On the high energy solitary waves solutions for a generalized KP equation in bounded domain
UDC 517.9 We are mainly concerned with the existence of infinitely many high energy solitary waves solutions for a class of generalized Kadomtsev – Petviashvili equation (KP equation) in bounded domain. The aim of this paper is to fill the gap in the relevant literature stated in a previous paper (J...
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2022
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512306852528128 |
|---|---|
| author | Rochdi, Jebari Rochdi, Jebari Rochdi, Jebari |
| author_facet | Rochdi, Jebari Rochdi, Jebari Rochdi, Jebari |
| author_sort | Rochdi, Jebari |
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| datestamp_date | 2025-03-31T08:44:52Z |
| description | UDC 517.9
We are mainly concerned with the existence of infinitely many high energy solitary waves solutions for a class of generalized Kadomtsev – Petviashvili equation (KP equation) in bounded domain. The aim of this paper is to fill the gap in the relevant literature stated in a previous paper (J. Xu, Z. Wei, Y. Ding, Stationary solutions for a generalized Kadomtsev – Petviashvili equation in bounded domain, Electron. J. Qual. Theory Differ. Equ., 2012, № 68, 1 – 18 (2012)). Under more relaxed assumption on the nonlinearity involved in KP equation, we obtain a new result on the existence of infinitely many high energy solitary waves solutions via a variant fountain theorems. |
| doi_str_mv | 10.37863/umzh.v74i3.6253 |
| first_indexed | 2026-03-24T03:26:42Z |
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| fulltext |
DOI: 10.37863/umzh.v74i3.6253
UDC 517.9
Rochdi Jebari (Dep. Math., College Sci. and Humanities-Al Quwayiyah, Shaqra Univ., Riyadh, Kingdom of Saudi
Arabia; Dep. Math., Faculté Sci. Tunis, Univ. Tunis El-Manar, Tunisie)
ON THE HIGH ENERGY SOLITARY WAVES SOLUTIONS
FOR A GENERALIZED KP EQUATION IN BOUNDED DOMAIN
РОЗВ’ЯЗКИ У ВИГЛЯДI СОЛIТОНОВИХ ХВИЛЬ ДЛЯ УЗАГАЛЬНЕНОГО
РIВНЯННЯ КАДОМЦЕВА – ПЕТВIАШВIЛI В ОБМЕЖЕНIЙ ОБЛАСТI
We are mainly concerned with the existence of infinitely many high energy solitary waves solutions for a class of
generalized Kadomtsev – Petviashvili equation (KP equation) in bounded domain. The aim of this paper is to fill the gap in
the relevant literature stated in a previous paper (J. Xu, Z. Wei, Y. Ding, Stationary solutions for a generalized Kadomtsev –
Petviashvili equation in bounded domain, Electron. J. Qual. Theory Differ. Equ., 2012, № 68, 1 – 18 (2012)). Under more
relaxed assumption on the nonlinearity involved in KP equation, we obtain a new result on the existence of infinitely many
high energy solitary waves solutions via a variant fountain theorem.
Розглядається, головним чином, iснування нескiнченної кiлькостi розв’язкiв у виглядi солiтонових хвиль для узагаль-
неного рiвняння Кадомцева – Петвiашвiлi в обмеженiй областi. Мета цiєї роботи — заповнити пробiли в результатах,
якi вказанi у попереднiй роботi (J. Xu, Z. Wei, Y. Ding, Stationary solutions for a generalized Kadomtsev – Petviashvili
equation in bounded domain, Electron. J. Qual. Theory Differ. Equ., 2012, № 68, 1 – 18 (2012)). При бiльш слабких обме-
женнях на нелiнiйнiсть у рiвняннi Кадомцева – Петвiашвiлi за допомогою варiанта теореми про фонтан отримано
новий результат щодо iснування нескiнченного числа розв’язкiв у виглядi солiтонових хвиль.
1. Introduction. The Kadomtsev – Petviashvili equation (KP equation) with variable coefficients has
been proposed some time ago [1 – 4]. The motivation was to describe water waves that propagate
in straits, or rivers, rather than on unbounded surfaces, like oceans. This equation appear in many
physic fields, see for example [5, 6] and the references therein. There are two distinct versions of the
KP equation, which can be written in normalized form as follows:
(ut + 6uux + uxxx)x + 3\sigma 2uyy = 0 (1.1)
or, in the ”integrated” form
ut + 6uux + uxxx + 3\sigma 2\partial - 1
x uyy = 0, (1.2)
where u = u(t, x, y) is a scalar function, x and y are respectively the longitudinal and transverse
spatial coordinates, subscripts x, y, t denote partial derivatives,
\partial - 1
x f(x) =
1
2
\left( x\int
- \infty
f(t) dt -
\infty \int
x
f(t) dt
\right)
and \sigma 2 =+
- 1. The case \sigma = 1 is known as the KPII equation, and models, for instance, water waves
with small surface tension. The case \sigma = i is known as the KPI equation, and may be used to model
waves in thin films with high surface tension. The presence of the nonlocal operator \partial - 1
x \partial 2y imposes a
constraint on the solution u of the KP equation, which, in some sense, has to be an x-derivative (see
[7, 8]). This last equation (among other completely integrable systems) was studied extensively by
means of algebro-geometric techniques [9], Hirota bilinear method [10] and reduction method [11].
c\bigcirc ROCHDI JEBARI, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3 311
312 ROCHDI JEBARI
A solitary wave or solitary wave solution of (1.2) is a solution of the form u(t, x, y) = v(x - ct, y),
where c > 0 is fixed, were studied by Ablowitz et al. [12]. Consequently, solitary wave solution
are important, because their properties can provide a useful platform for explaining many unusual
dynamical behaviors of various physical equations (see [13 – 16]).
