Ai In Cybersecurity

This project focuses on ai in cybersecurity using modern AI and machine learning techniques. The content below is adapted from research literature and practical implementation notes.

INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR

KIRCHHOFF TYPE PROBLEMS IN R3

Jijiang Sun

Lin Li

School of Mathematics and Statistics

Matija Cencelj

Faculty of Education and Faculty of Mathematics and Physics

Boštjan Gabrovšek

Faculty of Mechanical Engineering and Faculty of Mathematics and Physics

Abstract. In this paper, we consider the following nonlinear Kirchhoff type

 Z

− a+b |∇u|2 ∆u + V (x)u = f (u), in R3 ,  R3 u ∈ H 1 (R3 ), 

where a, b > 0 are constants, the nonlinearity f is superlinear at infinity with subcritical growth and V is continuous and coercive. For the case when f is odd in u we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik-

Schnirelman type minimax method. To the best of our knowledge, there are

only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity |u|p−2 u with p ∈ (2, 4].

Key words and phrases. Infinitely many sign-changing solutions, Kirchhoff type problems, in-

J. Sun was supported by NSFC (No.11501280, No.11861046) and the Natural Science Foun-

dation of Jiangxi Province (No.20181BAB201004). L. Li was supported by the National Natu- ral Science Foundation of China (No. 11601046), Chongqing Science and Technology Commis-

CXTDX201601026). M. Cencelj and B. Gabrovšek were supported by the Slovenian Research

Agency grants J1-8131, J1-7025, N1-0064 and N1-0083. ∗ Corresponding author.

2 JIJIANG SUN, LIN LI, M. CENCELJ, B. GABROVŠEK

1. Introduction and Main Results

In this paper we are interested in establishing the multiplicity of sign-changing

solutions to the following nonlinear Kirchhoff type problem

 Z

 2 − a+b |∇u| ∆u + V (x)u = f (u), in R3 ,

(1.1) R 3

where a, b > 0 are constants, V ∈ C(R3 , R), and f ∈ C(R, R).

Problems like (1.1) have been widely investigated because they have a strong

physical meaning. Indeed, (1.1) is related to the stationary analogue of the equation

Z

(1.2) utt − a + b |∇u|2 dx ∆u = f (x, u) Ω

proposed by Kirchhoff in as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. In , Lions proposed an abstract framework for the problem and after that, problem (1.2) began to receive a lot of attention. In (1.1), if we set V (x) = 0 and replace R3 and f (u) by a bounded domain Ω ⊂ RN and f (x, u), respectively, then we get the following Kirchhoff type equation

 Z

 − a+b |∇u|2 ∆u = f (x, u), x ∈ Ω,

(1.3) Ω

 u = 0, x ∈ ∂Ω. R The above problem is a nonlocal one as the appearance of the term Ω |∇u|2 dx implies that (1.3) is not a pointwise identity. This phenomenon causes some math- ematical difficulties, which make the study of (1.3) particularly interesting. In recent years, by using variational methods, the solvability of equation (1.3) with subcritical or critical growth nonlinearity has been paid much attention by various authors, see, e.g. [6, 24, 26, 27, 32, 33] and the references therein. For the results

concerning the existence of sign-changing solutions for (1.3), we refer the reader to papers [25, 29, 38] which depend heavily on the nonlinearity term with 4-superlinear growth at infinity in the sense that

F (x, t)

lim = +∞, uniformly in x ∈ Ω, |t|→∞ t4 Rt where F (x, t) = 0 f (x, s)ds and [23, 36] with the nonlinearity f (x, u) may not be 4-superlinear at infinity. If we replace f (u) by f (x, u) in (1.1), several authors have considered the follow- ing problem

