Existence of a weak solution for a class of nonlinear elliptic equations on the Sierpiński gasket
UDC 517.9 We study the existence of a weak (strong) solution of the nonlinear elliptic problem\begin{gather*} -\Delta u- \lambda ug_1 +h(u)g_2=f \quad\text{in}\quad V\setminus V_0,\\u=0 \quad\text{on}\quad V_0,\end{gather*} where $V$ is a Sierpi\'nski gasket in $\mathbb{R}^{N-1},$ $N\...
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| author | Badajena, A. K. Kar, R. Badajena, A. K. Kar, R. |
| author_facet | Badajena, A. K. Kar, R. Badajena, A. K. Kar, R. |
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UDC 517.9
We study the existence of a weak (strong) solution of the nonlinear elliptic problem\begin{gather*} -\Delta u- \lambda ug_1 +h(u)g_2=f \quad\text{in}\quad V\setminus V_0,\\u=0 \quad\text{on}\quad V_0,\end{gather*} where $V$ is a Sierpi\'nski gasket in $\mathbb{R}^{N-1},$ $N\geq 2,$ $V_0$ is its boundary (consisting of $N$ its corners), and $\lambda$ is a real parameter. Here, $f,g_1,g_2\colon V\to\mathbb{R}$ and $h\colon \mathbb{R}\to\mathbb{R}$ are functions satisfying suitable hypotheses. |
| doi_str_mv | 10.37863/umzh.v74i10.6248 |
| first_indexed | 2026-03-24T03:26:41Z |
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| fulltext |
DOI: 10.37863/umzh.v74i10.6248
UDC 517.9
A. K. Badajena, R. Kar1 (Nat. Inst. Technology Rourkela, Odisha, India)
EXISTENCE OF A WEAK SOLUTION FOR A CLASS
OF NONLINEAR ELLIPTIC EQUATIONS ON THE SIERPIŃSKI GASKET
IСНУВАННЯ СЛАБКОГО РОЗВ’ЯЗКУ ДЛЯ КЛАСУ НЕЛIНIЙНИХ
ЕЛIПТИЧНИХ РIВНЯНЬ НА ПРОКЛАДЦI СЕРПIНСЬКОГО
We study the existence of a weak (strong) solution of the nonlinear elliptic problem
- \Delta u - \lambda ug1 + h(u)g2 = f in V \setminus V0,
u = 0 on V0,
where V is a Sierpiński gasket in \BbbR N - 1, N \geq 2, V0 is its boundary (consisting of N its corners), and \lambda is a real
parameter. Here, f, g1, g2 : V \rightarrow \BbbR and h : \BbbR \rightarrow \BbbR are functions satisfying suitable hypotheses.
Дослiджується iснування слабкого (сильного) розв’язку нелiнiйної елiптичної задачi
- \Delta u - \lambda ug1 + h(u)g2 = f в V \setminus V0,
u = 0 на V0,
де V — прокладка Серпiнського в \BbbR N - 1, N \geq 2, V0 — її межа (що складається з її N кутiв) i \lambda — дiйсний параметр.
Тут f, g1, g2 : V \rightarrow \BbbR i h : \BbbR \rightarrow \BbbR — функцiї, що задовольняють вiдповiднi гiпотези.
1. Introduction. We study the existence of weak solutions for the following class of elliptic problem:
- \Delta u - \lambda ug1 + h(u)g2 = f in V \setminus V0
u = 0 on V0,
(1.1)
where V denotes the Sierpiński gasket in \BbbR N - 1 N \geq 2, V0 is its boundary (consisting of its N
corners). \Delta denotes the Laplacian operator on V, \lambda \in \BbbR and f, g1, g2 : V \rightarrow \BbbR , h : \BbbR \rightarrow \BbbR are
functions satisfying the following hypotheses:
(H1) h : \BbbR \rightarrow \BbbR is bounded (i.e., | h(t)| \leq A, t \in \BbbR , for a fixed A > 0) and continuous;
(H2) g1 \in L\infty (V ), g2 \in L2(V ) and f \in L2(V ).
Recently, there has been a considerable interest in the study of nonlinear partial differential
equations on fractal domains and in particular on the Sierpiński gasket. Many physical problems
on fractal regions such as reaction-diffusion problems, elastic properties of fractal media and flow
through fractal regions are modeled by nonlinear equations. Now, a natural question is whether the
classical existence results (we refer to [1, 24, 29]) in the standard framework of the Laplacian also
hold in the corresponding fractal framework. To answer this we have to overcome several difficulties
that arise due to the geometrical structure of fractal domains. One main difficulty is how to define
differential operators, like the Laplacian operator, on the fractal domains for there is no concept of
a generalized derivative of functions defined on such domains. However, a Laplacian is defined on
1 Corresponding author, e-mail: rasmitak6@gmail.com.
c\bigcirc A. K. BADAJENA, R. KAR, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10 1317
1318 A. K. BADAJENA, R. KAR
a few special fractals, we refer to [2, 3, 20, 21] and a Hilbert space structure is introduced in [20].