The generalized KP equation is written in the following form:
ut + uxxx + (f(u))x = D - 1
x \Delta yu, (1.3)
where (t, x, y) := (t, x, y1, . . . , yn - 1) \in \BbbR + \times \BbbR \times \BbbR n - 1, n \geq 2, D - 1
x h(x, y) =
\int x
- \infty
h(s, y)ds and
\Delta y =
\sum n - 1
k=1
\partial 2
\partial y2k
. The equation (1.3) were studied by many authors (see [17 – 23]).
In [17], B. Xuan studied the existence of multiple stationary solutions of Generalized KP equation
in a bounded domain with smooth boundary and for superlinear conditions of nonlinearity f(u) =
= \lambda | u| p - 2u+ | u| q - 2u where 1 \leq p, q < 2\ast =
2(2n - 1)
2n - 3
. The techniques used in [17] are based on
variational methods. In [18, 19], by means of constrained minimization method, Bouard et al. studied
the existence and nonexistence of solitary waves when f(u) = u
k
l , where k, l are relatively prime
and l is odd. In the Chapter 7 of [20], Willem extended the results of [18] to the case where n = 2
and with an autonomous continuous nonlinearity f(u). In [21], Xuan extended the result in [20] to
higher spatial dimension with f \in C(\BbbR ,\BbbR ). Their results were obtained by applying the mountain
pass theorem of Ambrosetti – Rabinowitz [28] and Lusternik – Schnirelman theory.
In [23], J. Xua et al. studied the existence of multiple solitary waves for the generalized KP
equation (1.3) in one-dimensional spaces when f(u) = \mu | u| \mu - 1 and 1 < \mu < 2. Their methods were
based on variant fountain Theorem [24].
To our knowledge, all known results are concerned with the case that f is autonomous. Except in
paper [22], Z. Liang et al. studied the existence of nontrivial solution for the limiting case f(x, y, u) =
= Q(x, y)up - 2u. Here, some compactness property for the energy functional like the Palais – Smale
condition [24] were used.
Inspired by the above facts, in the present paper we consider a more general problem (1.4)
ut + uxxx + (f(x, y, u))x = D - 1
x \Delta yu in \Omega ,
D - 1
x u| \partial \Omega = 0, u| \partial \Omega = 0.
(1.4)
Note here that the nonlinearity f is non autonomous. Such equation are of scientific and practical
interest because of the variety of applications involving solitary wave propagation in inhomogeneous
media [25 – 27]. We recall that in the above papers, the high energy solitary waves solutions have
not been studied. Under more general assumptions on the nonlinearity f which are much strong
assumptions than used in paper [23], we obtain a new result on the existence of infinitely many high
energy solitary waves solutions for the problem (1.4), (see Theorem 2 in Section 3). Such result are
obtained by using some special proof techniques.
This paper is organized as follows. In Section 2, we recall some basic preliminaries. In Section 3
we give some lemmas and finally, we prove our result.
2. Preliminaries and functional setting. In this section we introduce some preliminaries which
used in our paper. Let c > 0, substituting u(x - ct, y) in (1.4), we obtain
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
ON THE HIGH ENERGY SOLITARY WAVES SOLUTIONS FOR A GENERALIZED KP EQUATION . . . 313
- cux + uxxx + (f(x, y, u))x = D - 1
x \Delta yu (2.1)
or \bigl(
- uxx +D - 2
x \Delta yu+ cu - f(x, y, u)
\bigr)
x
= 0. (2.2)
Note that we can rewrite (1.4) in the following form (see [17, p. 12]):
- uxx +D - 2
x \Delta yu+ cu = f(x, y, u) in \Omega ,
D - 1
x u| \partial \Omega = 0, u| \partial \Omega = 0.
(2.3)
Definition 2.1. For \Omega \subset \BbbR n is a bounded domain with smooth boundary \partial \Omega on Y := \{ gx :
g \in C\infty
0 \} , we define the inner product
(u, v) =
\int
\Omega
\bigl[
uxvx +D - 1
x \nabla yu.D
- 1
x \nabla yv + cuv
\bigr]
dV (2.4)
where \nabla y =
\biggl(
\partial
\partial y1
, . . . ,
\partial
\partial yn - 1
\biggr)
, dV = dx dy and the corresponding norm
\| u\| :=
\left( \int
\Omega
\bigl[
u2x + | D - 1
x \nabla yu| 2 + cu2
\bigr]
dV
\right) 1
2
. (2.5)
A function u : \Omega \rightarrow \BbbR belongs to E, if there exists \{ um\} +\infty
m=1 \subset Y such that:
(a) um \rightarrow u a.e. on \Omega ,
(b) \| uj - uk\| \rightarrow 0 as j, k \rightarrow \infty .
Note that the space E with inner product (2.4) and norm (2.5) is a Hilbert space, see [22]
(Definition) and [17, p. 12, 13].
