The Nehari manifold approach for a $p(x)$ -Laplacian problem with nonlinear boundary conditions
We consider a class of $p(x)$-Laplacian equations that involve nonnegative weight functions with nonlinear boundary conditions. Our technical approach is based on the Nehari manifold, which is similar to the fibering method of Drabek and Pohozaev, together with the recent idea from Brown and Wu.
Saved in:
| Date: | 2017 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2017
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1678 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507511532027904 |
|---|---|
| author | Fallah, K. Rasouli, S. H. Фаллах, К. Расулі, С. Х. |
| author_facet | Fallah, K. Rasouli, S. H. Фаллах, К. Расулі, С. Х. |
| author_sort | Fallah, K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:23:35Z |
| description | We consider a class of $p(x)$-Laplacian equations that involve nonnegative weight functions with nonlinear boundary conditions. Our technical approach is based on the Nehari manifold, which is similar to the fibering method of Drabek and
Pohozaev, together with the recent idea from Brown and Wu. |
| first_indexed | 2026-03-24T02:10:29Z |
| format | Article |
| fulltext |
UDC 517.9
S. H. Rasouli, K. Fallah (Dep. Math., Babol Noshirvani Univ. Technology, Iran)
THE NEHARI MANIFOLD APPROACH FOR A \bfitp (\bfitx )-LAPLACIAN PROBLEM
WITH NONLINEAR BOUNDARY CONDITIONS
ПIДХIД НА ОСНОВI МНОГОВИДУ НЕХАРI ДО ПРОБЛЕМИ
\bfitp (\bfitx )-ЛАПЛАСIАНА З НЕЛIНIЙНИМИ ГРАНИЧНИМИ УМОВАМИ
We consider a class of p(x)-Laplacian equations that involve nonnegative weight functions with nonlinear boundary
conditions. Our technical approach is based on the Nehari manifold, which is similar to the fibering method of Drabek and
Pohozaev, together with the recent idea from Brown and Wu.
Розглянуто один клас p(x)-рiвнянь Лапласа, що включає невiд’ємнi ваговi функцiї з нелiнiйними граничними
умовами. Наш пiдхiд базується на многовидi Нехарi, що є подiбним до методу волокон Драбека та Похожаєва з
використанням нових iдей Брауна та Ву.
1. Introduction. The purpose of this paper is to study the existence and multiplicity of positive
solutions for the following nonlinear boundary-value problem involving the p(x)-Laplacian:
- \Delta p(x)u+m(x)| u| p(x) - 2u = \lambda f(x)| u| q(x) - 2u, x \in \Omega ,
| \nabla u| p(x) - 2 \partial u
\partial n
= g(x)| u| r(x) - 2u, x \in \partial \Omega ,
(1.1)
where \Omega \subset \BbbR N is a bounded domain, N \geq 2, p(x), q(x), r(x) \in C(\Omega ) such that 1 < q(x) <
< p(x) < r(x) < p\ast (x)
\biggl(
p\ast (x) =
Np(x)
N - p(x)
if N > p(x), p\ast (x) = \infty if N \leq p(x)
\biggr)
, 1 < p - :=
:= \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{i}\mathrm{n}\mathrm{f}x\in \Omega p(x) \leq p(x) \leq p+ := \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}x\in \Omega p(x) < \infty , 1 < q - \leq q+ < p - \leq p+ < r - \leq r+,
\lambda > 0 \in \BbbR the weight m(x) is a positive bounded function and f \in C(\Omega ), g \in C(\partial \Omega ) are
nonnegative weight functions with compact support in \Omega .
In this paper, we have generalized the articles of Afrouzi and Rasouli [1] and Wu [19 – 21], to the
p(x)-Laplacian by using the Nehari manifold under the certain conditions.
2. The space \bfitW \bfone ,\bfitp (\bfitx )(\bfOmega ). To discuss problem (1.1) we need some results on the space
W 1,p(x)(\Omega ) which we call variable exponent Sobolev space.
Let \Omega be a bounded domain of \BbbR n, we have
L\infty
+ (\Omega ) = \{ p \in L\infty (\Omega ) : p - > 1\} .
Let’s define by U(\Omega ) the set of all measurable real functions defined on \Omega . For any p \in L\infty
+ (\Omega ),
Lp(x)(\Omega ) =
\left\{ u \in U(\Omega ) :
\int
\Omega
| u(x)| p(x) dx < \infty
\right\} ,
we introduce a norm on Lp(x)(\Omega ), that so-called Luxemburg norm [11, 14],
| u| p(x) = \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ \delta > 0 :
\int
\Omega
\bigm| \bigm| \bigm| \bigm| u(x)\delta
\bigm| \bigm| \bigm| \bigm| p(x) dx \leq 1
\right\} ,
c\bigcirc S. H. RASOULI, K. FALLAH, 2017
92 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
THE NEHARI MANIFOLD APPROACH FOR A p(x)-LAPLACIAN PROBLEM . . . 93
then (Lp(x)(\Omega ), | \cdot | p(x)) becomes a Banach space, we call it as variable exponent Lebesgue space,
and
L
p(x)
c(x)(\Omega ) =
\left\{ u \in U(\Omega ) :
\int
\Omega
c(x)| u(x)| p(x) dx < \infty
\right\} ,
where c is a measurable real-valued function and c(x) > 0 for x \in \Omega .
We define
| u| (p(x)\cdot c(x)) = \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ \delta > 0 :
\int
\Omega
c(x)
\bigm| \bigm| \bigm| \bigm| u(x)\delta
\bigm| \bigm| \bigm| \bigm| p(x) dx \leq 1
\right\} .
Let us define the space
W 1,p(x)(\Omega ) = \{ u \in Lp(x)(\Omega ) : | \nabla u| \in Lp(x)(\Omega )\} .
In the Banach space W 1,p(x)(\Omega ) we introduce the norm which is equivalent to the standard one:
\| u\| p =
\left( \int
\Omega
(| \nabla u| p(x) +m(x)| u(x)| p(x)) dx
\right) p(x)
\forall u \in W 1,p(x)(\Omega ).