This enables us to investigate the existence of solutions for equations of type (1.1) defined on fractal
domains.
The study of nonlinear elliptic equations on the Sierpiński gasket was essentially initiated by
Falconer and Hu in the paper [15]. Since then many authors have contributed to the literature in this
direction. In [15], Falconer and Hu, considered the problem
\Delta u+ a(x)u = f(x, u), x \in V \setminus V0,
u| V0 = 0,
(1.2)
where V denotes the Sierpiński gasket with boundary V0 and a \in L1(V ) satisfies suitable condition.
The authors formulated the problem in a suitable function space over Sierpiński gasket and used the
Mountain Pass theorem [1] to prove the existence of a solution. In [5], Molica Bisci et al. considered
a similar problem
\Delta u+ \alpha (x)u = \lambda f(x, u), x \in V \setminus V0,
u| V0 = 0,
where \lambda is a positive real parameter and proved the existence of at least two solutions for small
values of \lambda . For this the authors used a variational result due to Ricceri [26]. In [8], Breckner et al.
studied the existence of infinitely many solutions of the problem
\Delta u(x) + \alpha (x)u(x) = g(x)f
\bigl(
u(x)
\bigr)
, x \in V \setminus V0,
u| V0 = 0.
The authors proved the existence of infinitely many solutions by extending a method introduced
by Faraci and Kristály [16] in the framework of Sobolev spaces to the case of function spaces on
fractal domains. For more results on the existence and multiplicity of solutions of nonlinear elliptic
equations on the Sierpiński gasket and on other fractals we refer to the papers [4, 6 – 11, 13, 14]
and [18, 19, 27, 28] as well as the references therein. The main tools used in these papers to prove
the existence of nontrivial solutions are basically Mountain Pass theorems, saddle-point theorems or
certain minimization procedures.
In [5, 15] on of the assumptions on the nonlinearity f(x, u) is
(f) there exist constants \nu > 2 and r \geq 0 such that, for | t| \geq r,
tf(x, t) < \nu F (x, t) < 0,
where F (x, t) =
\int t
0
f(x, s)ds.
If the condition (f) does not hold then the energy functional associated to the problem (1.2) in
general does not satisfy certain conditions needed to apply the Mountain Pass theorem in order to
prove the existence of solutions. In this paper, we show an application of demicontinuous operators to
nonlinear elliptic problems in the fractal setting. In particular, the main tool we used to establish the
existence of at least one solution is a result due to P. Hess [12] on linear demicontinuous operators.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
EXISTENCE OF A WEAK SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC . . . 1319
Our paper was inspired by a problem on bounded domains given in Section 29.9 of the book by
Zeidler [31, p. 661], we refer also to [25] where the existence of a weak solution is established for
the fractional counterpart of (1.1).
This paper is organized as follows. Section 2 deals with preliminaries and the weak formulation
of the problem. Section 3 concerns with the main result concerning the existence of a weak solution
of (1.1). Finally, Section 4 deals with an extension to a class of continuous functions h that are not
necessarily bounded.
2. Preliminaries. Let C(V ) denotes the space of real-valued continuous functions on V and
C0(V ) = \{ u \in C(V ) : u| V0 = 0\} both equipped with the usual supremum norm \| \cdot \| \infty . Let H1
0 (V )
be the Hilbert space structure defined on V with inner product denoted by \scrW (u, v). We refer to [15]
(see also [3 – 10, 13]), for more details. The space H1
0 (V ) with the inner product \scrW (u, v) is a
separable Hilbert space (we refer to [9]). The weak Laplacian \Delta of u on V is defined as
\langle \Delta u, v\rangle = - \scrW (u, v) for all v \in H1
0 (V ).
Now, we can define the weak solution for the problem (1.1).
Definition 2.1. We say that a function u \in H1
0 (V ) is a weak solution of (1.1) if it satisfies
\scrW (u, v) - \lambda
\int
V
g1(x)u(x)v(x) d\mu +
\int
V
h(u(x))g2(x)v(x) d\mu =
\int
V
f(x)v(x) d\mu
for all v \in H1
0 (V ).