For each v \in E, multiply the both sides of the above equation in (2.3) by v(x, y) and integrate
over \Omega to obtain\int
\Omega
\biggl(
- \partial 2
\partial x2
u
\biggr)
v dV +
\int
\Omega
(D - 2
x \Delta yu )v dV + c
\int
\Omega
uv dV =
\int
\Omega
f(x, y, u) v dV (2.6)
and then we obtain by Green formula and integration by parts,\int
\Omega
\partial
\partial x
u.
\partial
\partial x
v dV +
\int
\Omega
D - 1
x \nabla yu.D
- 1
x \nabla yv dV + c
\int
\Omega
uv dV =
\int
\Omega
f(x, y, u)v dV. (2.7)
Therefore, on E, define a functional \phi as
\phi (u) :=
1
2
\int
\Omega
\bigl[
u2x + | D - 1
x \nabla yu| 2 + cu2
\bigr]
dV -
\int
\Omega
F (x, y, u) dV =
=
1
2
\| u\| 2 - \psi (u) (2.8)
where F (x, y, u) :=
\int u
0
f(x, y, s) ds and \psi (u) :=
\int
\Omega
F (x, y, u) dV.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
314 ROCHDI JEBARI
Lemma 2.1 (see ([17], Lemma 1). The embedding from the space (E, \| .\| ) into the space
(Lp(\Omega ), \| .\| p) is compact for 1 \leq p < p with p =
2(2n - 1)
2n - 3
> 2. In add, there exists \tau p > 0
such that
\| u\| p \leq \tau p\| u\| , p \in [1, p), for all u \in E (2.9)
where \| u\| p =
\biggl( \int
\Omega
| u| p dV
\biggr) 1
p
.
We assume that the nonlinearity f, satisfying the following hypotheses:
(f1) f \in C(\Omega \times \BbbR ,\BbbR ), f(x, y, u)u \geq 0 for all u \in \BbbR , (x, y) \in \Omega and there exists a constants
C > 0 and p \in (2, p) such that
| f(x, y, u)| \leq C(1 + | u| p - 1), for all u \in \BbbR and (x, y) \in \Omega .
(f2) f(x, y, u) = o(| u| ) as | u| \rightarrow 0 uniformly for (x, y) \in \Omega .
(f3) \mathrm{l}\mathrm{i}\mathrm{m}| u| \rightarrow \infty
F (x, y, u)
| u| 2
= +\infty uniformly for (x, y) \in \Omega .
(f4) There exists \theta \geq 1 such that \theta \varphi (u) \geq \varphi (\tau u) for all \tau \in [0, 1] and (x, y, u) \in \Omega \times \BbbR
where \varphi (u) = u f(x, y, u) - 2F (x, y, u).
(f5) f(x, y, - u) = - f(x, y, u) for all u \in \BbbR and (x, y) \in \Omega .
Example of a function f satisfying the above assumptions is
f(x, y, t) = a(x, y)| t| \nu - 2t
for all (x, y) \in \Omega and t \in \BbbR where \nu \in (2, p) and a is a continuous bounded function with positive
lower bound.
Lemma 2.2 (see [23]). Let (f1) holds. Then \phi \in C1(E,\BbbR ). Moreover, we have
\langle \psi \prime (u), v\rangle =
\int
\Omega
f(x, y, u)v dV (2.10)
and
\langle \phi \prime (u), v\rangle = (u, v) - (\psi \prime (u), v) = (u, v) -
\int
\Omega
f(x, y, u)v dV (2.11)
for all u, v \in E. We note that a critical points of \phi on E are weak solutions of (2.3).
For the convenience of the readers, we recall some notation which will be used later.
Let X be a Banach space with the norm \| .\| and let \{ Xj\} be a sequence of subspaces of X with
dimXj <\infty for each j \in \BbbN .
Further, X = \oplus j\in \BbbN Xj the closure of the direct sum of all \{ Xj\} .
Set Wk := \oplus k
j=0Xj and Zk := \oplus \infty
j=k+1Xj , for \rho k > rk > 0
Bk = \{ u \in Wk : \| u\| \leq \rho k\} and Sk = \{ u \in Zk : \| u\| = rk\} .
Consider a family of C1-functionals \phi \lambda : X \rightarrow \BbbR defined by
\phi \lambda (u) = A(u) - \lambda B(u), \lambda \in [1, 2]. (2.12)
Theorem 2.1 (see [24]). Assume that the functional \phi \lambda defined above satisfies
(A1) \phi \lambda maps bounded sets into bounded sets uniformly for \lambda \in [0, 1], and \phi \lambda ( - u) = \phi \lambda (u)
for all (\lambda , u) \in [1, 2]\times X;
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
ON THE HIGH ENERGY SOLITARY WAVES SOLUTIONS FOR A GENERALIZED KP EQUATION . . . 315
(A2) B(u) \geq 0 for all u \in X, and B(u) \rightarrow \infty as \| u\| \rightarrow \infty on any finite dimensional subspace
of X, or
(A3) B(u) \leq 0 for all u \in X, and B(u) \rightarrow - \infty as \| u\| \rightarrow \infty .
