‖∞ = 4.6177 at

(−0.0005,−0.0032), sid
= a, c

J = 53.2727,
‖u‖∞ = 3.9597 at

(0.4978, 0.5000), sid
= a, c, d

(d) 2-saddle 3 (e) 3-saddle (f) 4-saddle 1

J = 65.1722,
‖u‖∞ = 4.3352 at

(−0.6056, 0.0019), sid
= a, b, c

4-saddle 2

Figure 6. Locally M-type saddles with J > 0

the method developed in the previous sections, we present some numerical examples
for solving W-type problems and, in particular, for mixed M-W-type problems
where both convex and concave nonlinearities are present, for multiple solutions.
As a trade-off for the generalization of the method, so far we can establish only
a one side bound (inequality) estimate for the Morse index of a solution found by
the local min-orthogonal method [3], not as an equality established by the local
min-max method [17].



EJDE-2023/SI/02 SHORT TITLE 191

J = 17.6073,
‖u‖∞ = 4.6509 at
(0.6375, 0.6408)

J = 38.7233,
‖u‖∞ = 4.7529 at

(−0.6686,−0.6668)

J = 39.8931,
‖u‖∞ = 4.8060 at
(−0.6855, 0.6760)

(a) 1-saddle (b) 2-saddle 1 2-saddle 2

J = 39.8933,
‖u‖∞ = 4.8022 at
(0.6838,−0.6831)

J = 56.6546,
‖u‖∞ = 4.9724 at

(−0.7159,−0.7175), sid
= a, c

J = 82.5416,
‖u‖∞ = 4.8665 at

(0.6910,−0.6946), sid
= a, c, d

(d) 2-saddle 3 (e) 3-saddle (f) 4-saddle

Figure 7. Locally M-type saddles with J > 0

J = 46.1140,
‖u‖∞ = 4.5370 at
(0.0208,−0.0104)

J = 29.4731,
‖u‖∞ = 5.7561 at
(0.6510,−0.0052)

J = 17.6390,
‖u‖∞ = 4.6441 at
(0.6458,−0.6354)

(a)NMO = 15 NMO = 90 NMO = 110

Figure 8. Locally M-type positive solutions with Newton’s method



192 M. LI, B. JI, J. ZHOU EJDE/SI/02

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Meiqin Li

School of Engineering, University of Virginia, Charlottesville, VA 22904, USA
Email address: ml2vq@virginia.edu

Bingbing Ji

MicroStrategy Inc., Tysons Corner, VA 22182 USA
Email address: bingbing.tamu@gmail.com

Jianxin Zhou

Texas A&M University, College Station, TX 77843, USA
Email address: jzhou@math.tamu.edu


	1. Introduction
	2. A new local min-orthogonal method
	3. A numerical local min-orthogonal algorithm
	4. Algorithm convergence analysis
	5. Two new variations to the min-orthogonal method
	5.1. Justification of the min-min-max algorithm
	5.2. Justification of the min-max-min algorithm

	6. Numerical examples
	6.1. Numerical examples by the mini-max-min method
	6.2. Numerical examples by the min-orthogonal method
	Final remarks

	References