For further details on the Laplacian operator on certain fractals, we refer to the paper [20].
We note that if the functions f, g1, g2 and h are continuous, then the weak solutions of the
equation (1.1) are also strong solutions of it; as shown by the following result.
Lemma 2.1. Assume that u \in H1
0 (V ) is a weak solution to the problem (1.1). If f, g1, g2 \in
\in C(V ) and h \in C(\BbbR ), then u is a strong solution of (1.1).
Proof is similar to [15] (Lemma 2.16), hence omitted.
At each step, a generic constant is denoted by C or c to avoid too many suffixes. We recall the
embedding properties of H1
0 (V ) into the spaces C0(V ) and L2(V, \mu ) (we refer to [15]), for the sake
of completeness.
Lemma 2.2. The embedding j : H1
0 (V ) \lhook \rightarrow C0(V ) is compact and, for every u \in H1
0 (V ),
| u(x)| \leq (2N + 3)\| u\| H1
0 (V ) for any x \in V.
Also, the embedding j : H1
0 (V ) \lhook \rightarrow L2(V ) is compact and
\| u\| 2 \leq C\| u\| H1
0 (V ), (2.1)
where \| u\| 2 =
\biggl( \int
V
| u(x)| 2d\mu
\biggr) 1
2
.
Let Y \ast denote the dual of the real Banach space Y . Let \| .\| and \| .\| Y \ast be the norms on Y and
Y \ast , respectively. For x \in Y and f \in Y \ast , we write (f | x) for f(x).
Definition 2.2. Let B, N : Y \rightarrow Y \ast be operators on the real separable reflexive Banach space
Y . Then:
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1320 A. K. BADAJENA, R. KAR
(i) B +N is asymptotically linear if B is linear and
\| Nu\|
\| u\|
\rightarrow 0 as \| u\| \rightarrow \infty ;
(ii) B satisfies condition (S) if
un \rightharpoonup u and \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
(Bun - Bu| un - u) = 0, implies un \rightarrow u.
We say that B is an (S)-operator if B satisfies condition (S).
The following definition deals with real Gårding forms (compare with [30, p. 364]).
Definition 2.3. Let X and Z be Hilbert spaces over \BbbR with the continuous embedding X \subseteq Z .
Then G : X \times X \rightarrow \BbbR is called a Gårding form if and only if G is bilinear and bounded, and there
exist a constant c > 0 and a real constant C such that
G(u, u) \geq c\| u\| 2X - C\| u\| 2Z for all u \in X. (2.2)
The relation (2.2) is called a Gårding inequality. If C = 0, then G is called a strict Gårding
form. The Gårding form G is called regular if and only if the embedding X \subseteq Z is compact.
In Section 3, we need the following result.
Proposition 2.1. Let B,N : Y \rightarrow Y \ast be operators on the real separable reflexive Banach
space Y . Assume that:
(i) the operator B : Y \rightarrow Y \ast is linear and continuous;
(ii) the operator N : Y \rightarrow Y \ast is demicontinuous and bounded;
(iii) B +N is asymptotically linear;
(iv) for each T \in Y \ast and for each t \in [0, 1], the operator At : Y \rightarrow Y \ast defined by At(u) =
= Bu+ t(Nu - T ) satisfies condition (S).
If Bu = 0 implies u = 0, then, for each T \in Y \ast , the equation Bu+Nu = T has a solution in Y .
For a detailed proof of the above theorem, we refer to [12] or [31] (Theorem 29.C).
We define the functionals B1, B2 : H1
0 (V )\times H1
0 (V ) \rightarrow \BbbR by
B1(u, \varphi ) = \scrW (u, \varphi ) - \lambda
\int
V
u(x)g1(x)\varphi (x)d\mu ,
B2(u, \varphi ) =
\int
V
h(u(x))g2(x)\varphi (x)d\mu .
Also define T : H1
0 (V ) \rightarrow \BbbR by
T (\varphi ) =
\int
V
f(x)\varphi (x)d\mu .
A function u \in H1
0 (V ) is a weak solution of (1.1) if
B1(u, \varphi ) +B2(u, \varphi ) = T (\varphi ) \forall \varphi \in H1
0 (V ). (2.3)
By applying the Cauchy – Schwarz inequality and the inequality (2.1), we note that, for every (u, \varphi ) \in
\in H1
0 (V )\times H1
0 (V ),
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
EXISTENCE OF A WEAK SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC . . . 1321
\bigm| \bigm| B1(u, \varphi )
\bigm| \bigm| \leq | \scrW (u, \varphi )| + | \lambda |
\int
V
| g1(x)\| u(x)\| \varphi (x)| d\mu \leq
\leq \| u\| H1
0 (V )\| \varphi \| H1
0 (V ) + | \lambda | \| g1\| \infty \| u\| 2\| \varphi \| 2 \leq
\leq (1 + C| \lambda | \| g1\| \infty )\| u\| H1
0 (V )\| \varphi \| H1
0 (V ).