Z

(1.4) − a+b |∇u| ∆u + V (x)u = f (x, u), x ∈ RN . RN

In recent years, there have been enormous results on existence, nonexistence and

multiplicity of nontrivial solutions for such problem depending on the assumptions of the potential V and f . See, for example, [13, 16, 35] and the references therein. Recently, replacing a and b by ε2 a and εb in (1.4), respectively, many researches have studied a certain concentration phenomena for the following Kirchhoff type

SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS 3

 Z

 −ε2 a + εb |∇u|2 ∆u + V (x)u = f (x, u), in R3 ,

 R3

u > 0, u ∈ H 1 (R3 ), see e.g. [8, 9, 10, 22, 34]. We mention that there are only few works concerning the existence of sign-changing solutions for (1.4). We are only aware of the works with prescribed numbers of nodal domains for (1.4) in Hr1 (R3 ), the subspace of radial functions of H 1 (R3 ) by using a Nehari manifold and gluing solution pieces together, when V (x) = V (|x|) satisfies (V1′ ) V ∈ C([0, +∞), R) is bounded below by a positive constant V0 ; and f (x, u) = f (|x|, u) satisfies the following hypotheses:

(f1′ ) f (r, u) ∈ C 1 ([0, +∞) × R, R) is odd in u for every r ≥ 0; (f2′ ) f (r, u) = o(|u|) as u → 0 uniformly in r ≥ 0; (f3′ ) for some constant p ∈ (4, 6), limu→+∞ fu(r,u) p−1 = 0 uniformly in r ≥ 0; ′ F (r,u) Ru (f4 ) limu→+∞ u4 = +∞, where F (r, u) = 0 f (r, t)dt; (r,u) (f5′ ) f|u| 3 is an increasing function of u ∈ R \ {0} for every r ≥ 0.

In , Huang and Liu studied the existence of least energy sign-changing solutions

with exactly two nodal domains for a variant of (1.4):

Z

2 2 − 1+λ (|∇u| + V (x)u ) [∆u + V (x)u] = |u|p−2 u, x ∈ RN . RN where λ > 0, p ∈ (4, 6) and V is assumed to guarantee the compactness. Ye proved the existence of least energy sign-changing solutions for equation (1.4) with f (x, u) = f (u) (i.e., (1.1)) by using constrained minimization of the sign-changing

Nehari manifold and Brouwer degree theory under the conditions that V satisfies

(V1 ) V ∈ C(R3 , R) satisfies inf R3 V (x) ≥ V0 > 0 for some positive constant V0 and is coercive, i.e., lim V (x) = ∞, |x|→∞

and the nonlinearity f ∈ C 1 (R, R) satisfies the following assumptions: (fe1 ) lims→0 f (s) |s|3 = 0;

(fe2 ) there exists 3 < q < 2∗ − 1 such that lim|s|→+∞ f|s|(s) q = 0, where 2 ∗ = +∞ ∗ if N = 2 and 2 = 6 if N = 3; Rs (fe3 ) lim|s|→+∞ Fs(s) 4 = +∞, where F (s) = 0 f (t)dt; (fe4 ) the function f|s|(s)

3 is nondecreasing on R \ {0}.

To the best of our knowledge, there is no result in the literature on the existence of multiple sign-changing solutions for problems (1.1) and (1.4) without any symmetry.

Motivated by the above works, in the present paper we study the existence of

infinitely many sign-changing solutions for problem (1.1) with coercive potential V , that is, (V1 ) holds and more general assumptions on f . More precisely, we assume that f satisfies the following assumptions: (f1 ) f ∈ C(R, R) and |f (u)| ≤ C(1 + |u|p−1 ) for some C > 0 and p ∈ (2, 6); (f2 ) f (u) = o(u) as u → 0; (f3 ) there exists µ > 2 such that µ1 f (u)u ≥ F (u) > 0 for all u ∈ R \ {0}, where Ru F (u) = 0 f (s)ds; (f4 ) f is odd, i.e., f (−u) = −f (u).

