#### Solutions for the quasi-linear elliptic problems involving the critical Sobolev exponent

Sang and Guo Journal of Inequalities and Applications
Solutions for the quasi-linear elliptic problems involving the critical Sobolev
Yanbin Sang 0 1 2 3
Siman Guo 0 1 2 3
0 of China , Taiyuan, Shanxi 030051 , China
1 School of Science, North University
2 Department of Mathematics
3 ,p In this paper, we use the following norm of W
and Nehari manifold. In this article, we study the existence and multiplicity of positive solutions for the quasi-linear elliptic problems involving critical Sobolev exponent and a Hardy term. The main tools adopted in our proofs are the concentration compactness principle ,p Definition . The function u ∈ W ( ) is called a weak solution of (.) if u satisfies
quasi-linear elliptic problems; Nehari manifold; positive solution; best
1 Introduction
In this article, we consider the following quasi-linear elliptic problem:
⎧
⎨ – pu – μ |u||xp|–pu = |u|p∗–u + β|x|α–p|u|p–u + λ|u|q–u in ,
⎩ u =
on ∂ ,
– pu – μ |u||xp|–pu = |x|α–p|u|p–u under Dirichlet boundary condition).
where
⊂ RN (N ≥ ) is a bounded domain with the smooth boundary ∂
such that
∈ . pu = div(|∇u|p–∇u) is the p-Laplacian operator of u, < p < N , λ > is a positive
real number. ≤ μ < μ (μ = (N–pp)p is the best Hardy constant). < q < p and p∗ = NN–pp
is the critical Sobolev exponent. < α < p – , < β < β (β is the first eigenvalue that
|∇u|p–∇u · ∇v – μ
|u|p–uv
dx
=
|u|p∗–uv + β|x|α–p|u|p–uv + λ|u|q–uv dx
,p
for all v ∈ W ( ).
|u|p
|x|p
dx
p
(.)
(.)
By the Hardy inequality (see [, ])
so this norm is equivalent to ( |∇u|p dx) p , the usual norm in W,p( ).
The norm in Lp( ) is represented by u p = ( |u|p dx) p . According to Hardy
inequality, the following best Sobolev constant is well defined for < p < N , and ≤ μ < μ:
(.)
(.)
(.)
Sμ, =
inf
u∈W,p( )\{}
(|∇u|p – μ ||ux||pp ) dx
p
( |u|p∗ dx) p∗
.
The quasi-linear problems on Hardy inequality have been studied extensively, either in
the smooth bounded domain or in the whole space RN . More and more excellent results
have been obtained, which provide us opportunities to understand the singular problems.
However, compared with the semilinear case, the quasi-linear problems related to Hardy
inequality are more complicated [–]. Abdellaoui, Felli and Peral [] considered the
extremal function which achieves the best constant Sμ,, and gave the properties of the
extremal functions. The conclusions obtained in [] can be applied in the problems with
critical Sobolev exponent and Hardy term.
Wang, Wei and Kang [] investigated the following problem:
⎧⎨ – pu – λ |u|p–
|x|p u = μf (x)|u|q–u + g(x)|u|p∗–u, x ∈ ,
⎩ u(x) = , x ∈ ∂ ,
where < q < p, μ > , f and g are non-negative functions and p∗ = NN–pp is the critical
Sobolev exponent. The property of the Nehari manifold was used to prove the existence
of multiple positive solutions for (.). Furthermore, Hsu [, ] improved and
complemented the main results obtained in []. Recently, Goyal and Sreenadh [] investigated
a class of singular N -Laplacian problems with exponential nonlinearities in RN . Very
recently, Xiang [] established the asymptotic estimates of weak solutions for p-Laplacian
equation with Hardy term and critical Sobolev exponent.