By the hypotheses (H1), (H2), Hölder’s inequality and (2.1), we have, for every (u, \varphi ) \in
\in H1
0 (V )\times H1
0 (V ),
| B2(u, \varphi )| \leq
\int
V
\bigm| \bigm| h(u(x))\bigm| \bigm| | \varphi (x)| | g2(x)| d\mu \leq
\leq A
\int
V
| \varphi (x)| | g2(x)| d\mu \leq
\leq A\| \varphi \| 2\| g2\| 2 \leq AC\| \varphi \| H1
0 (V )\| g2\| 2. (2.4)
Also, we have \bigm| \bigm| T (\varphi )\bigm| \bigm| \leq \int
V
| f(x)| | \varphi (x)| d\mu \leq
\bigm\| \bigm\| f\bigm\| \bigm\|
2
\| \varphi \| 2 \leq C
\bigm\| \bigm\| f\bigm\| \bigm\|
2
\| \varphi \| H1
0 (V ),
where C is a constant arising out of the inequality (2.1). Now, B1(u, \cdot ) and B2(u, \cdot ) are linear and
bounded for every u \in H1
0 (V ). We define the operators
B,N : H1
0 (V ) \rightarrow H - 1(V )
as
(Bu| \varphi ) = B1(u, \varphi ), (Nu| \varphi ) = B2(u, \varphi ) for u, \varphi \in H1
0 (V ).
Then, (2.3) is equivalent to the operator equation Bu+Nu = T, u \in H1
0 (V ).
3. Main results. In this section, we study the existence of a weak solution for (1.1).
Theorem 3.1. Assume that the hypotheses (H1) and (H2) hold. Let
1 > \lambda C2
\bigm\| \bigm\| g1\bigm\| \bigm\| \infty , (3.1)
where C is the constant in inequality (2.1). Then the BVP (1.1) has at least one weak solution
u \in H1
0 (V ). Moreover, every (weak) solution u of (1.1) satisfies
\| u\| H1
0 (V ) \leq
C
\bigl\{
A\| g2\| 2 + \| f\| 2\}
(1 - \lambda C2\| g1\| \infty )
,
where A is the constant from hypothesis (H1).
Proof. First we write the BVP (1.1) as the following operator equation in H - 1(V ):
u \in H1
0 (V ) : Bu+Nu = T.
We prove that T \in H - 1(V ), B,N : H1
0 (V ) \rightarrow H - 1(V ) satisfy all the conditions given in Proposi-
tion 2.1. For convenience, we divide the proof into five steps.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1322 A. K. BADAJENA, R. KAR
Step 1. From the previous section we know that the operator B is linear and continuous. By
Lemma 2.2 the embedding of H1
0 (V ) \lhook \rightarrow L2(V ) is compact which shows that B1(\cdot , \cdot ) is a regular
Gårding form. Furthermore, the inequality (3.1) and
B1(u, u) = \scrW (u, u) - \lambda
\int
V
u2(x)g1(x)d\mu \geq
\geq \| u\| 2H1
0 (V ) - \lambda \| g1\| \infty \| u\| 22 (3.2)
shows that B1(\cdot , \cdot ) is a strict Gårding form. Let \{ uk\} be any sequence in H1
0 (V ) and
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
(Buk| uk) = 0. (3.3)
Claim: B satisfies condition (S). Since B is linear, as in (3.2), we have
(Buk| uk) = (B(uk)| uk) = B1(uk, uk) \geq
\geq \| uk\| 2H1
0 (V ) - \lambda \| g1\| \infty \| uk\| 22 \geq
\geq (1 - \lambda C2\| g1\| \infty )\| uk\| 2H1
0 (V ). (3.4)
From (3.3) and (3.4), we note that
0 \leq
\bigl(
1 - \lambda C2\| g1\| \infty
\bigr)
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
\| uk\| 2H1
0 (V ) \leq \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
\bigl(
Buk| uk
\bigr)
= 0.
Since (1 - \lambda C2\| g1\| \infty ) > 0, we have \| uk\| 2H1
0 (V )
\rightarrow 0 as k \rightarrow \infty , which implies uk \rightarrow 0 as k \rightarrow \infty .