(A4) There exists \rho k > rk > 0 such that
bk(\lambda ) := \mathrm{i}\mathrm{n}\mathrm{f}
u\in Zk,\| u\| =rk
\phi \lambda (u) > ak(\lambda ) := \mathrm{m}\mathrm{a}\mathrm{x}
u\in Wk,\| u\| =\rho k
\phi \lambda (u) for all \lambda \in [1, 2].
Then bk(\lambda ) \leq ck(\lambda ) := \mathrm{i}\mathrm{n}\mathrm{f}\gamma \in \Gamma k
\mathrm{m}\mathrm{a}\mathrm{x}u\in Bk
\phi \lambda (\gamma (u)) for all \lambda \in [1, 2], where
\Gamma k =
\Bigl\{
\gamma \in C(Bk, X) : \gamma odd, \gamma | \partial Bk
= id
\Bigr\}
, k \geq 2.
Moreover, for almost every \lambda \in [1, 2] there exists a sequence ukn(\lambda ) such that
\mathrm{s}\mathrm{u}\mathrm{p}
n
\| ukn(\lambda )\| <\infty , \phi \prime \lambda (u
k
n(\lambda )) \rightarrow 0 and \phi \lambda (u
k
n(\lambda )) \rightarrow ck(\lambda ) as n\rightarrow \infty .
3. Existence of infinitely many high solitary waves energy solutions. In order to apply the
above theorem to prove our main results, we define the functional \phi \lambda on our working space E by
\phi \lambda (u) :=
1
2
\int
\Omega
\bigl[
u2x + | D - 1
x \nabla yu| 2 + cu2
\bigr]
dV - \lambda
\int
\Omega
F (x, y, u) dV =
1
2
\| u\| 2 - \lambda \psi (u) (3.1)
for all u \in E and \lambda \in [0, 1]. We use the some lemma to show the existence
Lemma 3.1. For the finite dimensional subspace F \subset E of E, there exists a constant \varepsilon 0 > 0
such that
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\bigl\{
(x, y) \in \Omega : | u(x, y)| \geq \varepsilon 0\| u\|
\bigr\}
\geq \varepsilon 0 \forall u \in F\setminus \{ 0\} . (3.2)
Proof. If not, for any n \in \BbbN \ast , there exists un \in F\setminus \{ 0\} such that
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\biggl\{
(x, y) \in \Omega : | un(x, y)| \geq
1
n
\| un\|
\biggr\}
<
1
n
\forall n \in \BbbN \ast . (3.3)
Let vn =
un
\| un\|
for all n \in \BbbN \ast , then \| vn\| = 1 for all n \in \BbbN \ast , and
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\biggl\{
(x, y) \in \Omega : | vn(x, y)| \geq
1
n
\biggr\}
<
1
n
\forall n \in \BbbN \ast . (3.4)
By the boundedness of \{ vn\} , passing to a subsequence if necessary, we may assume that vn \rightarrow v
with \| v\| = 1 in E for some v \in E since E is a finite dimension. By Lemma 2.1, we have\int
\Omega
| vn(x, y) - v(x, y)| 2 dV \rightarrow 0 as n\rightarrow \infty . (3.5)
Since v \not = 0, there exists a constant \delta 0 > 0 such that
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\bigl\{
(x, y) \in \Omega : | v(x, y)| \geq \delta 0
\bigr\}
\geq \delta 0. (3.6)
For any n \in \BbbN \ast , we set
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
316 ROCHDI JEBARI
Dn =
\biggl\{
(x, y) \in \Omega : | vn(x, y)| <
1
n
\biggr\}
, Dc
n =
\biggl\{
(x, y) \in \Omega : | vn(x, y)| \geq
1
n
\biggr\}
and D0 =
\bigl\{
(x, y) \in \Omega : | v(x, y)| \geq \delta 0
\bigr\}
. Thus for n large enough, by (3.4) and (3.6), we get
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(Dn \cap D0) \geq \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(D0) - \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(Dc
n) \geq
2\delta 0
3
. (3.7)
Consequently, for n large enough, we have\int
\Omega
| vn(x, y) - v(x, y)| 2 dV \geq
\int
Dn\cap D0
| vn(x, y) - v(x, y)| 2 dV \geq
\geq
\int
Dn\cap D0
\bigl[
| v(x, y)| 2 - 2vn(x, y)v(x, y)
\bigr]
dV \geq
\geq
\int
Dn\cap D0
\bigl[
| v(x, y)| 2 - 2| vn(x, y)| | v(x, y)|
\bigr]
dV \geq
\geq \delta 0
\biggl(
\delta 0 -
2
n
\biggr)
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(Dn \cap D0) \geq
2
9
\delta 30 > 0. (3.8)
This is in contradiction with (3.4). Therefore (3.2) holds.
Lemma 3.2. Assume that (f1) and (f3) hold. Then \psi (u) \geq 0 for all u \in E, and \psi (u) \rightarrow \infty
as \| u\| \rightarrow \infty on any finite dimensional subspace of E.
Proof. Evidently, from (f1), we have \psi (u) \geq 0 for all u \in E. Let H \subset E be any finite
dimensional subspace of E, next we will show that \psi (u) \rightarrow \infty as \| u\| \rightarrow \infty on H.
