Existence of nontrivial solution for fourth-order semilinear Δ γ -Laplace equation in ℝ N

Abstract

In this paper, we study existence of nontrivial solutions for a fourth-order
semilinear $\Delta_{\gamma}$-Laplace equation in $\mathbb{R}^N$
\begin{gather*}
\Delta_\gamma^{2}u-\Delta_\gamma u+\lambda b(x)u=f(x, u), \quad x\in \mathbb{R}^N,\quad
u\in \mathbf S_\gamma^{2}(\mathbb{R}^N),
\end{gather*}
where $\lambda >0$ is a parameter and $ \Delta_{\gamma}$ is the subelliptic operator of the type
$$
\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \quad \partial_{x_j}:
=\frac{\partial }{\partial x_{j}},\quad \gamma = (\gamma_1, \gamma_2, \dots, \gamma_N),\quad \Delta^2_\gamma: =\Delta_\gamma(\Delta_\gamma).
$$
Under some suitable assumptions on $b(x)$ and $f(x,\xi)$, we obtain the existence of nontrivial solution for $\lambda$ large enough.

Full text

Electronic Journal of Qualitative Theory of Differential Equations
2019, No. 78, 1–12; https://doi.org/10.14232/ejqtde.2019.1.78 www.math.u-szeged.hu/ejqtde/
Existence of nontrivial solution for fourth-order
semilinear ∆γ-Laplace equation in RN
Duong Trong LuyenB1, 2
1Division of Computational Mathematics and Engineering, Institute for Computational Science,
Ton Duc Thang University, Ho Chi Minh City, Vietnam
2Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Received 1 July 2019, appeared 24 October 2019
Communicated by Dimitri Mugnai
Abstract. In this paper, we study existence of nontrivial solutions for a fourth-order
semilinear ∆γ-Laplace equation in RN
∆2
γu−∆γu+λb(x)u=f(x,u),x∈RN,u∈S2
γ(RN),
where λ>0 is a parameter and ∆γis the subelliptic operator of the type
∆γ:=
N
∑
j=1
∂xjγ2
j∂xj,∂xj:=∂
∂xj
,γ= (γ1,γ2, . . . , γN),∆2
γ:=∆γ(∆γ).
Under some suitable assumptions on b(x)and f(x,ξ), we obtain the existence of non-
trivial solution for λlarge enough.
Keywords: fourth-order semilinear degenerate elliptic equations, ∆γ-Laplace operator,
nontrivial solutions, Cerami sequences, mountain pass theorem.
2010 Mathematics Subject Classification: 35J50, 35J60.
1 Introduction
In the last decades, the biharmonic elliptic equations
∆2u−∆u+λb(x)u=f(x,u),x∈RN,u∈H2(RN), (1.1)
has been studied by many authors see [12,19,20,26–30] and the references therein. The bi-
harmonic equations can be used to describe some phenomena appearing in physics and engi-
neering. For example, the problem of nonlinear oscillation in a suspension bridge [10,14,15]
and the problem of the static deflection of an elastic plate in a fluid [1]. In the last decades,
the existence and multiplicity of nontrivial solutions for biharmonic equations have begun to
receive much attention.
BEmail: duongtrongluyen@tdtu.edu.vn