4 JIJIANG SUN, LIN LI, M. CENCELJ, B. GABROVŠEK

Now we state our first main result. Theorem 1.1. Suppose that (V1 ) and (f1 )–(f4 ) hold and µ > 4. Then problem (1.1) admits infinitely many sign-changing solutions.

Remark. The assumptions (V1 ) plays a role only in guaranteeing the compactness

of the (PS) sequence for the energy functional I associated with (1.1). We point out that Theorem (1.1) also holds when working in Hr1 (R3 ) if V is a positive constant.

Recall that (f3 ) is the so-called Ambrosetti-Rabinowitz condition ((AR) for

short). It is easy to see that µ > 4 guarantees the Palais-Smale ((PS) for short) sequence for I at any c ∈ R is bounded. But if µ < 4, f may not be 4-superlinear at infinity, due to the effect of the nonlocal term, it is difficult to get a bounded (PS) sequence for I. Motivated by , to overcome this difficulty, in the case µ < 4, we suppose that V (x) satisfies the following additional condition (V2 ) V is weakly differentiable, (DV (x), x) ∈ Lr (R3 ) for some r ∈ [ 23 , ∞] and

µ−2 V (x) − (DV (x), x) ≥ 0 for a.e. x ∈ R3 , where µ is given by (f3 ). It is worth mentioning that this assumption is different from that of . Li and

Ye assumed

V (x) − (DV (x), x) ≥ 0 for a.e. x ∈ R3 , and then obtained a positive ground state solution to (1.1) with f (u) = |u|p−2 u (p ∈ (3, 6)) by using the constrained minimization on a suitable Pohozaev-Nehari manifold. We remark that the case 2 < p ≤ 3 is not included in their result. Then we have the following result. Theorem 1.2. Suppose that (V1 )–(V2 ) and (f1 )–(f4 ) hold. Then problem (1.1) admits infinitely many sign-changing solutions. Remark. (i) To the best of our knowledge, there is no existence result for sign-

changing solutions to (1.1) in the literature even in the special case f (u) = |u|p−2 u with 2 < p ≤ 4. (ii) There exists function V (x) satisfying the assumptions (V1 )–(V2 ). For exam- ple, let V (x) = ln(1 + |x|) + . µ−2 |x| Clearly, (V1 ) holds. Moreover, for x ∈ R3 \ {0}, (DV (x), x) = 1+|x| . Therefore, ∞ 3 3 (DV (x), x) ∈ L (R ) and for a.e. x ∈ R , µ−2 µ−2 |x| µ−2 V (x) − (DV (x), x) = ln(1 + |x|) + 1 − ≥ ln(1 + |x|) ≥ 0,

2 2 1 + |x| 2 |x| µ+2 and so condition (V2 ) holds. Another example is V (x) = ln(1 + |x|2 ) − 1+|x| 2 + µ−2 .

One can also check that V (x) satisfies (V1 )–(V2 ). Motivated by [15, 31], we will prove Theorems 1.1 and 1.2 by applying the usual

Ljusternik-Schnirelman type minimax method in conjunction with invariant set

method. More precisely, we will construct certain invariant sets of the gradient flow corresponding to the energy functional I such that all positive and negative solutions are contained in these invariant sets and then minimax arguments can be applied to construct sign-changing solutions outside these invariant sets. The

SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS 5

method of invariant sets of descending flow has been used widely in dealing with sign-changing solutions of elliptic problem, see [2, 3, 19] and the references therein. But for the nonlocal problem, there are serious technical difficulties to overcome. Here we would like to point out the difficulties we will encounter and our main ideas.

Due to the effect of the nonlocal term, the arguments of constructing invariant

sets of descending flow in [2, 20, 31] cannot be directly applied to problem (1.1).