By (f3), there exists R > 0 such that
F (x, y, u) \geq | u| 2 for all (x, y) \in \Omega and | u| \geq R. (3.9)
Let Du =
\bigl\{
(x, y) \in \Omega : | u(x, y)| \geq \varepsilon 0\| u\|
\bigr\}
for u \in E\setminus \{ 0\} . By Lemma 3.1, we see that for any
u \in E with \| u\| \geq R
\varepsilon 0
we have | u(x, y)| \geq R, for all (x, y) \in Du. Hence, for any u \in E with
\| u\| \geq R
\varepsilon 0
, from (f1) and (3.9), we get
\psi (u) \geq
\int
Du
F (x, y, u) dV \geq
\int
Du
| u| 2 dV \geq
\geq \varepsilon 20\| u\| 2\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(Du) \geq \varepsilon 30\| u\| 2. (3.10)
This implies that \psi (u) \rightarrow \infty as \| u\| \rightarrow \infty on any finite dimensional subspace of E.
The proof is completed.
Let \{ ej\} be a total orthonormal basis of E and Xj = \BbbR ej , Wk := \oplus k
j=0Xj and Zk :=
:= \oplus \infty
j=k+1Xj .
Lemma 3.3. If p \in [1, p), then one has \alpha k(p) := \mathrm{s}\mathrm{u}\mathrm{p}u\in Zk,\| u\| =1 \| u\| p \rightarrow 0 as k \rightarrow \infty .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
ON THE HIGH ENERGY SOLITARY WAVES SOLUTIONS FOR A GENERALIZED KP EQUATION . . . 317
Proof. Firstly, \alpha k(p) is convergent science \alpha k(p) \geq 0 and \alpha k(p) is decreasing in k. Further-
more, for any k \in \BbbN , by the definition of \alpha k(p), there exists uk \in Zk such that \| uk\| = 1 and
\| uk\| p \geq
\alpha k(p)
2
.
For any v \in E, v =
\sum \infty
n=1
anen, it has
| \langle uk, v\rangle | =
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggl\langle
uk,
\infty \sum
n=1
anen
\Biggr\rangle \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \| uk\|
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
k=n+1
anen
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
k=n+1
anen
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \rightarrow 0, as k \rightarrow \infty .
which implies that uk \rightharpoonup 0 weakly in E. By virtue of Lemma 2.1, we can conclude uk \rightarrow 0 strongly
in Lp(\Omega ). The combination with implies that \alpha k(p) \rightarrow 0.
Lemma 3.4. Assume that (f1) and (f2) hold. Then there exists a sequences rk > 0, k \in \BbbN
such that
bk(\lambda ) := \mathrm{i}\mathrm{n}\mathrm{f}
u\in Zk,\| u\| =rk
\phi \lambda (u) > 0 (3.11)
uniformly for \lambda \in [1, 2].
Proof. By (f1) and (f2), for any \epsilon > 0, there exists a C\epsilon > 0 such that
| f(x, y, u)| \leq \epsilon | u| + C\epsilon | u| p - 1 for all u \in \BbbR . (3.12)
Let \alpha k(p) := \mathrm{s}\mathrm{u}\mathrm{p}u\in Zk,\| u\| =1 \| u\| p, from Lemma 3.3, we see that \alpha k(p) \rightarrow 0. Therefore, for uk \in Zk
and \epsilon small enough, by (3.12), we have
\phi \lambda (u) \geq
1
2
\| u\| 2 - \lambda \epsilon
2
\| u\| 22 -
\lambda \epsilon
p
\| u\| pp \geq
\geq 1
4
\| u\| 2 - c4\| u\| pp \geq
1
4
\| u\| 2 - c4\alpha
p
k(p)\| u\|
p. (3.13)
If we choose rk =
\bigl(
8c4\alpha
p
k(p)
\bigr) 1
2 - p then for any u \in Zk with \| u\| = rk, we get that
\phi \lambda (u) \geq
1
8
\bigl(
8c4\alpha
p
k(p)
\bigr) 1
2 - p > 0. (3.14)
This inequality implies that
bk(\lambda ) := \mathrm{i}\mathrm{n}\mathrm{f}
u\in Zk,\| u\| =rk
\phi \lambda (u) \geq
1
8
\bigl(
8c4\alpha
p
k(p)
\bigr) 1
2 - p > 0 for all \lambda \in [1, 2]. (3.15)
Lemma 3.5. Assume that (f1), (f2), and (f3) hold. Then for the positive integer k1 and the
sequence rk obtained in Lemma 3.4, there exists \rho k > rk > 0 for any k \geq k1 such that
ak(\lambda ) := \mathrm{m}\mathrm{a}\mathrm{x}
u\in Wk,\| u\| =\rho k
\phi \lambda (u) < 0 (3.16)
uniformly for \lambda \in [1, 2].