Let p\ast \partial (x) =
(N - 1) p(x)
N - p(x)
if N > p(x).
Theorem 2.1 [11]. The space
\bigl(
Lp(x)(\Omega ), | \cdot | p(x)
\bigr)
is a separable, uniformly convex Banach space,
and has conjugate Lp\prime (x)(\Omega ), where
1
p\prime (x)
+
1
p(x)
= 1. For any u \in Lp(x)(\Omega ) and v \in Lp\prime (x)(\Omega ),
we have \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
uv dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\biggl(
1
p -
+
1
(p\prime ) -
\biggr)
| u| p(x)| v| p\prime (x) \leq 2| u| p(x)| v| p\prime (x).
Theorem 2.2 [11]. Let \rho (u) =
\int
\Omega
| u(x)| p(x) dx \forall u \in Lp(x)(\Omega ), then
(i) | u| p(x) < 1 (= 1; > 1) if and only if \rho (u) < 1 (= 1;> 1),
(ii) | u| p(x) > 1 implies | u| p
-
p(x) \leq \rho (u) \leq | u| p
+
p(x),
(iii) | u| p(x) < 1 implies | u| p
+
p(x) \leq \rho (u) \leq | u| p
-
p(x).
Theorem 2.3 [11]. Let p(x) and q(x) be measurable functions such that p(x) \in L\infty (\Omega ) and
1 \leq p(x)q(x) \leq \infty for a.e. x \in \Omega . Let u \in Lq(x)(\Omega ). Then
| u| p(x)q(x) \leq 1 implies | u| p
+
p(x)q(x) \leq
\bigm| \bigm| \bigm| | u| p(x)\bigm| \bigm| \bigm|
q(x)
\leq | u| p
-
p(x)q(x),
| u| p(x)q(x) \geq 1 implies | u| p
-
p(x)q(x) \leq
\bigm| \bigm| \bigm| | u| p(x)\bigm| \bigm| \bigm|
q(x)
\leq | u| p
+
p(x)q(x).
In particular, if p(x) = p is constant, then\bigm| \bigm| | u| p\bigm| \bigm|
q(x)
= | u| ppq(x).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
94 S. H. RASOULI, K. FALLAH
Theorem 2.4 [11]. If u, un \in Lp(x)(\Omega ), n = 1, 2, . . . , then
(i) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty | un - u| p(x) = 0,
(ii) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \rho (un - u) = 0,
(iii) un \rightarrow u in measure in \Omega and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \rho (un) = \rho (u)
are equivalent.
Theorem 2.5 [11]. If p - > 1 and p+ < \infty , then the spaces Lp(x)(\Omega ), L
p(x)
c(x)(\Omega ), and W 1,p(x)(\Omega )
are separable and reflexive Banach spaces.
Theorem 2.6 [11]. (i) Let p \in C(\Omega ) and \partial \Omega possesses the cone property. If q \in C(\Omega ) and
1 \leq q(x) < p\ast (x) for any x \in \Omega , then W 1,p(x)(\Omega ) \lhook \rightarrow \lhook \rightarrow Lq(x)(\Omega ).
(ii) Let p \in C(\Omega ) and \partial \Omega possesses the cone property. If q \in C(\Omega ) and 1 \leq q(x) < p\ast \partial (x) for
any x \in \Omega , then W 1,p(x)(\Omega ) \lhook \rightarrow \lhook \rightarrow Lq(x)(\partial \Omega ).
(iii) If p, q \in C(\Omega ) and p(x) \leq q(x) \leq p\ast (x) for any x \in (\Omega ), then W 1,p(x)(\Omega ) \lhook \rightarrow Lq(x)(\Omega )
and
| u| q(x) \leq c\| u\| \forall u \in W
1,p(x)
0 (\Omega ),
where c > 0 is constant.
Theorem 2.7. Let p \in C(\Omega ) and \partial \Omega possesses the cone property. Suppose that g \in L\beta (x)(\partial \Omega ),
g(x) > 0 for x \in \Omega , \beta \in C(\Omega ) and \beta - > 1, \beta -
0 \leq \beta 0(x) \leq \beta +
0 , where \beta 0(x) =
\beta (x)
\beta (x) - 1
. If
r \in C(\Omega ) and
1 < r(x) <
\beta (x) - 1
\beta (x)
p\ast \partial (x) \forall x \in \Omega , (2.1)
or
1 < \beta (x) <
N\beta (x)
N\beta (x) - r(x)(N - p(x))
,
then W 1,p(x)(\Omega ) \lhook \rightarrow L
r(x)
g(x)(\partial \Omega ) is compact. Moreover, there is a constant c5 > 0 such that the
inequality \int
\partial \Omega
g(x)| u| r(x) dS \leq c5
\bigl(
\| u\| r - + \| u\| r+
\bigr)
(2.2)
holds.
Proof. We must remark that our proof of the embedding W 1,p(x)(\Omega ) \lhook \rightarrow \lhook \rightarrow L
r(x)
g(x)(\partial \Omega ) is similar
to Fan [12]. Let u \in W 1,p(x)(\Omega ) and set h(x) =
\beta (x)
\beta (x) - 1
r(x) = \beta 0(x)r(x). Then (2.1) implies
h(x) < p\ast \partial (x). Hence, by Theorem 2.6 we have the embedding W 1,p(x)(\Omega ) \lhook \rightarrow \lhook \rightarrow Lh(x)(\partial \Omega ). So,
for u \in W 1,p(x)(\Omega ), we have | u| r(x) \in L\beta 0(x)(\partial \Omega ). By Theorem 2.1,\int
\partial \Omega
g(x)| u| r(x) dS \leq c1| g| \beta (x)
\bigm| \bigm| | u| r(x)\bigm| \bigm|
\beta 0(x)
< \infty .
