MULTIPLICITY RESULTS FOR GROUND STATE SOLUTIONS OF A SEMILINEAR EQUATION VIA ABRUPT CHANGES IN MAGNITUDE OF THE NONLINEARITY

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MULTIPLICITY RESULTS FOR GROUND STATE SOLUTIONS OF A

SEMILINEAR EQUATION VIA ABRUPT CHANGES IN

MAGNITUDE OF THE NONLINEARITY

CARMEN CORT ´

AZAR, MARTA GARC´

IA-HUIDOBRO, AND PILAR HERREROS

Abstract. Given k∈N, we deﬁne a class of continuous piecewise functions fhaving

abrupt but controlled magnitude changes so that the problem

∆u+f(u) = 0, x ∈RN, N > 2,

has at least kradially symmetric ground state solutions.

1. Introduction and main results

In this paper we deﬁne a class of continuous nonlinearities fso that the problem

∆u+f(u) = 0, x ∈RN, N > 2,

lim

|x|→∞ u(x) = 0,(1.1)

has multiple positive solutions. To this end we consider the radial version of (1.1), that

u00 +N−1

ru0+f(u) = 0, u(r)>0 for all r > 0, N > 2,

u0(0) = 0,lim

r→∞ u(r) = 0,

(1.2)

where all throughout this article u0denotes diﬀerentiation of uwith respect to r.

Any nonconstant solution to (1.1) is called a bound state solution. Bound state

solutions such that u(x)>0 for all x∈RNare referred to as a ﬁrst bound state

solution, or a ground state solution.

The uniqueness problem for positive solutions to problem (1.1) has been extensively

studied during the past decades, see for example [FLS, McLS, PeS2, PS, ST].

The multiplicity problem has been studied for the following non-autonomous problem

−∆u=f(x, u), u(x)→0 as |x|→∞

for fof the form f(x, u) = g(x, u)−a(x)uby [AT1, AT2, AW, CMP, CPS, CZ, DWY,

HL, WY, MP21]. Under diﬀerent assumptions on the nonnegative function gand the

coeﬃcient a, they have established existence of multiple ground state solutions. More

recently, Cerami and Molle [CM], considered f(x, u) = up−a(x)u−b(x)uq,q < p <

N+2

N−2, introducing the nonzero term b(x) and gave conditions to obtain inﬁnitely many

ground states. The autonomous case was studied in [DDG] for f(u) = −u+up+λuq

This research was supported by FONDECYT- 1190102 for the ﬁrst author, FONDECYT-1210241

for the second author and FONDECYT-1170665 for the third author.

arXiv:2210.03086v1 [math.AP] 6 Oct 2022

2 CARMEN CORT ´

AZAR, MARTA GARC´

IA-HUIDOBRO, AND PILAR HERREROS

with N= 3, 1 < q < 3, pnear 5, where they prove that there if λis large enough,

then there exist at least three radial ground state solutions to this problem. We also

mention the work by Wei and Wu, [WW], where the authors consider the nonlinearity

f(u) = |u|2∗−2u+λu+µ|u|q−2u, where 2∗=2N

N−2is the well known critical exponent and

among other results, they prove that if N= 3, 2 < q < 10/3, under some conditions in

µ > 0 the problem has a second ground state for some λ < 0.

In [CGHH] we established a multiplicity result for bound states by considering non-

linearities behaving like up−uat the start but having multiple sign changes so that

its primitive F(s) = Rs

0f(t)dt would have positive local maxima. In the present work,

we will deﬁne fpiecewise, starting with a function f1satisfying assumptions (H1)-(H4)

below so that problem (1.2) with f=f1has a unique ground state solution, and then

having abrupt but controlled “magnitude” changes. Therefore our results will consider

a nonlinearity of the form

f=f1χ[0,α∗+1]+

i=2

Li−1χ[αi−1,αi−1+i−1]+

k−1

i=2

ifiχ[αi−1+i−1,αi]+A2

kfkχ[αk−1+k−1,γ),

where k∈N,χEdenotes the characteristic function of the set E,Li−1is a linear function

deﬁned so that fis continuous, iand Aiare positive constants and the functions fi,

i≥2, are any positive continuous functions deﬁned in [α∗, γ), see Figure 1.