We should mention that Liu, Guo and Lei [] studied the existence and multiplicity
of positive solutions of Kirchhoff equation with critical exponential nonlinearity. Inspired
by [, ], we study the problem (.) on critical Sobolev exponent. Comparing with the
main results obtained in [, , –], in this paper, on the one hand, we will analysis the
effect of β|x|α–p|u|p–u, and the more careful estimates are needed. On the other hand, we
establish an lower bound for λ∗ (λ∗ is defined in Theorem .).
Define the energy functional associated to problem (.) as follows:
u p –
p
β
p
|u|p|x|α–p dx –
p∗
λ
q
Theorem . Suppose that < q < p, < α < p – . Then there exists λ∗ > such that
problem (.) admits at least two solutions and one of the solutions is a ground state solution
for all λ ∈ (, λ∗).
(.)
(.)
(.)
(.)
(.)
2 Preliminaries
Firstly, we introduce the Nehari manifold
,p
Nλ = u ∈ W ( )\{} : Iλ(u), u = .
Furthermore u ∈ Nλ if and only if
Let
then
u p –
|u|p∗ dx – β
|u|p|x|α–p dx – λ
Nλ can be divided into the following three parts:
Nλ+ = u ∈ Nλ : p u p – pβ
|x|α–p|u|p dx
– p∗
|u|p∗ dx – qλ
|u|q dx > ,
Nλ = u ∈ Nλ : p u p – pβ
|x|α–p|u|p dx
– p∗
|u|p∗ dx – qλ
|u|q dx = ,
Nλ– = u ∈ Nλ : p u p – pβ
|x|α–p|u|p dx
– p∗
|u|p∗ dx – qλ
|u|q dx < .
,p
Applying the Hölder inequality and the Sobolev inequality, for all u ∈ W ( )\{} we have
|u|q dx ≤
|u|
q· pq∗ dx
q
p∗
dx
– pq∗
p∗–q
= | | p∗
|u|p∗ dx
q
p∗
.
Lemma . Assume that λ ∈ (, T) with
T =
q–p∗
( (β–β)(p–p∗) ) p–p∗ ( q–p
β(q–p∗) p–p∗
q–p q–p∗
p–p∗
) p–p∗ Sμ,
p∗–q
| | p∗
.
Then (i) Nλ± = ∅, and (ii) Nλ = ∅.
Proof (i) We define a function
∈ C(R+, R) by
|u|p∗ dx) pq∗ ] pq––pp∗ –
p∗–q p–p∗
[λ| | p∗ ] q–p
(
u p
p
|u|p∗ dx) p∗
|u|p∗ dx
where < λ < T. Thus, there exist constants s+ and s– such that
< s+ = s (u) < smax < s = s (u), s u ∈ Nλ+ and s u ∈ Nλ .
+ – – + – –
(ii) We prove that Nλ = ∅ for all λ ∈ (, T). By contradiction, assume that there exists
u = such that u ∈ Nλ . From (.), we have
u p –
p∗
|u| dx – β
|u|p|x|α–p dx – λ
|u|q dx = ,
combining with (.), we obtain
Equations (.) and (.) imply that
p – p∗
u p = p – p∗ β
|u|p|x|α–p dx + p∗ – q λ
|u|q dx.
(p – q) u p – (p – q)β
|u|p|x|α–p dx = p∗ – q
p∗
|u| dx,
that is,
Similarly,
that is,
p∗
|u| dx ≥
p – q
u
It follows from (.) and (.) that
=
≤
=
< ,
Proof For u ∈ Nλ, we can deduce from (.) and (.) that
|u|p∗ dx <
p – q
p∗ – q
–
β
β
p
u .
(.)
u p –
p – q β
u p –
(q – p) qp – qp∗
p∗ – q
u p
Therefore, we have κλ ≤ κλ+ < .