This implies that W 1,p(x)(\Omega ) \subset L
r(x)
g(x)(\partial \Omega ). Now let \{ un\} \subset W 1,p(x)(\Omega ) and
un \rightharpoonup 0 (weakly) in W 1,p(x)(\Omega ).
Then we have
un \rightarrow 0 (strongly) in Lh(x)(\partial \Omega ).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
THE NEHARI MANIFOLD APPROACH FOR A p(x)-LAPLACIAN PROBLEM . . . 95
So, it follows that
\bigm| \bigm| | un| r(x)\bigm| \bigm| \beta 0(x)
\rightarrow 0. Thus,
\int
\partial \Omega
g(x)| un| r(x) dS \leq c1| g| \beta (x)
\bigm| \bigm| | un| r(x)\bigm| \bigm| \beta 0(x)
\rightarrow 0,
which implies | un| (r(x),g(x)) \rightarrow 0. Hence, W 1,p(x)(\Omega ) \lhook \rightarrow \lhook \rightarrow L
r(x)
g(x)(\partial \Omega ).
Now let’s show the inequality (2.2) holds. Since r - \leq r(x) \leq r+ and | u| r(x) \leq | u| r - + | u| r+ ,
thus \int
\partial \Omega
g(x)| u| r(x) dS \leq
\int
\partial \Omega
g(x)| u| r - dS +
\int
\partial \Omega
g(x)| u| r+ dS,
and r - \beta 0(x) \leq r+\beta 0(x) < p\ast \partial (x), we get\int
\partial \Omega
g(x)| u| r - dS \leq c2| g| \beta (x)
\bigm| \bigm| \bigm| | u| r - \bigm| \bigm| \bigm|
\beta 0(x)
= c2| g| \beta (x)| u| r
-
r - \beta 0(x)
\leq c3\| u\| r
-
. (2.3)
Moreover, \int
\partial \Omega
g(x)| u| r+ dS \leq c4\| u\| r
+
. (2.4)
As a result, from (2.3) and (2.4) it follows that\int
\partial \Omega
g(x)| u| r(x) dS \leq c5(\| u\| r
-
+ \| u\| r+).
Theorem 2.7 is proved.
Theorem 2.8. Let p \in C(\Omega ) and \partial \Omega possesses the cone property. Suppose that f \in L\alpha (x)(\Omega ),
f(x) > 0 for x \in \Omega , \alpha \in C(\Omega ) and \alpha - > 1, \alpha -
0 \leq \alpha 0(x) \leq \alpha +
0 , where \alpha 0(x) =
\alpha (x)
\alpha (x) - 1
. If
q \in C(\Omega ), p(x) <
\alpha (x)
\alpha (x) - 1
q(x) and
1 < q(x) <
\alpha (x) - 1
\alpha (x)
p\ast (x) \forall x \in \Omega (2.5)
or
Np(x)
Np(x) - q(x)(N - p(x))
< \alpha (x) <
p(x)
p(x) - q(x)
,
then W 1,p(x)(\Omega ) \lhook \rightarrow L
q(x)
f(x)(\Omega ) is compact. Moreover, there is a constant c7 > 0 such that the
following inequality is holds: \int
\Omega
f(x)| u| q(x) dx \leq c7(\| u\| q
-
+ \| u\| q+). (2.6)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
96 S. H. RASOULI, K. FALLAH
Proof. Let u \in W 1,p(x)(\Omega ). Set a(x) =
\alpha (x)
\alpha (x) - 1
q(x) = \alpha 0(x)q(x). Then (2.5) implies
a(x) < p\ast (x). Hence, by Theorem 2.6 there is the embedding W 1,p(x)(\Omega ) \lhook \rightarrow \lhook \rightarrow Lm(x)(\Omega ). For
u \in W 1,p(x)(\Omega ) we have | u| q(x) \in L\alpha 0(x)(\Omega ). By Theorem 2.1,\int
\Omega
f(x)| u| q(x) dx \leq c6| f | \alpha (x)
\bigm| \bigm| | u| q(x)\bigm| \bigm|
\alpha 0(x)
(\Omega ) \rightarrow 0.
This implies that W 1,p(x)(\Omega ) \subset L
q(x)
f(x)(\Omega ). Now let \{ un\} \subset W 1,p(x)(\Omega ) and
un \rightharpoonup 0 in W 1,p(x)(\Omega ).
Then we obtain
un \rightarrow 0 in Lm(x)(\Omega ).
So
\bigm| \bigm| | un| q(x)\bigm| \bigm| \alpha 0(x)
\rightarrow 0. Thus,\int
\Omega
f(x)| un| q(x) dx \leq c6| f | \alpha (x)
\bigm| \bigm| | un| q(x)\bigm| \bigm| \alpha 0(x)
\rightarrow 0,
which implies | un| (q(x),f(x)) \rightarrow 0. Hence, we have the embedding W 1,p(x)(\Omega ) \lhook \rightarrow \lhook \rightarrow L
q(x)
f(x)(\Omega ). Now,
by the above inequality we show that the inequality (2.6) holds. By q - \alpha 0(x) \leq q+\alpha 0(x) < p\ast (x)
and applying the similar steps as we did in proof of Theorem 2.7, we have\int
\Omega
f(x)| u| q(x) dx \leq c7(\| u\| q
-
+ \| u\| q+).
Theorem 2.8 is proved.
By Theorems 2.7 and 2.8, we conclude that for u \in W 1,p(x)(\Omega ), there exist positive constants
c8, c9, c10, c11 > 0 such that
(i)
\int
\partial \Omega
g(x)| u| r(x) dS \leq
\Biggl\{
c8\| u\| r
+
if \| u\| > 1,
c9\| u\| r
-
if \| u\| < 1,
(ii)
\int
\Omega
f(x)| u| q(x) dx \leq
\Biggl\{
c10\| u\| q
+
if \| u\| > 1,
c11\| u\| q
-
if \| u\| < 1,
hold.