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
318 ROCHDI JEBARI
Proof. By Lemma 3.1, for any k \in \BbbN , there exists \varepsilon k > 0 constant such that
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(Su) \geq \varepsilon k \forall u \in Wk\setminus \{ 0\} , (3.17)
where Su =
\bigl\{
(x, y) \in \Omega : | u(x, y)| \geq \varepsilon k\| u\|
\bigr\}
. By (f3), for any k \in \BbbN , there exists a constant
Rk > 0 such that
F (x, y, u) \geq 1
\varepsilon 3k
| u| 2 \forall u \geq Rk. (3.18)
Hence, by (3.17), we see that for any u \in Wk with \| u\| \geq Rk
\varepsilon k
, we have | u(x, y)| \geq Rk for all
(x, y) \in Su. Therefore, for any u \in Wk with \| u\| \geq Rk
\varepsilon k
and \lambda \in [1, 2], by (3.17) and (3.18), we
have
\phi \lambda (u) \leq
1
2
\| u\| 2 -
\int
\Omega
F (x, y, u) dV \leq 1
2
\| u\| 2 -
\int
Su
F (x, y, u) dV \leq
\leq 1
2
\| u\| 2 -
\int
Su
1
\varepsilon 3k
| u| 2 dV \leq 1
2
\| u\| 2 - \varepsilon 2k\| u\| 2
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(Su)
\varepsilon 3k
\leq
\leq 1
2
\| u\| 2 - \| u\| 2 = - 1
2
\| u\| 2. (3.19)
If we choose \rho k > \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
rk,
Rk
\varepsilon k
\biggr\}
, we get that
ak(\lambda ) := \mathrm{m}\mathrm{a}\mathrm{x}
u\in Wk,\| u\| =\rho k
\phi \lambda (u) \leq -
r2k
2
< 0 \forall k \in \BbbN and for all \lambda \in [1, 2].
The proof is completed.
By using (3.12) and Lemma 2.1 we can see that \phi maps bounded sets to bounded sets uniformly
for \lambda \in [1, 2]. Moreover, by (f5), \phi \lambda is even. Then condition (A1) in Theorem 2.1 is satisfied.
Condition (A2) is clearly true, while (A4) follows by Lemma 3.4 and Lemma 3.5. Then, by
Theorem 2.1, for any k \geq k1 and \lambda \in [1, 2] there exists a sequence \{ ukn(\lambda )\} n such that
\mathrm{s}\mathrm{u}\mathrm{p}
n
\| ukn(\lambda )\| <\infty , \phi \prime \lambda (u
k
n(\lambda )) \rightarrow 0 and \phi \lambda (u
k
n(\lambda )) \rightarrow ck(\lambda ) as n\rightarrow \infty ,
where ck(\lambda ) = \mathrm{i}\mathrm{n}\mathrm{f}\gamma \in \Gamma k
\mathrm{m}\mathrm{a}\mathrm{x}u\in Bk
\phi \lambda (\gamma (u)), \forall \lambda \in [1, 2] and Bk, \Gamma k are given by
Bk =
\bigl\{
u \in Wk : \| u\| \leq \rho k
\bigr\}
and \Gamma k =
\Bigl\{
\gamma \in C(Bk, X) : \gamma odd, \gamma | \partial Bk
= id
\Bigr\}
, k \geq 2.
In particular, from the proof of Lemma 3.3, we deduce that for any k \geq k1 and \lambda \in [1, 2]
1
8
\bigl(
8c4\alpha
p
k(p)
\bigr) 2
2 - p = : bk \leq bk \leq ck.
Also since
ck(\lambda ) = \mathrm{i}\mathrm{n}\mathrm{f}
\gamma \in \Gamma k
\mathrm{m}\mathrm{a}\mathrm{x}
u\in Bk
\phi \lambda (\gamma (u)) \leq \mathrm{m}\mathrm{a}\mathrm{x}
u\in Bk
\phi \lambda (\gamma (u)) = ck.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
ON THE HIGH ENERGY SOLITARY WAVES SOLUTIONS FOR A GENERALIZED KP EQUATION . . . 319
Hence,
bk \leq ck(\lambda ) \leq ck. (3.20)
As a consequence, for any k \geq k1, we can choose \lambda m \rightarrow 1 (depending on k) and get the corre-
sponding sequences satisfying
\mathrm{s}\mathrm{u}\mathrm{p}
n
\bigm\| \bigm\| ukn(\lambda m)
\bigm\| \bigm\| <\infty , \phi \prime \lambda m
\bigl(
ukn(\lambda m)
\bigr)
\rightarrow 0 and \phi \lambda m
\bigl(
ukn(\lambda m)
\bigr)
\rightarrow ck(\lambda m) (3.21)
as n\rightarrow \infty .
Lemma 3.6. For each \lambda m given in [1, 2] such that \lambda m \rightarrow 1, the sequence \{ ukn(\lambda m)\} \infty n=1 has
a strong convergent subsequence \{ uk(\lambda m)\} m such that \phi \prime \lambda m
(uk(\lambda m)) = 0 and \phi \lambda m(u
k(\lambda m)) \in
\in [bk, ck] for all m \in \BbbN , k \geq k1.
Proof. By (3.21) we may assume, without loss of generality, that as n\rightarrow \infty ,
ukn(\lambda m)\rightharpoonup uk(\lambda m) in E. (3.22)
By Lemma 2.1 we have
ukn(\lambda m) \rightarrow uk(\lambda m) in Lp(\Omega ). (3.23)
By (f1) and (f2), for any \epsilon > 0, there exists C\epsilon > 0 such that
| f(x, y, u)| \leq \epsilon | u| + C\epsilon | u| p - 1 for all u \in \BbbR (3.24)
and Hölder inequality it follows that\bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
f(x, y, ukn(\lambda m))(ukn(\lambda m) - uk(\lambda m)) dV
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \epsilon \| ukn(\lambda m)\| 2\| ukn(\lambda m) - uk(\lambda m)\| 2 + C\epsilon \| ukn(\lambda m)\| p - 1
p \| ukn(\lambda m) - uk(\lambda m)\| p
so, by using (3.23), we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\int
\Omega
f(x, y, ukn(\lambda m))(ukn(\lambda m) - uk(\lambda m)) dV = 0
and
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\int
\Omega
\bigl[
f(x, y, ukn(\lambda m)) - f(x, y, uk(\lambda m))
\bigr]
(ukn(\lambda m) - uk(\lambda m)) dV = 0.