I I IIII I I

α∗

α1

α1+ǫ1α2

α2+ǫ2

α3

α3+ǫ3

α4

α4+ǫ4

2f2

3f3

4f4

5f5

Figure 1. Figura 1

α∗

α∗−δ

r∗rarb

R2R

Figure 2. Figura 2

Figure 1. A function ffor k= 5

MULTIPLE EXISTENCE OF GROUND STATES 3

We start with the case k= 2, so that

f(s) = 









f1(s)s≤α∗+1=α1

L1(s)α1≤s≤α1+1

2f2(s)s≥α1+1,

(1.3)

where L1(s) is the line from (α1, f1(α1)) to (α1+1, A2f2(α1+1)) and α∗is given by

(H4). The constants 1and A2will be determined.

The continuity assumption on fis crucial to guarantee continuous dependence of the

solutions on initial conditions. We have chosen the transition functions Lito be linear

for simplicity, and it could be avoided at the cost of imposing that f2be monotone

nondecreasing.

We will assume the following conditions on the nonlinearity f:

(H1)f1∈C[0,∞)∩C1(0,∞), f1(0) = 0 and there exist b≥0 such that f1(s)>0 for

s > b,f1(s)≤0 for s∈[0, b] and moreover f1(s)<0 on (0, ) for some  > 0;

also, by setting F1(s) = Rs

0f1(t)dt, we assume that there exists a unique ﬁnite

β≥bsuch that F(β) = 0.

(H2) (F1/f1)0(s)>(N−2)/(2N) for all s>β;

(H3)f1/(s−b) is increasing for all s>b.

(H4) There is an initial condition α∗such that the problem

u00 +N−1

ru0+f1(u) = 0, r > 0, N > 2,

u(0) = α∗, u0(0) = 0,

(1.4)

is a ground state solution.

(H5)fiis a positive continuous function deﬁned on [α∗, γ) for some α∗< γ ≤ ∞ for

all i≥2.

Our ﬁrst result is the following.

Theorem 1.1. Assume that f1,α∗and f2satisfy the assumptions above. Then, there

exist positive constants ¯and ¯

Asuch that for any 0< 1<¯and A2>¯

A, problem (1.1)

with fgiven by (1.3) has at least two ground state solutions.

Moreover, if fsatisﬁes a subcritical type condition at inﬁnity, similar to the one

introduced ﬁrst by Castro and Kurepa see [CK] and used by Gazzola, Serrin and Tang

in [GST],

(H6) Let Q(s) := 2NF (s)−(N−2)sf(s), where F(s) = Rs

0f(t)dt. We assume that

Qis bounded from below in (0,∞) and that there exists θ∈(0,1)

lim

s→∞ inf

s1,s2∈[θs,s]Q(s2)s

f(s1)N/2=∞,(1.5)

we can obtain a third solution to (1.1). We have

Theorem 1.2. Assume that f1,α∗and f2be as in Theorem 1.1 with γ=∞, and let 1

and A2be as in its conclusion. If fsatisﬁes (H6), then problem (1.1) has at least three

ground state solutions.

4 CARMEN CORT ´

AZAR, MARTA GARC´

IA-HUIDOBRO, AND PILAR HERREROS

We now consider the case k= 3, that is fgiven by

f(s) = 









f1(s)s≤α1

L1(s)α1≤s≤α1+1

2f2(s)α1+1≤s≤α2

L2(s)α2≤s≤α2+2

3f3(s)s≥α2+2,

(1.6)

where L1(s) is the line from (α1, f1(α1)) to (α1+1, A2

2f2(α1+1)), L2(s) is the line from

(α2, A2

2f2(α2)) to (α2+2, A2

3f3(α2+2)) and α1, 1and A2are constants that satisfy

Theorem 1.1. The constants α2, 2and A3will be determined later.

Theorem 1.3. Under assumptions (H1)-(H5), there exist positive constants 1,2,

A2and A3such that problem (1.1) with fgiven by (1.6) has at least three ground state

solutions.

Finally, we address the general case k≥4.

Theorem 1.4. Let fi,i= 1, . . . , k satisfy assumptions (H1)-(H5)For i= 2...k, there

exists constants i>0,Ai>0and αiwith the condition α∗< αi−1+i−1< αisuch that

problem (1.1) with

f=f1χ[0,α∗+1]+

i=2

Li−1χ[αi−1,αi−1+i−1]+

k−1

i=2

ifiχ[αi−1+i−1,αi]+A2

kfkχ[αk−1+k−1,γ)

has at least kground state solutions.

Remark: It will follow from the proof of Theorem 1.2 that if kis even and fksatisﬁes

(H6), we obtain a (k+ 1)th ground state.