3. Assumptions and statement of main result. The Euler functional associated with (1.1) is
defined by
\scrJ \lambda (u) =
\int
\Omega
1
p(x)
(| \nabla u| p(x) +m(x)| u| p(x)) dx - \lambda
\int
\Omega
1
q(x)
f(x)| u| q(x) dx -
\int
\partial \Omega
1
r(x)
g(x)| u| r(x) dS.
Then
\scrJ \lambda (u) \geq
1
p+
\int
\Omega
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx - \lambda
q -
\int
\Omega
f(x)| u| q(x) dx - 1
r -
\int
\partial \Omega
g(x)| u| r(x) dS \geq
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
THE NEHARI MANIFOLD APPROACH FOR A p(x)-LAPLACIAN PROBLEM . . . 97
\geq 1
p+
\| u\| p - - \lambda
q -
c7(\| u\| q
-
+ \| u\| q+) - 1
r -
c5(\| u\| r
-
+ \| u\| r+).
Since q+ < p - \leq p+ < r - \leq h+, this shows \scrJ \lambda is not bounded below on whole W 1,p(x)(\Omega ).
However, it is useful to consider the functional on the Nehari manifold \scrN \lambda which is given by
\scrN \lambda =
\bigl\{
u \in W 1,p(x)(\Omega )\setminus \{ 0\} : \langle \scrJ \prime
\lambda (u), u\rangle = 0
\bigr\}
,
where \langle \cdot , \cdot \rangle denotes the duality between W 1,p(x)(\Omega ) and (W 1,p(x)(\Omega )) - 1. Clearly, the critical points
of \scrJ \lambda correspond to points on the Nehari manifold. In particular, u \in \scrN \lambda if and only if
K\lambda (u) = \langle \scrJ \prime
\lambda (u), u\rangle =
\int
\Omega
\bigl(
| \nabla u| p(x)+m(x)| u| p(x)
\bigr)
dx - \lambda
\int
\Omega
f(x)| u| q(x) dx -
\int
\partial \Omega
g(x)| u| r(x) dS = 0.
(3.1)
Then for u \in \scrN \lambda we have
\langle K \prime
\lambda (u), u\rangle =
\int
\Omega
p(x)
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx -
- \lambda
\int
\Omega
q(x)f(x)| u| q(x) dx -
\int
\partial \Omega
r(x)g(x)| u| r(x) dS \leq
\leq (p+ - q - )\lambda
\int
\Omega
f(x)| u| q(x) dx+ (p+ - r - )
\int
\partial \Omega
g(x)| u| r(x) dS.
We can write
\scrN +
\lambda = \{ u \in \scrN (\Omega ) : \langle K \prime
\lambda (u), u\rangle > 0\} ,
\scrN -
\lambda = \{ u \in \scrN (\Omega ) : \langle K \prime
\lambda (u), u\rangle < 0\} ,
\scrN 0
\lambda = \{ u \in \scrN (\Omega ) : \langle K \prime
\lambda (u), u\rangle = 0\} .
Lemma 3.1. There exists \lambda 1 > 0 such that for 0 < \lambda < \lambda 1 we have \scrN 0
\lambda (\Omega ) = \varnothing .
Proof. Let \scrN 0
\lambda (\Omega ) \not = \varnothing for all \lambda \in \BbbR \setminus \{ 0\} and u \in \scrN 0
\lambda (\Omega ) such that \| u\| > 1. Then using
(2.4), (3.1) and definition of \scrN 0
\lambda (\Omega ), we obtain
0 = \langle K \prime
\lambda (u), u\rangle =
\int
\Omega
p(x)
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx -
- \lambda
\int
\Omega
q(x)f(x)| u| q(x) dx -
\int
\partial \Omega
r(x)g(x)| u| r(x) dS \geq
\geq p -
\int
\Omega
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx -
- q+
\left( \int
\Omega
| \nabla u| p(x) dx -
\int
\partial \Omega
g(x)| u| r(x) dS
\right) - r+
\int
\partial \Omega
g(x)| u| r(x) dS \geq
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
98 S. H. RASOULI, K. FALLAH
\geq (p - - q+)
\int
\Omega
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx+ (q+ - r+)
\int
\partial \Omega
g(x)| u| r(x) dS.
Hence,
0 \geq (p - - q+)\| u\| p - + c8(q
+ - r+)\| u\| r+ ,
and then
\| u\| \geq c12
\biggl(
p - - q+
r+ - q+
\biggr) 1
r+ - p -
. (3.2)
Similarly,
0 = \langle K \prime
\lambda (u), u\rangle \leq
\leq p+
\int
\Omega
(| \nabla u| p(x) +m(x)| u| p(x)) dx - \lambda q -
\int
\Omega
f(x)| u| q(x) dx - r -
\int
\partial \Omega
g(x)| u| r(x) dS \leq
\leq p+
\int
\Omega
(| \nabla u| p(x) +m(x)| u| p(x)) dx - \lambda q -
\int
\Omega
f(x)| u| q(x) dx -
- r -
\left( \int
\Omega
(| \nabla u| p(x) +m(x)| u| p(x)) dx - \lambda
\int
\Omega
f(x)| u| q(x) dx
\right) .
Therefore,
0 \leq (p+ - r - )\| u\| p - + \lambda c10(r
- - q - )\| u\| q+
and
\| u\| \leq c13
\biggl(
\lambda
r - - q -
r - - p+
\biggr) 1
p - - q+
. (3.3)
If \lambda is sufficiently small
\Biggl(
e.g., \lambda =
\biggl(
h - - p+
r - - q -
\biggr) \biggl(
p - - q+
r+ - q+
\biggr) p - - q+
r+ - p -
\Biggr)
, then from (3.2) and (3.3) we
get \| u\| < 1 which contradicts with our assumption. Hence, we conclude \scrN 0
\lambda (\Omega ) = \varnothing .
Lemma 3.1 is proved.