2D. T. Luyen
In this paper, we consider the biharmonic equation as follows:
∆2
γu−∆γu+λb(x)u=f(x,u),x∈RN,u∈S2
γ(RN), (1.2)
where ∆γis a subelliptic operator of the form
∆γ:=
N
∑
j=1
∂xjγ2
j∂xj,γ= (γ1,γ2, . . . , γN):RN→RN,∆2
γ:=∆γ(∆γ).
The ∆γ-operator was considered by B. Franchi and E. Lanconelli in [6], and recently reconsid-
ered in [9] under the additional assumption that the operator is homogeneous of degree two
with respect to a group dilation in RN. The ∆γ-operator contains many degenerate elliptic
operators such as the Grushin-type operator
Gα:=∆x+|x|2α∆y,α≥0,
where (x,y)denotes the point of RN1×RN2(see [7,21,23]), and the operator of the form
Pα,β:=∆x+∆y+|x|2α|y|2β∆z,(x,y,z)∈RN1×RN2×RN3,
where α,βare nonnegative real numbers (see [22,24]).
We assume that the potential b(x)satisfies the following conditions:
(B1)b:RN→Ris a nonnegative continuous function on RN, there exists a constant C0>0
such that the set {b<C0}:={x∈RN:b(x)<C0}has finite positive Lebesgue measure
for e
N>4;
(B2)Ω=int{x∈RN:b(x) = 0}is nonempty and has smooth boundary with ¯
Ω={x∈
RN:b(x) = 0}.
Under the hypotheses (B1),(B2),λb(x)is called the steep potential well whose depth is
controlled by the parameter λ. Such potential is first suggested by Bartsch–Wang [3] in the
scalar Schrödinger equations. Later, the steep potential well is introduced to the study of
some other types of nonlinear differential equations by some researchers, such as Kirchhoff
type equations [16], Schrödinger–Poisson systems [8,18,31] and also biharmonic equations
[13,17,25].
Next, we can state the main theorem of the paper.
Theorem 1.1. Suppose that e
N>4and conditions (B1),(B2)hold. In addition, we assume that a
continuous function f (x,ξ) = α(x)g(ξ)satisfies:
(g1)g(ξ) = o(|ξ|)as ξ→0;
(g2)g(ξ) = o(|ξ|)as ξ→∞;
(α1)0<α(x)∈L1(RN)∩L∞(RN)and C1:=kαkL∞(RN)maxξ6=0g(ξ)
ξ<1
1+C2
2
;
(B3)Vol{b<C0}<1−C1(1+C2
2)
C2
3e
N
4,
where Vol(·)denotes the Lebesgue measure of a set in RNand where C2is the best constant in (2.2)
below.
Then there exists a constant Λ0>0such that the problem (1.2)has only the trivial solution for all
λ≥Λ0.

Existence of solutions for fourth-order degenerate elliptic equation 3
Theorem 1.2. Suppose that e
N>4and conditions (B1),(B2)hold. In addition, we assume that the
function f (x,ξ)satisfies:
(F1)f∈C(RN×R,R), and there exist a constant p ∈(2, 2γ
∗)and two functions f1(x),f2(x)∈
L∞(RN)satisfying 
f+
1
L∞(RN)<Θ−1
2and f2(x)>0on ¯
Ωsuch that
lim
ξ→0+
f(x,ξ)
|ξ|p−1=f1(x)and lim
ξ→∞
f(x,ξ)
|ξ|p−1=f2(x)uniformly in x ∈RN;
where f +
1:=max{f1, 0},Θ2is given in (2.5)below;
(F2)there exists are constants 1< ` < 2, µ>2and a nonnegative function f3∈L2
2−`(RN)such
that
µF(x,ξ)−f(x,ξ)≤f3(x)|ξ|`for all x ∈RNand ξ∈R,
where F(x,ξ) = Rξ
0f(x,τ)dτ.
Then there exists a constant Λ1>0such that the problem (1.2)admits at least a nontrivial solution
for all λ≥Λ1.
The paper is organized as follows. In Section 2 for convenience of the readers, we recall
some function spaces, embedding theorems and associated functional settings. We prove our
main results by using Ekeland’s variational principle and Gagliardo–Nirenberg’s inequality in
Section 3.
2 Preliminary results
2.1 Function spaces and embedding theorems
We recall the functional setting in [9]. We consider the operator of the form
∆γ:=
N
∑
j=1
∂xjγ2
j∂xj,∂xj=∂
∂xj
,j=1, 2, . . . , N.
Here, the functions γj:RN→Rare assumed to be continuous, different from zero and of
class C1in RN\Π, where
Π:=(x= (x1,x2, . . . , xN)∈RN:
N
∏
j=1
xj=0).
Moreover, we assume the following properties:
i) There exists a group of dilations {δt}t>0such that
δt:RN−→ RN
(x1, . . . , xN)7−→ δt(x1, . . . , xN)=(tε1x1, . . . , tεNxN),
where 1 =ε1≤ε2≤ · · · ≤ εN, such that γjis δt-homogeneous of degree εj−1, i.e.,
γj(δt(x)) =tεj−1γj(x),∀x∈RN,∀t>0, j=1, . . . , N.