Lemma . For u ∈ Nλ, there exist ε > and a differentiable function f = f (ω) : B(, ε) ⊂
W,p( ) −→ R+ such that
Lemma . tells us that Fs(, ) = . Thus, by the implicit function theorem at the point
(, ), there exist ε > , and a differentiable function
f : B(, ε) ⊂ W,p( ) −→ R+
Proof It follows from Lemma . that Iλ is coercive on Nλ. Using the Ekeland variational
principle [], we can find a minimizing sequence {un} ⊂ Nλ of Iλ satisfying
Without loss of generality, we can assume that un ≥ . By Lemma ., we know that {un} is
bounded in W,p( ). As a consequence, there exist a subsequence (still denoted by {un})
and u∗ in W,p( ) such that
⎧⎪⎪ un u∗ weakly in W,p( ),
⎨ un → u∗ strongly in Lp( ) ( ≤ p < p∗),
⎪⎪⎩ un(x) → u∗(x) a.e. in .
From Lemma ., for s > sufficiently small and φ ∈ W,p( ), and set u = un, ω = sφ ∈
W,p( ), we can find that fn(s) = fn(sφ) such that fn() = and fn(s)(un + sφ) ∈ Nλ. Since
n fn(s) – un + sfn(s) φ
≥ n fn(s)(un + sφ) – un
≥ Iλ(un) – Iλ fn(s)(un + sφ) .
Notice that
Therefore
Iλ(un) – Iλ fn(s)(un + sφ)
= p un p – fnpp(s) un p + fnpp∗∗(s)
+ λq fnq(s) |un + sφ|q dx – λ
q
|un + sφ|p∗ dx –
p∗
|un + sφ|q dx + βp fnp(s)
|un + sφ|p∗ dx
|x|α–p|un + sφ|p dx
β
– p
– p∗
β
+ p
+ p∗
β
+ p
+ μ
+ λ
|x|α–p|un + sφ|p dx + fnpp(s) un p – fnpp(s) un + sφ p + p∗
|un|p∗ dx + λ
q
|un + sφ|q dx – λ
q
|un + sφ|p∗ dx + λq fnq(s) –
|un + sφ|q dx
un p – un + sφ p
|un + sφ|q – |un|q dx
Dividing by s > and taking the limit for s → , combining with (.) and (.), we have
|fn()| un + φ
n
≥ –fn() un p + fn()
|un|p∗ dx + λfn()
|un|p–φ|x|α–p dx
= – Iλ, φ .
Consequently
for every φ ∈ W,p( ). Note that (.) holds equally for –φ, we see that (.) holds.
has radially symmetric ground states
V (x) =
p–N
p Up,μ
x =
p–N
p Up,μ
|x|
∀ > ,
such that
RN
∇V (x) p – μ |V (x)|p
|x|p
dx =
RN
N
V (x) p∗ dx = Sμp,,
where the function Up,μ(x) = Up,μ(|x|) is the unique radial solution of the above limiting
problem with
= –fn() un p –
D = p –p q pp∗∗–qq | | p∗p∗–q Sμ–,pq βN–ββ – pq p–pq
.
Then there exists u ∈ W,p( ) such that un → u in Lp∗ ( ).
Proof Since
Iλ(un) → κλ–
as n → +∞.
Lemma . Let {un} ⊂ Nλ– be a minimizing sequence for Iλ with κλ– <
p
– Dλ p–q , where
By Lemma ., we know that {un} is bounded in W,p( ). In fact, we can deduce from (.)
and (.) that
β
p
|un|p|x|α–p dx –
p∗
= p
– p∗
≥ p – p∗
p∗ – q λ
p – p∗ β
p∗ – q λ| | p∗p∗–q
|un|p∗ dx
pq∗
β
– β
β
– β
p∗ – q λ
⎧⎪⎪ un u weakly in W,p( ),
⎨ un → u strongly in Lp( ) ( ≤ p < p∗).
⎪⎪⎩ un(x) → u(x) a.e. in .
(.)