For 0 < \lambda < \lambda 1, we can write \scrN \lambda (\Omega ) = \scrN +
\lambda (\Omega ) \cup \scrN -
\lambda (\Omega ) and
\alpha +
\lambda = \mathrm{i}\mathrm{n}\mathrm{f}
u\in \scrN +
\lambda (\Omega )
\scrJ \lambda (u), \alpha -
\lambda = \mathrm{i}\mathrm{n}\mathrm{f}
u\in \scrN -
\lambda (\Omega )
\scrJ \lambda (u).
Theorem 3.1. Suppose that u0 is a local maximum or minimum for \scrJ on \scrN \lambda (\Omega ). If u0 /\in
/\in \scrN 0
\lambda (\Omega ), then u0 is a critical point of \scrJ .
Proof. The proof of Theorem 3.1 can be obtained directly from the following lemmas.
Lemma 3.2. The energy functional \scrJ is coercive and bounded below on \scrN \lambda (\Omega ).
Proof. Let u \in \scrN \lambda (\Omega ) and \| u\| > 1. Then using (3.1) and Theorem 2.2 we have
\scrJ \lambda (u) =
\int
\Omega
1
p(x)
(| \nabla u| p(x) +m(x)| u| p(x)) dx -
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
THE NEHARI MANIFOLD APPROACH FOR A p(x)-LAPLACIAN PROBLEM . . . 99
- \lambda
\int
\Omega
1
q(x)
f(x)| u| q(x) dx -
\int
\partial \Omega
1
r(x)
g(x)| u| r(x) dS \geq
\geq 1
p+
\int
\Omega
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx - \lambda
q -
\int
\Omega
f(x)| u| q(x) dx -
- 1
r -
\left( \int
\Omega
(| \nabla u| p(x) +m(x)| u| p(x)) dx - \lambda
\int
\Omega
f(x)| u| q(x) dx
\right) \geq
\geq
\biggl(
1
p+
- 1
r -
\biggr) \int
\Omega
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx+ \lambda
\biggl(
1
r -
- 1
q -
\biggr) \int
\Omega
f(x)| u| q(x) dx \geq
\geq
\biggl(
r - - p+
r - p+
\biggr)
\| u\| p - - c10\lambda
\biggl(
r - - q -
r - q -
\biggr)
\| u\| q+ .
Since p - > q+ so, \scrJ (u) \rightarrow \infty as \| u\| \rightarrow \infty . This implies \scrJ \lambda is coercive and bounded below on
\scrN \lambda (\Omega ).
Lemma 3.3. If 0 < \lambda < \lambda 1, then
(i) \scrJ \lambda (u) < 0 for all u \in \scrN +
\lambda (\Omega ),
(ii) \scrJ \lambda (u) > 0 for all u \in \scrN -
\lambda (\Omega ).
Proof. (i) Let u \in \scrN +
\lambda (\Omega ). By definition of \scrJ \lambda (u), we can write
\scrJ \lambda (u) \leq
1
p -
\int
\Omega
(| \nabla u| p(x) +m(x)| u| p(x)) dx - \lambda
q+
\int
\Omega
f(x)| u| q(x) dx - 1
r+
\int
\partial \Omega
g(x)| u| r(x) dS.
Since u \in \scrN +
\lambda (\Omega ), we have
p+
\int
\Omega
(| \nabla u| p(x) +m(x)| u| p(x)) dx - \lambda q -
\int
\Omega
f(x)| u| q(x) dx - r -
\int
\partial \Omega
g(x)| u| r(x) dS > 0.
We get \int
\partial \Omega
g(x)| u| r(x) dS <
p+ - q -
r - - q -
\int
\Omega
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx.
Moreover,
\scrJ \lambda (u) \leq
\biggl(
1
p -
- 1
q+
\biggr) \int
\Omega
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx+
\biggl(
1
q+
- 1
r+
\biggr) \int
\partial \Omega
g(x)| u| r(x) dS.
Therefore,
\scrJ \lambda (u) < - (p - - q+)(r+ - p - )
r+p - q+
\| u\| p - < 0.
Hence, we have \alpha +
\lambda = \mathrm{i}\mathrm{n}\mathrm{f}u\in \scrN +
\lambda
(\Omega )\scrJ \lambda (u) < 0.
(ii) Let u \in \scrN -
\lambda (\Omega ). By definition of \scrJ \lambda (\Omega ) and (3.1), we obtain
\scrJ \lambda (u) \geq
1
p+
\int
\Omega
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx - \lambda
q -
\int
\Omega
f(x)| u| q(x) dx - 1
r -
\int
\partial \Omega
g(x)| u| r(x) dS
and
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
100 S. H. RASOULI, K. FALLAH\int
\partial \Omega
g(x)| u| r(x) dS =
\int
\Omega
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx - \lambda
\int
\Omega
f(x)| u| q(x) dx.
Therefore
\scrJ \lambda (u) \geq
1
p+
\int
\Omega
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx - \lambda
q -
\int
\Omega
f(x)| u| q(x) dx -
- 1
r -
\left( \int
\Omega
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx - \lambda
\int
\Omega
f(x)| u| q(x) dx
\right) \geq
\geq
\biggl(
1
p+
- 1
r -
\biggr) \int
\Omega
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx+ \lambda
\biggl(
1
r -
- 1
q -
\biggr) \int
\Omega
f(x)| u| q(x) dx.
By Theorem 2.2 and the condition p - > q+, we obtain
\scrJ \lambda (u) \geq
\biggl(
1
p+
- 1
r -
\biggr)
\| u\| p - + c10\lambda
\biggl(
1
r -
- 1
q -
\biggr)
\| u\| q+ \geq
\geq
\biggl(
r - - p+
p+r -
+ c10\lambda
\biggl(
q - - r -
q - r -
\biggr) \biggr)
\| u\| p - .
So, if we choose \lambda <
q - (r - - p+)
c10p+(r - - q - )
, we get \scrJ \lambda (u) > 0. If we consider the facts \scrN \lambda (\Omega ) =
= \scrN +
\lambda (\Omega )\cup \scrN -
\lambda (\Omega ) (see Lemma 3.1), \scrN +
\lambda (\Omega )\cap \scrN -
\lambda (\Omega ) = \varnothing , and by the Lemma 3.3, we must have
u \in \scrN -
\lambda (\Omega ).