4D. T. Luyen
The number
e
N:=
N
∑
j=1
εj(2.1)
is called the homogeneous dimension of RNwith respect to {δt}t>0.
ii)
γ1=1, γj(x)=γjx1,x2, . . . , xj−1,j=2, . . . , N.
iii) There exists a constant ρ≥0 such that
0≤xk∂xkγj(x)≤ργj(x),∀k∈{1, 2, . . . , j−1},∀j=2, . . . , N,
and for every x∈RN
+:=(x1, . . . , xN)∈RN:xj≥0, ∀j=1, 2, . . . , N.
iv) Equalities γj(x)=γj(x∗)(j=1, 2, . . . , N)are satisfied for every x∈RN, where
x∗=(|x1|, . . . , |xN|)if x= (x1,x2, . . . , xN).
Definition 2.1. By S2
γ(RN)we will denote the set of all functions u∈L2(RN)such that
γj∂xju∈L2(RN)for all j=1, . . . , Nand ∆γu∈L2(RN). We define the norm in this space as
follows
kukS2
γ(RN)=ZRN|∆γu|2+|∇γu|2+|u|2dx1
2
,
where ∇γu=(γ1∂x1u,γ2∂x2u, . . . , γN∂xNu).
Let
Eλ=u∈S2
γ(RN):ZRN|∆γu|2+λb(x)u2dx<∞.
For λ>0, the inner product and norm of Eλare given by
(u,v)Eλ=ZRN(∆γu∆γv+λb(x)uv)dx,kukEλ= (u,u)1
2
Eλ.
Lemma 2.2. The following embeddings are continuous:
i) S2
γ(RN),→Lp(RN)for all 2≤p<2γ
∗:=2e
N
e
N−4.
ii) Assume that (B1)and (B2)hold, for every λ≥Λ, the embedding Eλ,→S2
γ(RN)and Eλ,→
Lp(RN),p∈[2, 2γ
∗).
Proof. i)We follow the ideas in the case of bounded domains (see the proofs of Theorem 3.3,
Proposition 3.2 in [9] and Lemma 2.2 in [2]). More precisely, we first embed S2
γ(RN)into an
anisotropic Sobolev-type space, and then use an embedding theorem for classical anisotropic
Sobolev-type spaces of fractional orders. Because the proof is very similar to the case of
bounded domains [2,9], so we omit it here.
ii)For all u∈C∞
0(RN), with slight modification, the proof is similar to the one of Theorems
12.85 and 12.87 in [11], there exists C2,C3>0 such that
ZRN|∇γu|2dx≤C2
2ZRN|∆γu|2dx1
2ZRNu2dx1
2
, (2.2)