To overcome this difficulty, we will adopt some ideas from which studied the

existence of infinitely many sign-changing solutions for a Schrödinger-Poisson sys- construct an auxiliary operator A (see Lemma 3.4 below) from which we can con- struct closed convex sets containing all the positive and negative solutions in their interior. However, A itself cannot be used to defined the desired invariant sets of the flow, because the operator A is merely continuous under our assumptions. Inspired by , from A, we can get a locally Lipschitz continuous operator B (see Lemma

3.7 below) which inherits the main properties of A. Then, we can use B to define

the flow (see Lemma 3.8 below). Finally, by using a suitable deformation lemma in the presence of invariant sets (see Lemma 3.9 below) and minimax procedures, we prove that problem (1.1) has infinitely many sign-changing solutions. As mentioned above, if µ < 4, the nonlinearity term f may not be 4-superlinear at infinity. It prevents us from obtaining a bounded (PS) sequence, let alone (PS) condition holds for I. Therefore, the above arguments cannot be applied directly to prove Theorem 1.2. To overcome the obstacle, inspired by , we consider the

perturbed functionals Iλ : E → R (see (4.1) below) defined by Z λ Iλ (u) = I(u) − |u|r , λ ∈ (0, 1], r R3 where r ∈ (max{4, p}, 6), here p is from (f1 ). It will be shown that Iλ admits infinitely many sign-changing critical points {uλk }k≥2 by using the above framework. Then, by using a Pohozaev identity and (V2 ), we can prove that uλk → uk strongly in E as λ → 0+ and then the existence of infinitely many sign-changing solutions for (1.1) are obtained. The remainder of this paper is organized as follows. In Section 2 we derive a

variational setting for problem (1.1) and give some preliminary lemmas. We will prove Theorem 1.1 in Section 3. Section 4 is devoted to the proof of Theorem 1.2.

2. Variational setting and preliminary lemmas

Throughout this paper, we use the standard notations. We denote by C, ci , Ci , i = 1, 2, · · · for various positive constants whose exact value may change from lines to lines but are not essential to the analysis of problem.

R k·kq denotes the usual norm of

Lq (R3 ) for q ∈ [2, ∞]. For simplicity, we write R3 h to mean the Lebesgue integral of h(x) over R3 . For a functional J : E → R, we set J b := {u ∈ E : J(u) ≤ b}. We use “→” and “⇀” to denote the strong and weak convergence in the related function space respectively. We will write o(1) to denote quantity that tends to 0 as n → ∞.

Our argument is variational. In the paper, we work in the following Hilbert space

Z

E := u ∈ H (R ) : V (x)u < ∞

6 JIJIANG SUN, LIN LI, M. CENCELJ, B. GABROVŠEK

Z 12

2 2 kukE = (a|∇u| + V (x)|u| ) . R3

We denote its inner product by (·, ·)E . It is well known that E ֒→ Ls (R3 ) is continuous for s ∈ [2, 6]. Moreover, as in , we have the following result which plays an important role in our proof. Lemma 2.1. Under (V1 ), the embedding E ֒→ Ls (R3 ) is compact for any s ∈ [2, 6).

Remark. Indeed, as in , (V1 ) can be replaced by the more general condition. (V1′ ) V ∈ C(R3 , R) satisfies inf R3 V (x) ≥ V0 > 0 and there exists r0 > 0 such that for any M > 0, lim m {x ∈ R3 : V (x) ≤ M } ∩ {x ∈ R3 : |x − y| ≤ r0 } = 0. |y|→∞

Since E ֒→ L2 (R3 ) and L2 (R3 ) is a separable Hilbert space, E has a count- able orthogonal basis {ei }. In the following, for any m ∈ N, we denote Em := span{e1 , e2 , · · · , em }.

Under our assumptions, it is standard to show that the weak solutions to (1.1)

correspond to the critical points of the energy functional I ∈ C 1 (E, R) defined by

Z Z 2 Z

1 b (2.1) I(u) = a|∇u|2 + V (x)|u|2 + |∇u|2 − F (u),

2 R3 4 R3 R3

Ru where F (u) = 0 f (s)ds. Moreover, for any u, v ∈ E, we have

Z Z Z Z

hI ′ (u), vi = (a∇u · ∇v + V (x)uv) + b |∇u|2 ∇u · ∇v − f (u)v.