Theorem 3.2. If 0 < \lambda < \lambda 1, then the functional \scrJ \lambda has a minimizer u+0 in \scrN +
\lambda (\Omega ) and
\scrJ \lambda (u
+
0 ) = \alpha +
\lambda .
Proof. Since \scrJ \lambda is bounded below on \scrN \lambda (\Omega ). Then there exist a minimizing sequence \{ u+n \} \subseteq
\subseteq \scrN +
\lambda (\Omega ) such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrJ \lambda (u
+
n ) = \mathrm{i}\mathrm{n}\mathrm{f}
u\in \scrN +
\lambda (\Omega )
\scrJ \lambda (u) = \alpha +
\lambda < 0.
Since \scrJ \lambda is coercive, u+n is bounded in W 1,p(x)(\Omega ). Let u+n \rightharpoonup u+0 in W 1,p(x)(\Omega ),
u+n \rightarrow u+0 in L
q(x)
f(x)(\Omega ),
and
u+n \rightarrow u+0 in L
r(x)
g(x)(\Omega ).
Now, we prove that u+n \rightarrow u+0 in W 1,p(x)(\Omega ). Otherwise, suppose u+n \nrightarrow u+0 in W 1,p(x)(\Omega ). Then\int
\Omega
\bigl(
| \nabla u+0 |
p(x) +m(x)| u+0 |
p(x)
\bigr)
dx < \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{i}\mathrm{n}\mathrm{f}
\int
\Omega
(| \nabla u+n | p(x) +m(x)| u+n | p(x)) dx.
Moreover, \int
\Omega
f(x)| u+0 |
q(x) dx = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{i}\mathrm{n}\mathrm{f}
\int
\Omega
f(x)| u+n | q(x) dx,
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
THE NEHARI MANIFOLD APPROACH FOR A p(x)-LAPLACIAN PROBLEM . . . 101\int
\partial \Omega
g(x)| u+0 |
r(x) dS = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{i}\mathrm{n}\mathrm{f}
\int
\partial \Omega
g(x)| u+n | r(x) dS.
By \langle \scrJ \prime
\lambda (u
+
n ), (u
+
n )\rangle = 0 and Theorem 2.8, we have
\scrJ \lambda (u
+
n ) \geq
\biggl(
1
p+
- 1
r -
\biggr) \int
\Omega
\bigl(
| \nabla u+n | p(x) +m(x)| u+n | p(x)
\bigr)
dx+ \lambda
\biggl(
1
r -
- 1
q -
\biggr) \int
\Omega
f(x)| u+n | q(x) dx,
and then
\alpha +
\lambda = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrJ \lambda (u
+
n ) \geq
\geq
\biggl(
1
p+
- 1
r -
\biggr)
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\int
\Omega
\bigl(
| \nabla u+n | p(x) +m(x)| u+n | p(x)
\bigr)
dx+
+\lambda
\biggl(
1
r -
- 1
q -
\biggr)
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\int
\Omega
f(x)| u+n | q(x) dx >
>
\biggl(
1
p+
- 1
r -
\biggr)
\| u+0 \|
p - + c7\lambda
\biggl(
1
r -
- 1
q -
\biggr) \bigl(
\| u+0 \|
q - + \| u+0 \|
q+
\bigr)
.
Since p - > q+, for \| u+0 \| > 1, we obtain
\alpha +
\lambda = \mathrm{i}\mathrm{n}\mathrm{f}
u\in \scrN +
\lambda
\scrJ \lambda (u) > 0.
By Lemma 3.3, for any u \in \scrN +
\lambda (\Omega ),\scrJ \lambda (u) < 0.
So, this is a contradiction. Hence, u+n \rightarrow u+0 in W
1,p(x)
0 (\Omega ) and
\scrJ \lambda (u
+
0 ) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrJ \lambda (u
+
n ) = \mathrm{i}\mathrm{n}\mathrm{f}
u\in \scrN +
\lambda (\Omega )
\scrJ \lambda (u).
Thus, u+0 is a minimizer for \scrJ \lambda on \scrN +
\lambda (\Omega ).
Theorem 3.2 is proved.
Theorem 3.3. If 0 < \lambda < \lambda 1, then the functional \scrJ \lambda has a minimizer u - 0 in \scrN -
\lambda (\Omega ) and
\scrJ \lambda (u
-
0 ) = \alpha -
\lambda .
Proof. Since \scrJ \lambda is bounded below on \scrN \lambda (\Omega ) and so on \scrN -
\lambda (\Omega ), then there exists a minimizing
sequence \{ u - n \} \subseteq \scrN -
\lambda (\Omega ) such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrJ \lambda (u
-
n ) = \mathrm{i}\mathrm{n}\mathrm{f}
u\in \scrN -
\lambda (\Omega )
\scrJ \lambda (u) = \alpha -
\lambda > 0.
Since \scrJ \lambda is coercive, u - n is bounded in W 1,p(x)(\Omega ). Thus, u - n \rightharpoonup u - 0 in W 1,p(x)(\Omega ) and
u - n \rightarrow u - 0 in L
q(x)
f(x)(\Omega ),
u - n \rightarrow u - 0 in L
r(x)
g(x)(\Omega ).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
102 S. H. RASOULI, K. FALLAH
Moreover, if u - 0 \in \scrN -
\lambda (\Omega ), then there is a constant t > 0 such that tu - 0 \in \scrN -
\lambda (\Omega ) and \scrJ \lambda (u
-
0 ) \geq
\geq \scrJ \lambda (tu
-
0 ). Since
K \prime
\lambda (u) =
\int
\Omega
p(x)
\bigl(
| \nabla u| p(x) +m(x)| u| p(x)
\bigr)
dx - \lambda
\int
\Omega
q(x)f(x)| u| q(x) dx -
\int
\partial \Omega
r(x)g(x)| u| r(x) dS,
then
K \prime
\lambda (tu
-
0 ) =
\int
\Omega
p(x)
\bigl(
| \nabla tu - 0 |
p(x) +m(x)| u - 0 |
p(x)
\bigr)
dx -
- \lambda
\int
\Omega
q(x)f(x)| tu - 0 |
q(x) dx -
\int
\partial \Omega
r(x)g(x)| tu - 0 |
r(x) dS \leq
\leq tp
+
p+
\int
\Omega
(| \nabla u - 0 |
p(x) +m(x)| u - 0 |
p(x)) dx -
- \lambda tq
-
q -
\int
\Omega
f(x)| u - 0 |
q(x) dx - tr
-
r -
\int
\partial \Omega
g(x)| u - 0 |
r(x) dS.