Existence of solutions for fourth-order degenerate elliptic equation 5
ZRN|u|2e
N
e
N−4dxe
N−4
e
N≤C3ZRN|∆γu|2dx. (2.3)
This shows that
ZRN|∆γu|2+u2dx≤kuk2
S2
γ(RN)≤1+C2
2
2ZRN|∆γu|2+u2dx. (2.4)
From (B1), using Hölder’s inequality and (2.2), we obtain
ZRNu2dx=Z{b≥C0}u2dx+Z{b<C0}u2dx
≤1
C0Z{b≥C0}b(x)u2dx+(Vol({b<C0}))4
e
NZRN|u|2e
N
e
N−4dxe
N−4
e
N
≤1
C0ZRNb(x)u2dx+C2
3(Vol({b<C0})) 4
e
NZRN|∆γu|2dx,
where C3is the best constant in (2.3). Combining the above inequality with (2.4) yields
kukS2
γ(RN)≤1+C2
2
21+C2
3(Vol({b<C0})) 4
e
NZRN|∆γu|2dx+1
C01+C2
2
2ZRNb(x)u2dx.
Then for λ≥1+C2
3Vol({b<C0})C0, we have
kuk2
S2
γ(RN)≤1+C2
2
21+C2
3(Vol({b<C0}))4
e
Nkuk2
Eλ.
This implies that the embedding Eλ,→S2
γ(RN)is continuous. By using Hölder ’s inequality,
we obtain
ZRN|u|pdx≤ZRN|u|2dx2e
N−p(e
N−4)
8ZRN|u|2γ
∗dxe
N(p−2)
4e
N−4
2e
N
≤kuk
2e
N−p(e
N−4)
8
L2(RN)Ce
N(p−2)
4
3k∆γuke
N(p−2)
4
L2(RN)
≤kuk
2e
N−p(e
N−4)
8
S2
γ(RN)Ce
N(p−2)
4
3kuke
N(p−2)
4
S2
γ(RN)
≤Ce
N(p−2)
4
3kukp
S2
γ(RN)
≤Ce
N(p−2)
4
31+C2
2
2p
21+C2
3(Vol({b<C0}))4
e
Np
2kukp
Eλ,
where p∈[2, 2γ
∗). We get
Θp=Ce
N(p−2)
4
31+C2
2
2p
21+C2
3(Vol({b<C0}))4
e
Np
2, (2.5)
and
Λ=1+C2
3Vol({b<C0})C0.
Thus, for any p∈[2, 2γ
∗)and λ≥Λ, there holds
ZRN|u|pdx≤Θpkukp
Eλ,
which implies that the embedding Eλ,→Lp(RN)is continuous.

6D. T. Luyen
Definition 2.3. A function u∈S2
γ(RN)is called a weak solution of the problem (1.2) if u∈Eλ
and
ZRN(∆γu∆γϕ+∇γu· ∇γϕ+λb(x)uϕ)dx−ZRNf(x,u(x))ϕdx=0, ∀ϕ∈Eλ.
2.2 Mountain Pass Theorem
Definition 2.4. Let Xbe a real Banach space with its dual space X∗and Φ∈C1(X,R). For
c∈Rwe say that Φsatisfies the (C)ccondition if for any sequence {xn}∞
n=1⊂Xwith
Φ(xn)→cand (1+kxnkX)
Φ0(xn)
X∗→0,
then there exists a subsequence {xnk}∞
k=1that converges strongly in X. If Φsatisfies the (C)c
condition for all c>0 then we say that Φsatisfies the Cerami condition.
We will use the following version of the Mountain Pass Theorem.
Lemma 2.5 (see [4,5]).Let Xbe an infinite dimensional Banach space and let Φ∈C1(X,R)satisfy
the (C)ccondition for all c ∈R,Φ(0) = 0, and
(i)There are constants ρ,α>0such that Φ(u)≥αfor all u ∈Xsuch that kukX=ρ;
(ii)There is an e ∈X,kukX>ρsuch that Φ(e)≤0.
Then β=infθ∈Γmax0≤t≤1Φ(θ(t)) ≥αis a critical value of Φ, where
Γ={θ∈C([0, 1],X):θ(0) = 0, θ(1) = e}.
3 Proofs of the main results
Define the Euler–Lagrange functional associated with the problem (1.2) as follows
Φ(u)=1
2ZΩ|∆γu|2+|∇γu|2+λb(x)u2dx−ZΩF(x,u)dx.
By fsatisfies (f1),(f2),(α1)or (F1), hence its not difficult to prove that the functional Φis
of class C1in Eλ, and that
Φ0(u)(v) = ZΩ(∆γu∆γv+∇γu· ∇γv+λb(x)uv)dx−ZΩf(x,u)vdx
for all v∈Eλ. One can also check that the critical points of Φare weak solutions of the
problem (1.2).
3.1 Proof of Theorem 1.1
By condition (g1), for all ε>0, there exists δ(ε)>0, we have
|g(u)|≤ε|u|for all |u|<δ(ε).
By condition (g2), there exists M>0, we obtain
|g(u)|≤|u|for all |u|>M.