Our results are based on the study of initial value problems of the form

u00 +N−1

ru0+f(u) = 0, r > 0, N > 2,

u(r0) = α, u0(r0) = ¯α

(1.7)

for some r0≥0, α > 0 and ¯α∈R−

0. Under our assumptions on fthe solution to this

problem is unique. In case that r0= 0, we set ¯α= 0 so it corresponds to a radially

symmetric solution of problem (1.1). In section 2 we give some preliminary properties

of these solutions.

In section 3 we study the solutions to (1.7) when u(r0) = α∗and u0(r0) = ¯α < 0, so

that f=f1in its range. Under our assumptions (H1) through (H4) we will use the well

known functional Pintroduced by Erbe and Tang in [ET], to compare our solutions to

the known positive solutions of (1.7). We prove that, under some conditions on r0and

¯α, these solutions stay positive.

Next, in section 4, we will study how the constants A2and 1aﬀect the solutions of

(1.7) with r0= 0, α > α∗and ¯α= 0, proving that for small enough 1the eﬀect can

be controlled. The interesting part is that for A2big enough, solutions will reach α∗

MULTIPLE EXISTENCE OF GROUND STATES 5

satisfying the conditions found in the previous section. Thus, there is a positive solution

with initial condition α∗

1> α∗, which implies the existence of a ground state solution

between them and proves Theorem 1.1.

When falso satisﬁes (H6), using results from [GST] we show that solutions with large

enough initial value αchange sign. This will prove the existence of a third ground state

solution with initial condition larger than the one before, proving Theorem 1.2.

We ﬁnish this section adding another magnitude change in f, making it small for values

of αlarger than the α∗

1found in Theorem 1.1. We prove that if A3is small enough there

will be another ground state solution with initial condition α∗

2> α∗

1, proving Theorem

1.3.

In section 5 we prove the general case, Theorem 1.4. Alternating between big and

small Aiwe can use the arguments from the previous theorems to determine recursively

the constants αi, iand Aiso that the solutions with initial condition αiare positive if

iis even, and change sign if iis odd, hence obtaining kground state solutions.

We ﬁnish the paper with some examples in section 6. For the case k= 2 they show

the diﬀerent behavior of solutions with large α, showing that some condition on f2is

necessary for solutions with large αto change sign.

We also include an Appendix, where we sketch the proofs of some of the properties

of solutions to the initial value problem (1.7) given in section 2. These are known facts,

but, to our knowledge there is no paper where the results are proven with our conditions.

2. Preliminaries

The aim of this section is to establish several properties of the solutions to the initial

value problem (1.7) in the case r0= 0 and ¯α= 0. with f∈C[0,∞) and such that (H1)

is satisﬁed. This problem has a unique solution deﬁned for all r > 0 for any α > 0 and

we denote it by u(·, α).

It can be seen that for α∈(b, ∞), one has u(r, α)>0 and u0(r, α)<0 for rsmall

enough, and thus we can deﬁne

R(α) := sup{r > 0|u(s, α)>0 and u0(s, α)<0 for all s∈(0, r)}.

Following [PeS1], [PeS2] we set

N={α∈(b, ∞) : u(R(α), α) = 0 and u0(R(α), α)<0}

G={α∈(b, ∞) : u(R(α), α) = 0 and u0(R(α), α) = 0}

P={α∈(b, ∞) : u(R(α), α)>0}.

The following propositions state some known facts, but, to our knowledge there is no

paper where the result is proven with our conditions. So, for the sake of completeness,

we will give a sketch of the proof in the appendix.

Proposition 2.1.

i) If fsatisﬁes (H1)and (H5), then the sets Nand Pare open sets.

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MULTIPLICITYRESULTSFORGROUNDSTATESOLUTIONSOFASEMILINEAREQUATIONVIAABRUPTCHANGESINMAGNITUDEOFTHENONLINEARITYCARMENCORTAZAR,MARTAGARCIA-HUIDOBRO,ANDPILARHERREROSAbstract.Givenk2N,wedeneaclassofcontinuouspiecewisefunctionsfhavingabruptbutcontrolledmagnitudechangessothattheproblemu+f(u)=0;x2RN;N>2;h...

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MULTIPLICITY RESULTS FOR GROUND STATE SOLUTIONS OF A SEMILINEAR EQUATION VIA ABRUPT CHANGES IN MAGNITUDE OF THE NONLINEARITY

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