In term of the concentration compactness principle, going if necessary to a subsequence,
there exist an at most countable set J , a set of points {xj}j∈J ⊂ \ {}, and real numbers
μj, νj, χ such that
|∇un|p
dμ ≥ |∇u|p +
μjδxj + μδ,
|un|p∗
|un|p
|x|p
j∈J
dν = |u|p∗ +
νjδxj + νδ,
j∈J
dχ = ||ux||pp + χδ,
where δxj is the Dirac mass at xj.
Let be sufficient small satisfying ∈/ B(xj, ) and B(xj, ) ∩ B(xi, ) = ∅ for i = j, i, j =
, , . . . , k. Let ψ ,j(x) be a smooth cut-off function centered at xj such that ≤ ψ ,j(x) ≤ ,
ψ ,j(x) = for x ∈ B(xj, ), ψ ,j(x) = for x ∈
\B(xj, ) and |∇ψ ,j(x)| ≤
. Note that
Iλ(un), unψ ,j(x)
Furthermore, we have
lim
n→∞
lim
n→∞
lim lim
→ n→∞
lim lim
→ n→∞
By (.), we deduce that
|un|qψ ,j dx ≤
and
|un|p|x|α–pψ ,j(x) dx ≤
p∗–q
p∗
B(xj, )
B(xj, )
p∗(α–p)
|x| p∗–p dx
p∗–p
p∗
p∗(α–p)
|x – xj| p∗–p dx
p∗–p
p∗
p∗–p
p∗
B(xj, )
B(xj, )
q
– p
≤ Sμ, un
q
≤
=
q
– p
≤ Sμ,
N
p∗–q
p∗
≤
p∗
|un|q q dx
B(xj, )
r
N–
dr
q
p∗
dx
p∗–q
p∗
B(xj, )
dx
p∗–q
p∗
un
q
q N(p∗–q)
– p p∗
Sμ,
un
q
B(xj, )
B(xj, )
p∗
p∗
|un|p p dx
|un|p p dx
p
p∗
p
p∗
≤ Sμ–, un
= Sμ–, un
p
p
p
Since {un} is bounded in W,p( ), and un
u weakly in Lp∗ ( ), we conclude that
lim lim
→ n→∞
lim lim
→ n→∞
and
By (.), we have
|un|qψ ,j(x) dx =
|un|p|x|α–pψ ,j(x) dx = .
= lim lim Iλ(un), unψ ,j(x) ≥ μj – νj.
→ n→∞
p
N
Since S,νjp∗ ≤ μj, we have μj = νj = or μj ≥ (S,) p .
On the other hand, let > be sufficiently small satisfying xj ∈/ B(, ), ∀j ∈ J . Let ψ ,(x)
a smooth cut-off function centered at the origin such that ≤ ψ ,(x) ≤ , ψ ,(x) = for
|x| ≤ , ψ ,(x) = for |x| ≥
and |∇ψ ,(x)| ≤ . Hence, we have
Therefore
|∇un|pψ ,(x) dx =
ψ ,(x) dμ ≥
|∇u|pψ ,(x) dx + μ,
|un|p∗ ψ ,(x) dx =
|un|p ψ ,(x) dx =
|x|p
ψ ,(x) dν =
ψ ,(x) dχ =
|u|p∗ ψ ,(x) dx + ν,
|u|p ψ ,(x) dx + χ,
|x|p
un|∇un|p–∇un · ∇ψ ,(x) dx = ,
|un|qψ ,(x) dx =
|un|p|x|α–pψ ,(x) dx = .
= lim lim Iλ(un), unψ ,(x) ≥ μ – μχ – ν.