R3 R3 R3 R3

3. proof of Theorem 1.1

In this section, we devote to prove the existence of infinitely many sign-changing

solutions to problem (1.1) with µ > 4 by using a combination of invariant sets method and Ljusternik-Schnirelman type minimax results.

3.1. Some technical lemmas. In order to construct the minimax values for the

functional I defined in (2.1), the following three technical lemmas are needed.

Lemma 3.1. Under the assumptions (f1 )-(f3 ), the functional I satisfies the (PS)

Proof. By (f3 ), it is easy to check that any (PS) sequence for I at level c ∈ R is bounded. Thus, by Lemma 2.1, one can follow the same way as in the proof of

Lemma 4 in to complete the present proof.

Lemma 3.2. Suppose (f1 )-(f3 ) hold and m ≥ 1. Then there exists Rm = R(Em ) > 0, such that sup I < 0, c

BR ∩Em

c where BR := E \ BR and BR := {u ∈ E : kukE ≤ R}.

SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS 7

Proof. By (f1 )-(f3 ), there exists c1 , c2 > 0 such that F (t) ≥ c1 |t|µ − c2 |t|2 for all t ∈ R. Thus, Z

1 b

I(u) ≤ kuk2E + 2 kuk4E − F (u)

Z Z

1 b

≤ kuk2E + 2 kuk4E − c1 |u|µ + c2 |u|2 . 2 4a R3 R3 Since µ > 4 and any norm in finite dimensional space is equivalent, one concludes that lim I(u) = −∞ u ∈ Em kukE → ∞

for any fixed m ≥ 1. Therefore the result follows. Lemma 3.3. Assume (f1 ) and (f2 ) hold. Then there exist ρ > 0 and α > 0 such that inf I ≥ α. ∂Bρ

Proof. By (f1 ) and (f2 ), for any δ > 0, there exists Cδ > 0 such that (3.1) |f (t)| ≤ δ|t| + Cδ |t|p−1 for t ∈ R. Then, for u ∈ E, we have

Z 2 Z

1 b

I(u) = kuk2E + |∇u|2 − F (u)

2 4 R3 R3

(3.2) Z Z

1 δ Cδ ≥ kuk2E − |u|2 − |u|p . 2 2 R3 p R3 By the Sobolev embedding theorem, for any r ∈ [2, 6], there exists C(r) > 0 such that kukr ≤ C(r)kukE . Choose δ satisfying that C(2)δ < 21 . Then, it follows from (3.2) that

1 Cδ C(p)

I(u) ≥ kuk2E − kukpE .

4 p

Noting that p > 2, we conclude that there exist ρ, α > 0 such that inf ∂Bρ I ≥ α, as required.

3.2. Invariant subsets of descending flow. In order to construct a descending

flow guaranteeing existence of the desired invariant sets for the functional I, we introduce an auxiliary operator A : E → E. Precisely, for any u ∈ E, we define v = A(u) the unique solution to the following equation

Z

(3.3) − a+b |∇u| ∆v + V (x)v = f (u), v ∈ E. R3 Clearly, the set of fixed points of A is the same as the set of critical points of I. Lemma 3.4. The operator A is well defined and continuous.

Proof. Let u ∈ E, and define

Z Z Z Z

1 1 J(v) = a+b |∇u|2 |∇v|2 + V (x)v 2 − f (u)v, v ∈ E.

2 R3 R3 2 R3 R3

Obviously, J ∈ C 1 (E, R). And it is easy to check that J is weakly lower semicon- tinuous.