Observe that
\| ukn(\lambda m) - uk(\lambda m)\| 2 =
\Bigl\langle
\phi \prime \lambda m
(ukn(\lambda m) - \phi \prime \lambda m
(uk(\lambda m)
\Bigr\rangle
+
+
\int
\Omega
\bigl[
\lambda mf(x, y, u
k
n(\lambda m)) - f(x, y, uk(\lambda m))
\bigr]
(ukn(\lambda m) - uk(\lambda m)) dV (3.25)
it is clear that \Bigl\langle
\phi \prime \lambda m
(ukn(\lambda m) - \phi \prime \lambda m
(uk(\lambda m), ukn(\lambda m) - uk(\lambda m)
\Bigr\rangle
\rightarrow 0 (3.26)
as n\rightarrow \infty .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
320 ROCHDI JEBARI
By (3.25), we have \| ukn(\lambda m) - uk(\lambda m)\| \rightarrow 0 as n\rightarrow \infty .
As a consequence, we obtain
\phi \prime \lambda m
(uk(\lambda m)) = 0 and \phi \lambda m(u
k(\lambda m)) \in [bk, ck] (3.27)
for all m \in \BbbN , k \geq k1.
Lemma 3.7. For any k \geq k1, the sequence
\bigl\{
uk(\lambda m)
\bigr\} \infty
m=1
is bounded in E.
Proof. For simplicity we set um = uk(\lambda m). We suppose by contradiction that, up to a subse-
quence,
\| um\| \rightarrow \infty as m\rightarrow \infty . (3.28)
Let wm =
um
\| um\|
for any m \in \BbbN . Then, up to subsequence, we may assume that
wm \rightharpoonup w in E,
wm \rightarrow w in Lp(\Omega ), (3.29)
wm \rightarrow w a.e. in \Omega .
Now we distinguish two cases.
Case \bfitw = \bfzero . As in [29], we can say that for any m \in \BbbN there exists tm \in [0, 1] such that
\phi \lambda m(tmum) = \mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
\phi \lambda m(tum). (3.30)
Since (3.28) holds, for any j \in \BbbN , we can choose rj = 2
\surd
jwm such that
rj\| um\| - 1 \in (0, 1) (3.31)
provided m is large enough. By (3.29), F (., 0) = 0 and the continuity of F, we can see that
F (x, y, rjwm) \rightarrow F (x, y, rjw) = 0 a.e. (x, y) \in \Omega (3.32)
as m \rightarrow \infty for any j \in \BbbN . Then, taking into account (3.24), (3.29), (3.32), (A4) and by using the
Dominated Convergence Theorem we deduce that
F (x, y, rjwm) \rightarrow 0 in L1(\Omega ) (3.33)
as m\rightarrow \infty for any j \in \BbbN . Then (3.30), (3.31) and (3.33) yield
\phi \lambda m(tmum) \geq \phi \lambda m(rjwm) \geq 2j - \lambda m
\int
\Omega
F (x, y, rjwm) dV \geq j
for m is large enough and for any j \in \BbbN . As a consequence
\phi \lambda m(tmum) \rightarrow \infty as m\rightarrow \infty . (3.34)
Since \phi \lambda m(0) = 0 and \phi \lambda m(um) \in
\bigl[
bk, ck
\bigr]
, we deduce that tm \in (0, 1) for m large enough. Thus,
by (3.30) we have \bigl\langle
\phi \prime \lambda m
(tmum), tnum
\bigr\rangle
= tm
d
dt | t=tm
\phi \lambda m(tum) = 0. (3.35)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 3
ON THE HIGH ENERGY SOLITARY WAVES SOLUTIONS FOR A GENERALIZED KP EQUATION . . . 321
Taking into account (f4), (3.35) and (2.11) we obtain
1
\theta
\phi \lambda m(tmum) =
1
\theta
\biggl(
\phi \lambda m(tmum) - 1
2
\bigl\langle
\phi \prime \lambda m
(tmum), tmum
\bigr\rangle \biggr)
=
=
\lambda m
2\theta
\int
\Omega
\varphi (tmum) dV \leq \lambda m
2
\int
\Omega
\varphi (um) dV =
= \phi \lambda m(um) - 1
2
\bigl\langle
\phi \prime \lambda n
(um), um
\bigr\rangle
= \phi \lambda m(um)
which contradicts (3.27) and (3.34).
Case \bfitw \not = \bfzero . Thus the set \Omega \prime :=
\bigl\{
(x, y) \in \Omega : w(x, y) \not = 0
\bigr\}
has positive Lebesgue measure.