Hence, B satisfies condition (S).
Step 2. Claim: B +N is asymptotically linear. By (2.4), we have
\| Nu\| H - 1(V ) \leq C \prime ,
where C \prime = AC\| g2\| 2 is a constant depending on V . Consequently,
\| Nu\| H - 1(V )
\| u\| H1
0 (V )
\rightarrow 0 as \| u\| H1
0 (V ) \rightarrow \infty ,
which shows that B + N is asymptotically linear. Next we show that N is strongly continuous.
Since h is continuous and is a function of u only, the hypotheses (H1) and (H6\ast ) of [31] (Corol-
lary 26.14) are satisfied. The hypothesis (H2) follows from the fact that h is bounded. Hence, by
[31] (Corollary 26.14), N is a strongly continuous operator.
Step 3. From Step 2, we note that the operator B satisfies condition (S). Since, N is strongly
continuous, we note that the operator u \in H1
0 (V ) \mapsto \rightarrow t(Nu - T ) \in
\bigl(
H1
0 (V )
\bigr) \ast
is strongly continuous
for t \in [0, 1]. For each t \in [0, 1], the operator At(u) = Bu+t(Nu - T ) is thus a strongly continuous
perturbation of the (S)-operator B . So, the operator At(u) satisfies condition (S) (we refer to [31],
Problem 27.1).
Step 4. Now, Bu = 0, with u \in H1
0 (V ), implies
\scrW (u, u) - \lambda
\int
V
u2(x)g1(x)d\mu = 0.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
EXISTENCE OF A WEAK SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC . . . 1323
Consequently, we have \bigl(
1 - \lambda C2\| g1\| \infty
\bigr)
\| u\| 2H1
0 (V ) \leq 0,
which shows that u = 0, since 1 - \lambda C2\| g1\| \infty > 0.
By Proposition 2.1, Bu + Nu = T has a solution u \in H1
0 (V ) which equivalently shows, the
BVP (1.1) has a solution u \in H1
0 (V ).
Step 5. Let u \in H1
0 (V ) be a weak solution of (1.1). As in (3.4) (with the help of the embedding
in Lemma 2.2), we obtain
B1(u, u) \geq
\bigl(
1 - \lambda C2\| g1\| \infty
\bigr)
\| u\| 2H1
0 (V ).
Since, 1 > \lambda C2\| g1\| \infty , we have
\| u\| 2H1
0 (V ) \leq
1
1 - \lambda C2\| g1\| \infty
B1(u, u). (3.5)
Also, we note that
| B1(u, u)| \leq C
\bigl\{
A\| g2\| 2 + \| f\| 2
\bigr\}
\| u\| H1
0 (V ). (3.6)
By (3.5) and (3.6), we get
\| u\| H1
0 (V ) \leq
C
\bigl\{
A\| g2\| 2 + \| f\| 2\} \bigl(
1 - \lambda C2\| g1\| \infty
\bigr) .
Theorem 3.1 is proved.
Next, we dispense with the condition (3.1) when g1 does not change sign. The two results are
related to the cases when g1 \geq 0 with \lambda \leq 0 and g1 \leq 0 with \lambda > 0. These results are similar to
that of Theorem 3.1 but with suitable changes.
Theorem 3.2. Suppose that the hypotheses (H1) and (H2) hold. Let g1 \geq 0 and \lambda \leq 0, then
the BVP (1.1) has at least one weak solution. For every weak solution u \in H1
0 (V ) of (1.1) the
inequality
\| u\| H1
0 (V ) \leq C
\bigl\{
A\| g2\| 2 + \| f\| 2\}
holds, where C is the constant in inequality (2.1).
Proof. As in Theorem 3.1, the basic idea is to reduce the problem (1.1) to the operator equation
Bu+Nu = T and then to apply Proposition 2.1. For this, we define B,N and T, as in Theorem 3.1.