Existence of solutions for fourth-order degenerate elliptic equation 7
Since is a continuous function, gachieves its maximum and minimum on [δ(ε),M], so there
exists a positive number C(ε), we have that
|g(u)|≤C(ε)≤C(ε)|u|
δ(ε)for all δ(ε)≤|u|≤M.
Then we obtain that
|g(u)|≤1+ε+C(ε)
δ(ε)|u|for all u∈R.
Hence maxξ6=0g(ξ)
ξis well defined.
Let uis a nontrivial solution of the problem (1.2), we get
kuk2
Eλ=ZRNα(x)g(u)udx,
hence
kuk2
Eλ≤kαkL∞(RN)ZRN
g(u)
uu2dx≤C1ZRNu2dx.
By Lemma 2.2 and condition (B3), we have
kuk2
Eλ
<kuk2
Eλ,
which is a contradiction, thus u≡0. The proof of Theorem 1.1 is therefore complete.
3.2 Proof of Theorem 1.2
Lemma 3.1. Assume that conditions (B1),(B2)and (F1)hold. Then for each λ≥Λ, there exists
ρ,β>0such that
inf{Φ(u):u∈Eλ,kukEλ=ρ}>α.
Proof. For any ε>0, it follows from the condition (F1)that there exists Cε>0 and p∈(2, 2γ
∗)
such that
f(x,ξ)≤
f+
1
L∞(RN)+εξ+Cεξp−1for all ξ∈R(3.1)
and
F(x,ξ)≤
f+
1
L∞(RN)+ε
2ξ2+Cε
pξpfor all ξ∈R.
From Lemma 2.2, we have for all u∈Eλ,
ZRNF(x,u)dx≤
f+
1
L∞(RN)+ε
2ZRNu2dx+Cε
pZRNupdx
≤
f+
1
L∞(RN)+εΘ2
2kuk2
Eλ+CεΘp
pkukp
Eλ. (3.2)