8 JIJIANG SUN, LIN LI, M. CENCELJ, B. GABROVŠEK

By (3.1) and the Sobolev embeddings, one has

Z Z Z

f (u)v ≤ δ |u||v| + Cδ |u|p−1 |v| ≤ δkuk2 kvk2 + Cδ kukp−1 p kvkp

(3.4) R 3 R 3 R 3

Z Z Z Z

1 2 1

J(v) = a+b |∇u| |∇v|2 + V (x)v 2 − f (u)v

2 R3 R3 2 R3 R3

≥ kvk2E − CkvkE → +∞, as kvkE → ∞, where C = C(ε, u) is a constant depending on ε and u. Therefore, J is coercive. By (3.4), it is easy to see that J is bounded below and maps bounded sets into bounded sets. Now we prove J is also strictly convex. In fact,

Z Z Z

′ ′ 2 2 hJ (v) − J (w), v − wi = a + b |∇u| |∇(v − w)| + V (x)|v − w|2

R3 R3 R3

≥ kv − wkE > 0, if v 6= w, where

Z Z Z Z

′ 2 hJ (v), ϕi = a + b |∇u| ∇v · ∇ϕ + V (x)vϕ − f (u)ϕ, ϕ ∈ E.

R3 R3 R3 R3

Thus, by Theorems 1.5.6 and 1.5.8 in , J admits a unique minimizer v = A(u) ∈ E, which is the unique solution to (3.3). Moreover, by (3.4), A maps bounded sets into bounded sets. In the following, we prove that the operator A is continuous. Let {un } ⊂ E with un → u in E strongly. Denote v = A(u) and vn = A(un ). Then we have

Z Z Z Z

2 2 2 kvn − vkE = b |∇u| ∇v∇(vn − v) − b |∇un | ∇vn · ∇(vn − v)

3 R3 R3 R3

ZR + (f (un ) − f (u))(vn − v) R3

Z Z Z

≤b |∇u|2 − |∇un |2 ∇v · ∇(vn − v)

3 R3 R3

Z R

+ (f (un ) − f (u))(vn − v) R3

, I1 + I2 .

By the Hölder inequality and Sobolev embedding theorem,

Z Z

I1 ≤ b |∇u| − |∇un |2 kvk2 kvn − vk2

R3 R3

(3.5) Z Z

≤C |∇u| − |∇un |2 kvkE kvn − vkE .

R3 R3

Now, we estimate the second term I2 . The proof is similar to that of . Let φ ∈ C0∞ (R) be such that φ(t) ∈ [0, 1] for t ∈ R, φ(t) = 1 for ktk ≤ 1 and φ(t) = 0 for ktk ≥ 2. Let g1 (t) = φ(t)f (t), g2 (t) = f (t) − g1 (t). Then, by (f1 ) and (f2 ), there

SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS 9

exists C > 0 such that |g1 (s)| ≤ C|s| and |g2 (s)| ≤ C|s|p−1 for s ∈ R. Thus,

Z Z

I2 = (g1 (un ) − g1 (u))(vn − v) + (g2 (un ) − g2 (u))(vn − v)

R3 R3

Z 12 Z 12

≤ |g1 (un ) − g1 (u)|2 |vn − v|2

R3 R3

Z p−1 p Z p1 + |g2 (un ) − g2 (u)|p |vn − v|p

R3 R3

"Z 12 Z p−1 # p

2 p

≤ Ckvn − vkE |g1 (un ) − g1 (u)| + |g2 (un ) − g2 (u)| ,

R3 R3

which, jointly with (3.5), implies " Z Z Z 12 2 2 2 kvn − vkE ≤ C |∇u| − |∇un | kvkE + |g1 (un ) − g1 (u)|

R3 R3 R3

Z p−1 # p p + |g2 (un ) − g2 (u)| . R3

Therefore, noting that un → u in E and by the dominated convergence theorem, one has kvn − vkE → 0 as n → ∞. This proof is completed.