The compact embedding of H1
0 (V ) \lhook \rightarrow L2(V ) and (3.1) shows that B1(\cdot , \cdot ) is a strict regular Gårding
form. Also, \lambda \leq 0 and g1 \geq 0 yields
B1(u, u) = \scrW (u, u) - \lambda
\int
V
u2(x)g1(x)d\mu \geq \| u\| 2H1
0 (V ). (3.7)
Let \{ uk\} be any sequence in H1
0 (V ) and
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
(Buk| uk) = 0. (3.8)
We claim that B satisfies condition (S). Since B is linear, as in (3.7), we have
(Buk| uk) = (B(uk)| uk) = B1(uk, uk) \geq \| uk\| 2H1
0 (V ). (3.9)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1324 A. K. BADAJENA, R. KAR
From (3.8) and (3.9), we note that
0 \leq \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
\| uk\| 2H1
0 (V ) \leq \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
(Buk| uk) = 0
which implies \| uk\| H1
0 (V ) \rightarrow 0 as k \rightarrow \infty and, consequently, B satisfies condition (S). Next, we
show that B+N is asymptotically linear and, N is strongly continuous. The proof is similar to that
of Theorem 3.1 and we omit it for brevity. Since \lambda \leq 0, we get from (3.7) that Bu = 0 implies
u = 0. By Proposition 2.1, Bu+Nu = T has a solution u \in H1
0 (V ) which equivalently shows that
the BVP (1.1) has at least one weak solution. Let u \in H1
0 (V ) be such a solution. Then, by (3.7) and
a similar argument as in Theorem 3.1, we have
\| u\| H1
0 (V ) \leq C
\bigl\{
A\| g2\| 2 + \| f\| 2
\bigr\}
,
where C is the constant in inequality (2.1).
Theorem 3.2 is proved.
With suitable modifications in the proof of Theorem 3.2, we have the following result.
Theorem 3.3. Suppose that the hypotheses (H1) and (H2) hold. Let g1 \leq 0 and \lambda > 0, then
(1.1) has a weak solution u \in H1
0 (V ) and there is a constant k0 such that \| u\| H1
0 (V ) \leq k0 for every
(weak) solution u.
4. Extensions. In Section 3, the nonlinearity h is assumed to be continuous and bounded.
In this section, we extend these results for a class of functions h which are continuous only. The
generalized Hölder inequality comes handy for getting suitable estimates. We establish the existence
of a weak solution for (1.1), where h : \BbbR \rightarrow \BbbR is required to be continuous and to satisfy | h(t)| \leq | t| \epsilon ,
0 < \epsilon < 1, for all t \in \BbbR . Again, we consider the cases \lambda \leq 0 and \lambda > 0 separately. Since the proofs
are similar to the ones in Section 3, we restrict ourselves to sketch only the differences wherever
needed. The Corollary 26.14 in [31] is not applicable here since h is not bounded. We collect the
common hypotheses for convenience:
(H \prime
1) the function h : \BbbR \rightarrow \BbbR is continuous and satisfies | h(t)| \leq | t| \epsilon , t \in \BbbR , for a fixed
0 < \epsilon < 1;
(H \prime
2) g1 \in L\infty (V ), g2 \in L
2
1 - \epsilon (V ), 0 < \epsilon < 1 and f \in L2(V ).
Theorem 4.1. Assume that the hypotheses (H \prime
1), (H
\prime
2) hold. If g1 \geq 0 and \lambda \leq 0, then (1.1)
has at least one weak solution and there is a constant k0 such that \| u\| H1
0 (V ) \leq k0 for every (weak)
solution u \in H1
0 (V ).
Proof. We give only a sketch of the proof since it is similar to the proof of Theorem 3.2. By
Lemma 2.2 and generalized Hölder’s inequality [23, p. 67], we have
| B2(u, \varphi )| \leq
\int
V
| h(u(x))| | \varphi (x)| | g2| d\mu \leq \| u\| \epsilon 2\| \varphi \| 2\| g2\| 2
1 - \epsilon
for every \varphi \in H1
0 (V ).
By a similar argument as in Theorem 3.2 we observe that the operator B1 satisfies condition (S). We
also observe that
| (Nu| \varphi )| = | B2(u, \varphi )| \leq C\epsilon +1\| u\| \epsilon H1
0 (V )\| \varphi \| H1
0 (V )\| g2\| 2
1 - \epsilon
,
which implies
\| Nu\| H - 1(V ) \leq C\epsilon +1\| u\| \epsilon H1
0 (V )\| g2\| 2
1 - \epsilon
= c\| u\| \epsilon H1
0 (V ),
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
EXISTENCE OF A WEAK SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC . . . 1325
where the constant c = C\epsilon +1\| g2\| 2
1 - \epsilon
. So
\| Nu\| H - 1(V )
\| u\| H1
0 (V )
\leq
c\| u\| \epsilon
H1
0 (V )
\| u\| H1
0 (V )
\rightarrow 0 as \| u\| H1
0 (V ) \rightarrow \infty . (4.1)
This shows that B +N is asymptotically linear. Also, u \in L2(V ) implies that h(u) \in L
2
\epsilon (V ) and
define the Nemytskii operator
F : L2(V ) \rightarrow L
2
\epsilon (V )
by (Fu)(x) = h(u(x)); we have that F is continuous (by [22], Theorem 2.1). Now, the hypotheses
(H \prime
1), (H
\prime
2) and the generalized Hölder inequality imply that\bigm| \bigm| (Nun| \varphi ) - (Nu| \varphi )
\bigm| \bigm| \leq \int
V
| h(un) - h(u)\| g2\| \varphi | d\mu \leq
\leq C
\bigm\| \bigm\| h(un) - h(u)
\bigm\| \bigm\|
2
\epsilon
\| g2\| 2
1 - \epsilon
\| \varphi \| H1
0 (V ).