By using (3.28) and that w \not = 0, we have
| um(x, y)| \rightarrow \infty a.e. (x, y) \in \Omega \prime as m\rightarrow \infty . (3.36)
Putting together (3.27), (3.36) and (f3), and by applying Fatou’s Lemma, we can easily deduce that
1
2
- \phi \lambda m(um)
\| um\| 2
= \lambda m
\int
\Omega
F (x, y, um)
\| um\| 2
dV \geq
\geq \lambda m
\int
\Omega \prime
| wm| 2F (x, y, um)
| um| 2
dV \rightarrow \infty as m\rightarrow \infty
which gives a contradiction because of (3.27). Then, we have proved that the sequence \{ um\} is
bounded in E.
Theorem 3.1. Assume that (f2), (f3) – (f5) hold. Then problem (2.3) possesses infinitely many
high energy solutions uk \in E for every k \in \BbbN , in the sense that
1
2
\int
\Omega
\bigl[
(uk)
2
x + | D - 1
x \nabla yuk| 2 + cu2k
\bigr]
dV -
\int
\Omega
F (x, y, uk) dV \rightarrow +\infty (3.37)
as k \rightarrow \infty .
Proof. Taking into account Lemma 3.7 and (3.27), for each k \geq k1, we can use similar
arguments to those in the proof of Lemma 3.6, to show that the sequence
\bigl\{
uk(\lambda m)
\bigr\} \infty
m=1
admits
a strong convergent subsequence with the limit uk being just a critical point of \phi 1 = \phi . Clearly,
\phi (uk) \in
\bigl[
bk, ck
\bigr]
for all k \geq k1. Since bk \rightarrow \infty as k \rightarrow \infty in (3.20), we deduce the existence
of infinitely many nontrivial critical points of \phi . As a consequence, we have that (2.3) possesses
infinitely many nontrivial weak solutions.
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|
| id | umjimathkievua-article-6253 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:42Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f2/d6b3eb74c2a1231edf2556a73adce5f2.pdf |
| spelling | umjimathkievua-article-62532025-03-31T08:44:52Z On the high energy solitary waves solutions for a generalized KP equation in bounded domain On the high energy solitary waves solutions for a generalized KP equation in bounded domain Rochdi, Jebari Rochdi, Jebari Rochdi, Jebari Generalized KP equation, variant fountain theorems, variational methods, infinitely many high energy solitary waves solutions. UDC 517.9 We are mainly concerned with the existence of infinitely many high energy solitary waves solutions for a class of generalized Kadomtsev – Petviashvili equation (KP equation) in bounded domain. The aim of this paper is to fill the gap in the relevant literature stated in a previous paper (J. Xu, Z. Wei, Y. Ding, Stationary solutions for a generalized Kadomtsev – Petviashvili equation in bounded domain, Electron. J. Qual. Theory Differ. Equ., 2012, № 68, 1 – 18 (2012)). Under more relaxed assumption on the nonlinearity involved in KP equation, we obtain a new result on the existence of infinitely many high energy solitary waves solutions via a variant fountain theorems. УДК 517.9 Розв'язки у вигляді солітонових хвиль для узагальненого рівняння кадомцева–петвіашвілі в обмеженій області Розглядається, головним чином, існування нескінченної кількості розв'язків у вигляді солітонових хвиль для узагальненого рівняння Кадомцева–Петвіашвілі в обмеженій області. Мета цієї роботи — заповнити пробіли в результатах, які вказані у попередній роботі (J. Xu, Z. Wei, Y. Ding, Stationary solutions for a generalized Kadomtsev–Petviashvili equation in bounded domain, Electron. J. Qual. Theory Differ. Equ., 2012, No. 68, 1–18 (2012)). При більш слабких обмеженнях на нелінійність у рівнянні Кадомцева–Петвіашвілі за допомогою варіанта теореми про фонтан отримано новий результат щодо існування нескінченного числа розв'язків у вигляді солітонових хвиль. Institute of Mathematics, NAS of Ukraine 2022-04-26 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6253 10.37863/umzh.v74i3.6253 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 3 (2022); 311-322 Український математичний журнал; Том 74 № 3 (2022); 311-322 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6253/9199 Copyright (c) 2022 Rochdi Jebari |
| spellingShingle | Rochdi, Jebari Rochdi, Jebari Rochdi, Jebari On the high energy solitary waves solutions for a generalized KP equation in bounded domain |
| title | On the high energy solitary waves solutions for a generalized KP equation in bounded domain |
| title_alt | On the high energy solitary waves solutions for a generalized KP equation in bounded domain |
| title_full | On the high energy solitary waves solutions for a generalized KP equation in bounded domain |
| title_fullStr | On the high energy solitary waves solutions for a generalized KP equation in bounded domain |
| title_full_unstemmed | On the high energy solitary waves solutions for a generalized KP equation in bounded domain |
| title_short | On the high energy solitary waves solutions for a generalized KP equation in bounded domain |
| title_sort | on the high energy solitary waves solutions for a generalized kp equation in bounded domain |
| topic_facet | Generalized KP equation variant fountain theorems variational methods infinitely many high energy solitary waves solutions. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6253 |
| work_keys_str_mv | AT rochdijebari onthehighenergysolitarywavessolutionsforageneralizedkpequationinboundeddomain AT rochdijebari onthehighenergysolitarywavessolutionsforageneralizedkpequationinboundeddomain AT rochdijebari onthehighenergysolitarywavessolutionsforageneralizedkpequationinboundeddomain |