→ n→∞
p
= lim un p +
n→∞ N
p∗ – q λ| | p∗p∗–q Sμ–,pq un q
≥ N
u p +
μj + μ – μχ +
j∈J
p∗ – q λ| | p∗p∗–q Sμ–,pq u q
≥ N SμNp, + N u p +
= N SμNp, + N u p – pp∗∗–qq λ| | p∗p∗–q Sμ–,pq u q
p∗ – q λ| | p∗p∗–q Sμ–,pq u q
≥ N SμNp, – Dλ p–q ,
p
where D is defined in (.). Hence, we conclude that
is a contradiction. It follows that νj = for j ∈ {} ∪ J , which means that
|u|p∗ dx as n → ∞. The proof is completed.
p
– Dλ p–q ≤ κλ– <
p
– Dλ p–q , which
|un|p∗ dx →
f (ξ ) = (p – )ξ p – (N – p)ξ p– + μ, ξ ≥ , ≤ μ < μ,
where b(μ) is the zero of the function
satisfying < Np–p < b(μ) < Np––p .
Lemma . There exists λ > such that
sup Iλ(sv ) <
s≥
p
– Dλ p–q , for λ ∈ (, λ),
where
and D are defined in Lemma ..
Proof For two positive constants s and s (independent of , λ), we show that there
exists s > with < s ≤ s ≤ s < ∞ such that sups≥ Iλ(sv ) = Iλ(s v ). In fact, since
lims→+∞ Iλ(sv ) = –∞, we can deduce that
sp– v p – βsp–
|v |p|x|α–p dx – sp∗–
|v |p∗ dx – λsq–
|v |q dx =
(.)
(.)
(.)
(.)
and
That is,
(p – )sp– v p – (p – )βsp–
|v |p|x|α–p dx
– p∗ – sp∗–
|v |p∗ dx – (q – )λsq–
|v |q dx < .
(.)
Equations (.) and (.) imply that
(p – )sp– v p – (p – )βsp–
|v |p|x|α–p dx – p∗ – sp∗–
|u |p∗ dx
< (q – )sp– v p – (q – )βsp–
|v |p|x|α–p dx – (q – )sp∗–
|v |p∗ dx.
(p – q)sp– v p – (p – q)βsp–
|v |p|x|α–p dx < p∗ – q sp∗–
|v |p∗ dx.
(.)
Hence, we can obtain from (.) that s is bounded below. Moreover, it is clear to see from
(.) that s is bounded above for all > small enough. Therefore, our claim holds.
Set
h(s v ) = sp
p
v p – sp∗
p∗
|v |p∗ dx.
In the following, we prove that
h(s v ) ≤
+ O
p(b(μ)– Np +) .
Let
h(s) = sp
p
v p – sp∗
p∗
|v |p∗ dx.
Direct computations give us that lims→∞ h(s) = –∞ and h() = . Thus sups≥ h(s) is
obtained at some S > , and
S =
v p
|v |p∗ dx
p∗–p
.
Since h (s)|S = , that is,
Sp– v p – Sp∗–
|v |p∗ dx = .
It is easy to check that h(s) is increasing in [, S ), according to (.) and (.), we have
h(s v ) ≤ h(S )
Therefore, by (.), we have Iλ(s v ) = h(s v ) –
p(b(μ)– Np +)
.
βsp
p
(i) b(μ) < q < p. Choose = λ (p–q)(b(μ)– Np +) , for λ < λ := ( C+D ) N–qb(μ) , we have
N
C
C
p(b(μ)– Np +)
p
– λC N+q(– Np ) = Cλ p–q – λCλ (p–q)(b(μ)– Np +)
|v |q dx.
(p–q)(b(μ)– Np +)
N+q(– Np )
N+q(– Np )
N–qb(μ)
p
= Cλ p–q – Cλ (p–q)(b(μ)– Np +)
+
p
= λ p–q C – Cλ (p–q)(b(μ)– Np +)
< –Dλ p–q .
p
N+q(– Np )
N+q(– Np )
Consequently, for λ < λ := min{λ, λ, λ}, we deduce that
Iλ(s v ) <
≥ nl→im∞ p – p∗
≥
p∗
p∗ – p p∗–p
pp∗ Sμ,
= SμNp,,
N
un p
= nl→im∞ p un p – βp
|un|p|x|α–p –
p∗
which contradicts with κλ <
p
– Dλ p–q (from Lemma .).