Let un \rightharpoonup u weakly in H1
0 (V ). Then, by the continuity of F in L
2
\epsilon (V ) and by the compact
embedding H1
0 (V ) \lhook \rightarrow L2(V ), we have
\| Nun - Nu\| H - 1(V ) \rightarrow 0 as n \rightarrow \infty .
By a similar argument as in Theorem 3.1, we can show that the operator At(u) = Bu+ t(Nu - T )
satisfies condition (S). If \lambda \leq 0, then Bu = 0 implies as in the proof of Theorem 3.2 that u = 0.
By Proposition 2.1, problem (1.1) has at least one weak solution u \in H1
0 (V ), which completes the
proof of existence result.
Let u \in H1
0 (V ) be a weak solution of (1.1). Then\bigm| \bigm| B1(u, u)
\bigm| \bigm| \leq C
\bigl\{
C\epsilon \| u\| \epsilon H1
0 (V )\| g2\| 2
1 - \epsilon
+ \| f\| 2
\bigr\}
\| u\| H1
0 (V ). (4.2)
By (3.7) and (4.2), we have
\| u\| H1
0 (V ) \leq C
\bigl\{
C\epsilon \| u\| \epsilon H1
0 (V )\| g2\| 2
1 - \epsilon
+ \| f\| 2
\bigr\}
. (4.3)
If \| u\| H1
0 (V ) \geq 1, then, from (4.3), we have
\| u\| H1
0 (V ) \leq C
\bigl(
C\epsilon \| g2\| 2
1 - \epsilon
+ \| f\| 2
\bigr)
\| u\| \epsilon H1
0 (V ),
which implies that
\| u\| 1 - \epsilon
H1
0 (V )
\leq c or \| u\| H1
0 (V ) \leq c
1
1 - \epsilon ,
where c = C
\bigl(
C\epsilon \| g2\| 2
1 - \epsilon
+ \| f\| 2
\bigr)
. If \| u\| H1
0 (V ) \leq 1, we have nothing to prove. Let k0 =
= \mathrm{m}\mathrm{a}\mathrm{x}\{ 1, c
1
1 - \epsilon \} . Hence, we have
\| u\| H1
0 (V ) \leq k0.
Theorem 4.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1326 A. K. BADAJENA, R. KAR
Remark. Theorem 4.1 also holds if g1 \leq 0 and \lambda > 0. But when \lambda > 0 and g1 changes
sign, we need additional conditions on \lambda and g1 (stated below) as in Theorem 3.1. We state these
results below in Theorem 4.2 for which we give only a sketch of the proof. We note that in (4.1) the
asymptotic linearity of B +N is a consequence of \epsilon lying between 0 and 1.
Theorem 4.2. Let the hypotheses (H \prime
1), (H
\prime
2) hold. Also, let 1 > \lambda C2\| g1\| \infty . Then the BVP
(1.1) has at least one weak solution and there is a constant k0 such that \| u\| H1
0 (V ) \leq k0 for every
(weak) solution u \in H1
0 (V ).
Proof. The proof for the existence of at least one weak solution u \in H1
0 (V ) for (1.1) is similar
to the arguments in the proof of Theorem 4.1 and Theorem 3.1 and hence it is omitted. As in
Theorem 3.1, we note that
(1 - \lambda C2\| g1\| \infty )\| u\| 2H1
0 (V ) \leq C
\bigl\{
C\epsilon \| u\| \epsilon H1
0 (V )\| g2\| 2
1 - \epsilon
+ \| f\| 2\} \| u\| H1
0 (V ),
where C is a constant. Since 1 > \lambda C2\| g1\| \infty , we obtain
\| u\| H1
0 (V ) \leq
C
\bigl(
C\epsilon \| u\| \epsilon
H1
0 (V )
\| g2\| 2
1 - \epsilon
+ \| f\| 2
\bigr)
(1 - \lambda C2\| g1\| \infty )
. (4.4)
If \| u\| H1
0 (V ) \geq 1, from (4.4), we have
\| u\| H1
0 (V ) \leq
C
\bigl(
C\epsilon \| g2\| 2
1 - \epsilon
+ \| f\| 2
\bigr)
\| u\| \epsilon
H1
0 (V )\bigl(
1 - \lambda C2\| g1\| \infty
\bigr) ,
which implies that
\| u\| 1 - \epsilon
H1
0 (V )
\leq c or \| u\| H1
0 (V ) \leq c
1
1 - \epsilon ,
where c =
C
\bigl(
C\epsilon \| g2\| 2
1 - \epsilon
+ \| f\| 2
\bigr)
(1 - \lambda C2\| g1\| \infty )
and 0 < \epsilon < 1.