Now we summarize some properties of the operator A which are useful to study

our problem. Lemma 3.5. (i) hI ′ (u), u − A(u)i ≥ ku − A(u)k2E for all u ∈ E. (ii) kI ′ (u)k ≤ ku − A(u)kE (1 + Ckuk2E ) for some C > 0 and all u ∈ E . (iii) For M > 0 and α > 0, there exists β > 0 such that ku − A(u)kE ≥ β for any u ∈ E with |I(u)| ≤ M and kI ′ (u)k ≥ α. (iv) If f is odd, then so is A. Proof. (i) Noting that A(u) is the solution to (3.3), for u ∈ E, it is easy to see that

Z Z

′ 2 hI (u), u − A(u)i = a + b |∇u| |∇(u − A(u))|2

R3 R3

Z (3.6) + V (x)|u − A(u)|2 R3 ≥ ku − A(u)k2E . (ii) For any ϕ ∈ E, By the Hölder inequality, one has

Z Z

hI ′ (u), ϕi = (u − A(u), ϕ)E + b |∇u|2 ∇(u − A(u)) · ∇ϕ

R3 R3

≤ ku − A(u)kE kϕkE + Ckuk2E ku − A(u)kE kϕkE , which implies kI ′ (u)k ≤ ku − A(u)kE (1 + Ckuk2E ) for all u ∈ E, here C = b/a2 > 0 is a constant. (iii) Since

Z Z

′ 2 hI (u), ui = (u − A(u), u)E + b |∇u| ∇(u − A(u))∇u,

R3 R3

10 JIJIANG SUN, LIN LI, M. CENCELJ, B. GABROVŠEK

I(u) − (u − A(u), u)E

Z 2

1 1 2 1 1 2 = − kukE + − b |∇u| 2 µ 4 µ R3

Z Z Z

b 2 1 + |∇u| ∇(u − A(u))∇u + f (u)u − F (u) µ R3 R3 R3 µ

Z 2

1 1 1 1 ≥ − kuk2E + − b |∇u|2 2 µ 4 µ R3

Z Z

b + |∇u|2 ∇u · ∇(u − A(u)). µ R3 R3

Z 2

Z Z !

≤ C |I(u)| + kukE ku − A(u)kE + |∇u| ∇u · ∇(u − A(u)) .

R3 R3

By the Hölder inequality and Young inequality, for any ε > 0,

Z Z

R3 R3

Z Z 12 Z 21

2 2 2 ≤ |∇u| |∇u| |∇(u − A(u))|

R3 R3 R3

Z 2

≤ε |∇u|2 + C(ε)kuk2E ku − A(u)k2E . R3

Then, for ε small enough, from (3.7), we have

(3.8) kuk2E ≤ C1 (|I(u)| + kukE ku − A(u)kE + kuk2E ku − A(u)k2E ). Arguing indirectly, suppose that there exists {un } ⊂ E with |I(un )| ≤ M and kI ′ (un )k ≥ α such that kun − A(un )kE → 0 as n → ∞. Then it follows from (3.8) that {kun kE } is bounded. Thus, by (ii), one concludes that kI ′ (un )k → 0 as n → ∞, which is a contradiction.

The conclusion (iv) is obviously, and the proof is completely.

Here and in the sequel, define the convex cones

P + := {u ∈ E : u ≥ 0} and P − := {u ∈ E : u ≤ 0}.

For ε > 0, we denote

Pε+ := {u ∈ E : dist(u, P + ) < ε} and Pε− := {u ∈ E : dist(u, P − ) < ε}, where dist(u, P ± ) = inf v∈P ± ku − vkE . Obviously, Pε− = −Pε+ . Let W := Pε+ ∪ Pε− . It is easy to see that W is an open and symmetric subset of E and Q := E \ W contains only sign-changing functions. The following result shows that for ε small, all sign-changing solutions to (1.1) are contained in Q.

SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS 11

Lemma 3.6. There exists ε0 > 0 such that for any ε

Frequently Asked Questions

What is this project about?

This project covers practical implementation and research aspects of the topic using AI/ML techniques.