(.)
(.)
κλ < Iλ s+uλ < Iλ s–uλ = Iλ(uλ) = κλ,
Iλ(uλ) = κλ+ = κλ < .
Secondly, we prove that uλ ∈ Nλ+. Suppose that this is not true, i.e., uλ ∈ Nλ–. From
Lemma ., we can find positive numbers s+ and s– with s+ < smax < s– = such that
s+uλ ∈ Nλ+, s–uλ ∈ Nλ– and
which is a contradiction. Hence uλ ∈ Nλ+. Furthermore, combining with Lemma ., we
can obtain
Therefore, we see that uλ is a non-negative ground state solution of problem (.).
–
In the following, we prove that problem (.) has a second solution vλ with vλ ∈ Nλ .
Since Iλ is coercive on Nλ–, according to the Ekeland variational principle and Lemma .,
there exists a minimizing sequence {vn} ⊂ Nλ– of Iλ such that
(i) Iλ(vn) < κλ– + n ;
(ii) Iλ(u) ≥ Iλ(vn) – n u – vn for all u ∈ Nλ .
–
Note that {vn} is bounded in W,p( ), there exist a subsequence (still denoted by {vn})
and vλ ∈ W,p( ) such that
⎧⎪ vn vλ weakly in W,p( ),
⎨⎪ vn → vλ strongly in Ls( ) ( ≤ s < p∗),
⎪⎪⎩ vn(x) → vλ(x) a.e. in ,
as n → ∞.
Similar to the above discussion, we can deduce that vn → vλ in W,p( ) and vλ is a
nonnegative solution of (.). Thirdly, we show that vλ = in . According to vn ∈ Nλ–, we
obtain
(p – q) vn p = p∗ – q
|vn|p∗ dx + (p – q)β
|vn|p|x|α–p dx
p∗ β
< p∗ – q Sμ–,p vn p∗ + (p – q) β
vn p,
hence
vn >
(p – q)( – ββ )Sμpp∗, p∗–p
p∗ – q
,
–
∀vn ∈ Nλ ,
together with vn → vλ in W,p( ) means that vλ ≡ .
Lastly, we show that vλ ∈ Nλ–. We only need to prove that Nλ– is closed. In fact, for
{vn} ⊂ Nλ–, it follows from Lemmas . and . that
lim
n→∞
|vn|p∗ dx =
|vλ|p∗ dx.
(.)
(.)
In addition Thus
(p – q) vn p – p∗ – q
|vn|p∗ dx – (p – q)β
|vn|p|x|α–p dx < .
(p – q) vλ p – p∗ – q
|vλ|p∗ dx – (p – q)β
|vλ|p|x|α–p dx ≤ ,
which means that vλ ∈ Nλ ∪ Nλ–. Combining with Lemma . and vλ ≡ , we see that Nλ–
+
is closed. Note that Nλ ∩ Nλ– = ∅, we know that uλ and vλ are different.
4 Conclusions
In this paper, we study the existence and multiplicity of positive solutions for the
quasilinear elliptic problem which consists of critical Sobolev exponent and a Hardy term.
The main conclusions of this work:
() Adding a linear perturbation in the nonlinear term of elliptic equation.
() The main challenge of this study is the lack of compactness of the embedding
W,p → Lp∗ . We overcome it by the concentration compactness principle.
() We apply the Ekeland variational principle to obtain a minimizing sequence with
good properties.
5 Discussion
In the future, a natural question is whether the multiplicity of positive solutions for (.)
can be established with negative exponent uγ ( < γ < ).
Acknowledgements
This project is supported by the Natural Science Foundation of Shanxi Province (2016011003), Science Foundation of
North University of China (110246), NSFC (11401583) and the Fundamental Research Funds for the Central Universities
(16CX02051A).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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