Since q - < p+ < r - , and by the assumptions on f and g, it follows K \prime
\lambda (tu
-
0 ) < 0. Hence, by the
definition of \scrN -
\lambda (\Omega ), tu - 0 \in \scrN -
\lambda (\Omega ).
Now, we prove u - n \rightarrow u - 0 in W
1,p(x)
0 (\Omega ). Let u - n \nrightarrow u - 0 in W
1,p(x)
0 (\Omega ). By\int
\Omega
(| \nabla u - 0 |
p(x) +m(x)| u - 0 |
p(x)) dx < \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{i}\mathrm{n}\mathrm{f}
\int
\Omega
(| \nabla u - n | p(x) +m(x)| u - n | p(x)) dx,
we have
\scrJ \prime
\lambda (tu
-
0 ) \leq
tp
+
p -
\int
\Omega
(| \nabla u - 0 |
p(x) +m(x)| u - 0 |
p(x)) dx -
- \lambda
tq
-
q+
\int
\Omega
f(x)| u - 0 |
q(x) dx - tr
-
r+
\int
\partial \Omega
g(x)| u - 0 |
r(x) dS <
< \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\Biggl[
tp
+
p -
\int
\Omega
(| \nabla u - n | p(x) +m(x)| u - n | p(x)) dx -
- \lambda
tq
-
q+
\int
\Omega
f(x)| u - n | q(x) dx - tr
-
r+
\int
\partial \Omega
g(x)| u - n | r(x) dS
\Biggr]
\leq
\leq \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrJ \lambda (tu
-
n ) \leq \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrJ \lambda (u
-
n ) = \mathrm{i}\mathrm{n}\mathrm{f}
u\in \scrN -
\lambda (\Omega )
\scrJ \lambda (u).
Thus, u - 0 is a minimizer for \scrJ \lambda on \scrN -
\lambda (\Omega ).
Theorem 3.3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
THE NEHARI MANIFOLD APPROACH FOR A p(x)-LAPLACIAN PROBLEM . . . 103
Conclusions. By Theorems 3.2 and 3.3 we conclude that there exist u+0 \in \scrN +
\lambda (\Omega ) and u - 0 \in
\in \scrN -
\lambda (\Omega ) such that \scrJ \lambda (u
+
0 ) = \mathrm{i}\mathrm{n}\mathrm{f}u\in \scrN +
\lambda (\Omega ) \scrJ \lambda (u) and \scrJ \lambda (u
-
0 ) = \mathrm{i}\mathrm{n}\mathrm{f}u\in \scrN -
\lambda (\Omega ) \scrJ \lambda (u). Moreover,
since \scrJ \lambda (u
\pm
0 ) = \scrJ \lambda (| u\pm 0 | ) and | u\pm 0 | \in \scrN \pm
\lambda (\Omega ), we may assume u\pm 0 \geq 0. By Theorem 3.1, u\pm 0 are
critical points \scrJ \lambda on W
1,p(x)
0 (\Omega ) and hence are weak solutions of (1.1). Finally, by the Harnack
inequality due to [22], we obtain that u\pm 0 are positive solutions of (1.1).
References
1. Afrouzi G. A., Rasouli S. H. A variational approach to a quasilinear elliptic problem involving the p-Laplacian and
nonlinear boundary condition // Nonlinear Anal. – 2009. – 71. – P. 2447 – 2455.
2. Brown K. J., Wu T. F. A semilinear elliptic system involving nonlinear boundary condition and sign changing weight
function // J. Math. Anal. and Appl. – 2008. – 337. – P. 1326 – 1336.
3. Drabek P., Pohozaev S. I. Positive solutions for the p(x)-Laplacian: application of the fibering method // Proc. Roy.
Soc. Edinburgh A. – 1997. – 127. – P. 721 – 747.
4. Chen Y., Levine S., Rao M. Variable exponent, linear growth functionals in image restoration // SIAM J. Appl. Math. –
2006. – 66, № 4. – P. 1383 – 1406.
5. Dai G. Infinitely many solutions for a Neumann-type differential inclusion problem involving the p(x)-Laplacian //
Nonlinear Anal. – 2009. – 70. – P. 2297 – 2305.
6. Dai G. Infinitely many solutions for a hemivariational inequality involving the p(x)-Laplacian // Nonlinear Anal. –
2009. – 71. – P. 186 – 195.
7. Dai G. Three solutions for a Neumann-type differential inclusion problem involving the p(x)-Laplacian // Nonlinear
Anal. – 2009. – 70. – P. 3755 – 3760.
8. Dai G. Infinitely many solutions for a differential inclusion problem in \mathrm{R}N involving the p(x)-Laplacian // Nonlinear
Anal. – 2009. – 71. – P. 1116 – 1123.
9. Fan X. L. On the sub-supersolution methods for p(x)-Laplacian equations // J. Math. Anal. and Appl. – 2007. –
330. – P. 665 – 682.
10. Fan X. L., Han X. Y. Existence and multiplicity of solutions for p(x)-Laplacian equations in \mathrm{R}N // Nonlinear Anal. –
2004. – 59. – P. 173 – 188.
11. Fan X. L., Shen J. S., Zhao D. Sobolev embedding theorems for spaces W k,p(x)(\Omega ) // J. Math. Anal. and Appl. –
2001. – 262. – P. 749 – 760.