This section provides additional detailed analysis and supporting information derived from the research paper content to ensure comprehensive coverage of the topic with expanded discussion on key concepts, methods, and findings.

Speaker Verification Xvector

Ai In Cybersecurity

INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR

KIRCHHOFF TYPE PROBLEMS IN R3

Jijiang Sun

Lin Li

School of Mathematics and Statistics

Matija Cencelj

Faculty of Education and Faculty of Mathematics and Physics

Boštjan Gabrovšek

Faculty of Mechanical Engineering and Faculty of Mathematics and Physics

Abstract. In this paper, we consider the following nonlinear Kirchhoff type

  Z 

Schnirelman type minimax method. To the best of our knowledge, there are

Key words and phrases. Infinitely many sign-changing solutions, Kirchhoff type problems, in-

J. Sun was supported by NSFC (No.11501280, No.11861046) and the Natural Science Foun-

CXTDX201601026). M. Cencelj and B. Gabrovšek were supported by the Slovenian Research

2 JIJIANG SUN, LIN LI, M. CENCELJ, B. GABROVŠEK

1. Introduction and Main Results

In this paper we are interested in establishing the multiplicity of sign-changing

  Z 

(1.1) R 3

Problems like (1.1) have been widely investigated because they have a strong

 Z 

  Z 

(1.3) Ω

F (x, t)

 Z 

In recent years, there have been enormous results on existence, nonexistence and

SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS 3

  Z 

 R3

In , Huang and Liu studied the existence of least energy sign-changing solutions

 Z 

Nehari manifold and Brouwer degree theory under the conditions that V satisfies

3 is nondecreasing on R \ {0}.

Motivated by the above works, in the present paper we study the existence of

4 JIJIANG SUN, LIN LI, M. CENCELJ, B. GABROVŠEK

Remark. The assumptions (V1 ) plays a role only in guaranteeing the compactness

Recall that (f3 ) is the so-called Ambrosetti-Rabinowitz condition ((AR) for

Ye assumed

Ljusternik-Schnirelman type minimax method in conjunction with invariant set

SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS 5

Due to the effect of the nonlocal term, the arguments of constructing invariant

To overcome this difficulty, we will adopt some ideas from which studied the

3.7 below) which inherits the main properties of A. Then, we can use B to define

2. Variational setting and preliminary lemmas

R k·kq denotes the usual norm of

Our argument is variational. In the paper, we work in the following Hilbert space

 Z 

E := u ∈ H (R ) : V (x)u < ∞

6 JIJIANG SUN, LIN LI, M. CENCELJ, B. GABROVŠEK

Z  12

Under our assumptions, it is standard to show that the weak solutions to (1.1)

Z Z 2 Z

2 R3 4 R3 R3

Z Z Z Z

R3 R3 R3 R3

3. proof of Theorem 1.1

In this section, we devote to prove the existence of infinitely many sign-changing

3.1. Some technical lemmas. In order to construct the minimax values for the

Lemma 3.1. Under the assumptions (f1 )-(f3 ), the functional I satisfies the (PS)

Lemma 4 in to complete the present proof.

BR ∩Em

SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS 7

1 b

I(u) ≤ kuk2E + 2 kuk4E − F (u)

Z Z

1 b

Z 2 Z

1 b

I(u) = kuk2E + |∇u|2 − F (u)

2 4 R3 R3

(3.2) Z Z

1 Cδ C(p)

4 p

3.2. Invariant subsets of descending flow. In order to construct a descending

 Z 

Proof. Let u ∈ E, and define

 Z Z Z Z

 Z

 Z

Z

 Z

Z

 Z

Z

Z

Z 12

Z Z 2 Z

Z 2 Z

Z

Z Z Z Z

Z Z Z Z

Z Z Z

Z Z Z Z

Z Z Z

Z 12 Z 12

Z Z

Z 2

Z Z Z

Z 2

Z 2

Z Z 12 Z 21

Z 2