If \| u\| H1
0 (V ) \leq 1, we have nothing to prove. Let k0 = \mathrm{m}\mathrm{a}\mathrm{x}\{ 1, c
1
1 - \epsilon \} . Then we have
\| u\| H1
0 (V ) \leq k0.
Theorem 4.2 is proved.
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ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
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| id | umjimathkievua-article-6248 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:41Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/12/69f6925d4ab9aa972f9f2ac36d0e8312.pdf |
| spelling | umjimathkievua-article-62482023-01-07T13:45:37Z Existence of a weak solution for a class of nonlinear elliptic equations on the Sierpiński gasket Existence of a weak solution for a class of nonlinear elliptic equations on the Sierpński gasket Badajena, A. K. Kar, R. Badajena, A. K. Kar, R. Sierpinski gasket, Nonlinear elliptic equations, Fractal domains, Demicontinu- ous operators. 35J60; 28A80; 47H99. UDC 517.9 We study the existence of a weak (strong) solution of the nonlinear elliptic problem\begin{gather*} -\Delta u- \lambda ug_1 +h(u)g_2=f \quad\text{in}\quad V\setminus V_0,\\u=0 \quad\text{on}\quad V_0,\end{gather*} where $V$ is a Sierpi\'nski gasket in $\mathbb{R}^{N-1},$ $N\geq 2,$ $V_0$ is its boundary (consisting of $N$ its corners), and $\lambda$ is a real parameter. Here, $f,g_1,g_2\colon V\to\mathbb{R}$ and $h\colon \mathbb{R}\to\mathbb{R}$ are functions satisfying suitable hypotheses. УДК 517.9 Існування слабкого розв’язку для класу нелінійних еліптичних рівнянь на прокладці Серпінського Досліджується існування слабкого (сильного) розв’язку нелінійної еліптичної задачі \begin{gather*} -\Delta u- \lambda ug_1 +h(u)g_2=f \quad\text{в}\quad V\setminus V_0,\\u=0 \quad\text{на}\quad V_0,\end{gather*} де $V$ – прокладка Серпінського в $\mathbb{R}^{N-1},$ $N\geq 2,$ $V_0$ – її межа (що складається з її $N$ кутів) і $ \lambda$ – дійсний параметр. Тут $f,g_1,g_2\colon V\to\mathbb{R}$ і $h\colon \mathbb{R}\to\mathbb{R}$ – функції, що задовольняють відповідні гіпотези. Institute of Mathematics, NAS of Ukraine 2022-11-27 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6248 10.37863/umzh.v74i10.6248 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 10 (2022); 1317 - 1327 Український математичний журнал; Том 74 № 10 (2022); 1317 - 1327 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6248/9307 Copyright (c) 2022 Rasmita |
| spellingShingle | Badajena, A. K. Kar, R. Badajena, A. K. Kar, R. Existence of a weak solution for a class of nonlinear elliptic equations on the Sierpiński gasket |
| title | Existence of a weak solution for a class of nonlinear elliptic equations on the Sierpiński gasket |
| title_alt | Existence of a weak solution for a class of nonlinear elliptic equations on the Sierpński gasket |
| title_full | Existence of a weak solution for a class of nonlinear elliptic equations on the Sierpiński gasket |
| title_fullStr | Existence of a weak solution for a class of nonlinear elliptic equations on the Sierpiński gasket |
| title_full_unstemmed | Existence of a weak solution for a class of nonlinear elliptic equations on the Sierpiński gasket |
| title_short | Existence of a weak solution for a class of nonlinear elliptic equations on the Sierpiński gasket |
| title_sort | existence of a weak solution for a class of nonlinear elliptic equations on the sierpiński gasket |
| topic_facet | Sierpinski gasket Nonlinear elliptic equations Fractal domains Demicontinu- ous operators. 35J60 28A80 47H99. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6248 |
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