12. Fan X. L., Zhang Q. H. Existence of solutions for p(x)-Laplacian Dirichlet problems // Nonlinear Anal. – 2003. –
52. – P. 1843 – 1852.
13. Fan X. L., Zhang Q. H., Zhao D. Eigenvalues of p(x)-Laplacian Dirichlet problem // J. Math. Anal. and Appl. –
2005. – 302. – P. 306 – 317.
14. Fan X. L., Zhao D. On the spaces Lp(x) and W k,p(x) // J. Math. Anal. and Appl. – 2001. – 263. – P. 424 – 446.
15. Fan X. L., Zhao Y. Z., Zhang Q. H. A strong maximum principle for p(x)-Laplace equations // Chinese J. Contemp.
Math. – 2003. – 24, № 3. – P. 277 – 282.
16. Harjulehto P., Hasto P. An overview of variable exponent Lebesgue and Sobolev spaces // Future Trends in Geometric
Function Theory / Ed. D. Herron. – Jyvaskyla: RNC Workshop, 2003. – P. 85 – 93.
17. He X., Zou W. Infinitely many positive solutions for Kirchhoff-type problems // Nonlinear Anal. – 2009. – 70. –
P. 1407 – 1414.
18. Rasouli S. H., Afrouzi G. A. The Nehari manifold for a class of concave-convex elliptic systems involving the
p-Laplacian and nonlinear boundary condition // Nonlinear Anal. – 2010. – 73. – P. 3390 – 3401.
19. Wu T. F. Multiplicity results for a semilinear elliptic equation involving sign-changing weight function // Rocky
Mountain J. Math. – 2009. – 39, № 3. – P. 995 – 1011.
20. Wu T. F. A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential // Electron.
J. Different. Equat. – 2006. – 131. – P. 1 – 15.
21. Wu T. F. On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function //
J. Math. Anal. and Appl. – 2006. – 318. – P. 253 – 270.
22. Zhang X., Liu X. The local boundedness and Harnack inequality of p(x)-Laplace equation // J. Math. Anal. and
Appl. – 2007. – 332. – P. 209 – 218.
Received 30.01.13,
after revision — 08.10.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
|
| id | umjimathkievua-article-1678 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:10:29Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/57/da94fad9a55998412e1dfeaafce15c57.pdf |
| spelling | umjimathkievua-article-16782019-12-05T09:23:35Z The Nehari manifold approach for a $p(x)$ -Laplacian problem with nonlinear boundary conditions Ппiдхiд на основi многовиду нехарi до проблеми $p(x)$-лапласiана з нелiнiйними граничними Fallah, K. Rasouli, S. H. Фаллах, К. Расулі, С. Х. We consider a class of $p(x)$-Laplacian equations that involve nonnegative weight functions with nonlinear boundary conditions. Our technical approach is based on the Nehari manifold, which is similar to the fibering method of Drabek and Pohozaev, together with the recent idea from Brown and Wu. Розглянуто один клас $p(x)$-рiвнянь Лапласа, що включає невiд’ємнi ваговi функцiї з нелiнiйними граничними умовами. Наш пiдхiд базується на многовидi Нехарi, що є подiбним до методу волокон Драбека та Похожаєва з використанням нових iдей Брауна та Ву. Institute of Mathematics, NAS of Ukraine 2017-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1678 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 1 (2017); 92-103 Український математичний журнал; Том 69 № 1 (2017); 92-103 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1678/660 Copyright (c) 2017 Fallah K.; Rasouli S. H. |
| spellingShingle | Fallah, K. Rasouli, S. H. Фаллах, К. Расулі, С. Х. The Nehari manifold approach for a $p(x)$ -Laplacian problem with nonlinear boundary conditions |
| title | The Nehari manifold approach for a $p(x)$ -Laplacian problem with
nonlinear boundary conditions |
| title_alt | Ппiдхiд на основi многовиду нехарi до проблеми
$p(x)$-лапласiана з нелiнiйними граничними |
| title_full | The Nehari manifold approach for a $p(x)$ -Laplacian problem with
nonlinear boundary conditions |
| title_fullStr | The Nehari manifold approach for a $p(x)$ -Laplacian problem with
nonlinear boundary conditions |
| title_full_unstemmed | The Nehari manifold approach for a $p(x)$ -Laplacian problem with
nonlinear boundary conditions |
| title_short | The Nehari manifold approach for a $p(x)$ -Laplacian problem with
nonlinear boundary conditions |
| title_sort | nehari manifold approach for a $p(x)$ -laplacian problem with
nonlinear boundary conditions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1678 |
| work_keys_str_mv | AT fallahk theneharimanifoldapproachforapxlaplacianproblemwithnonlinearboundaryconditions AT rasoulish theneharimanifoldapproachforapxlaplacianproblemwithnonlinearboundaryconditions AT fallahk theneharimanifoldapproachforapxlaplacianproblemwithnonlinearboundaryconditions AT rasulísh theneharimanifoldapproachforapxlaplacianproblemwithnonlinearboundaryconditions AT fallahk ppidhidnaosnovimnogoviduneharidoproblemipxlaplasianaznelinijnimigraničnimi AT rasoulish ppidhidnaosnovimnogoviduneharidoproblemipxlaplasianaznelinijnimigraničnimi AT fallahk ppidhidnaosnovimnogoviduneharidoproblemipxlaplasianaznelinijnimigraničnimi AT rasulísh ppidhidnaosnovimnogoviduneharidoproblemipxlaplasianaznelinijnimigraničnimi AT fallahk neharimanifoldapproachforapxlaplacianproblemwithnonlinearboundaryconditions AT rasoulish neharimanifoldapproachforapxlaplacianproblemwithnonlinearboundaryconditions AT fallahk neharimanifoldapproachforapxlaplacianproblemwithnonlinearboundaryconditions AT rasulísh neharimanifoldapproachforapxlaplacianproblemwithnonlinearboundaryconditions |