8D. T. Luyen
Hence
Φ(u) = 1
2kuk2
Eλ+1
2ZRN|∇γu|2dx−ZRNF(x,u)dx
≥1
2kuk2
Eλ−ZRNF(x,u)dx
≥1
2kuk2
Eλ−
f+
1
L∞(RN)+εΘ2
2kuk2
Eλ−CεΘp
pkukp
Eλ
=1
2h1−
f+
1
L∞(RN)+εΘ2ikuk2
Eλ−CεΘp
pkukp
Eλ.
So, fixing ε∈(0, Θ−1
2−
f+
1
L∞(RN))and letting kukEλ=ρ>0 small enough, it is easy to see
that there exists α>0 such that this lemma holds.
Lemma 3.2. Assume that conditions (B1),(B2)and (F1)hold. Let ρ>0be as in Lemma 3.1. Then
there exists e ∈Eλwith kekEλ
>ρsuch that Φ(e)<0for λ>0.
Proof. Since f2>0 on Ω, we can choose a nonnegative function φ∈Eλsuch that
ZRNf2(x)φp(x)dx>0. (3.3)
From (3.3), the condition (F1)and Fatou’s lemma, we get
lim
t→∞
Φ(tφ)
tp=lim
t→∞1
2tp−2kφk2
Eλ+1
2tp−2ZRN|∇γφ|2dx−ZRN
F(x,tφ)
(tφ)pφpdx
=−ZRN
F(x,tφ)
(tφ)pφpdx
≤ − 1
pZ
RN
f2(x)φp(x)dx<0.
Let t→+∞we have Φ(tφ)→ −∞. The proof of Lemma 3.2 is therefore complete.
Lemma 3.3. Assume that the assumptions of Theorem 1.2 hold. Then there exists a constant Λ1>0
such that Φsatisfies the (C)c-condition in Eλfor all c ∈R,λ≥Λ1.
Proof. Let {un}be a sequence in Eλsuch that
Φ(un)→cand 1+kunkEλ
Φ0(un)
E∗
λ→0.
We first show that {un}is bounded in Eλ. Indeed, for nlarge enough, by the condition (F2),
we have
c+1≥Φ(un)−1
µΦ0(un)(un)
=µ−2
2µkunk2
Eλ+µ−2
2µZRN|∇γun|2dx+ZRN1
µf(x,un)un−F(x,un)dx
≥µ−2
2µkunk2
Eλ−
kf3kL2
2−`(RN)Θ`
2
µkunk`
Eλ.
Since 1 <`<2, hence {un}is bounded in Eλfor every λ>Λ.

Existence of solutions for fourth-order degenerate elliptic equation 9
Because of the above result, without loss of generality, we can suppose that
un*u0in Eλ,
un→u0strongly in Lp
loc(RN), for 2 ≤p<2γ
∗,
un→u0a.e. in RN,
and Φ0(u0) = 0. Now we prove that un→u0strongly in Eλ. Let vn=un−u0. Then vn*0 in
Eλhence {vn}is bounded in Eλ. By the condition (B2), we get
ZRNv2
ndx=Z{b≥C0}v2
ndx+Z{b<C0}v2
ndx
≤1
λC0ZRNλb(x)v2
ndx+Z{b<C0}v2
ndx
≤1
λC0
kvnk2
Eλ+o(1). (3.4)
Using (3.4), together with Hölder’s inequality and Lemma 2.2, for any λ>Λ, we obtain
ZRN|u|pdx≤ZRN|u|2dx2γ
∗−p
2γ
∗−2ZRN|u|2γ
∗dxp−2
2γ
∗−2
≤1
λC0
kvnk2
Eλ2γ
∗−p
2γ
∗−2
C2γ
∗
3ZRN|∆γv(n)|2γ
∗dx2γ
∗
2

p−2
2γ
∗−2
+o(1)
≤C
2γ
∗(p−2)
2γ
∗−2
31
λC02γ
∗−p
2γ
∗−2kvnkp
Eλ+o(1). (3.5)
Set
Πλ=C
2γ
∗(p−2)
2γ
∗−2
31
λC02γ
∗−p
2γ
∗−2.
By the condition (F1)and (3.4) and (3.5), we get
o(1) = Φ0(vn)(vn) = kvnk2
Eλ+ZRN|∇γvn|2dx−ZRNf(x,vn)vndx
≥kvnk2
Eλ−εZRNv2
ndx−CεZRN|vn|pdx
≤kvnk2
Eλ−ε
λC0
kvnk2
Eλ−CεΠλkvnkp
Eλ+o(1). (3.6)
Since Πλ→0 as λ→∞, by (3.6), there exists Λ1≥Λsuch that for λ>Λ1,
vn→0 strongly in Eλ.
This completes the proof.
Proof of Theorem 1.2.Combining Lemmas 3.1–3.3, we deduce that the problem (1.2) has a non-
trivial weak solution.
Acknowledgements
This research is funded by Vietnam National Foundation for Science and Technology Devel-
opment (NAFOSTED) under grant number 101.02–2017.21.

10 D. T. Luyen
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