Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms
Boundary Value Problems
Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms
Ying Chu 0 1
Wenjie Gao 0 1
0 University , Changchun, 130012 , PR China
1 Institute of Mathematics , Jilin
terms; quasilinear elliptic problem; nonlinear singular term; existence

⎩ u = ,
x ∈ ∂ ,
1 Introduction
⎩ u = ,
on ∂ .
⎩ u = ,
on ∂ ,
MSC: 35J62; 35B25; 76D03
The authors of this paper deal with the existence of weak solutions to the
In this paper, we study the existence of solutions for the following quasilinear elliptic
Model (.) may describe many physical phenomena such as chemical heterogeneous
catalysts, nonlinear heat transfers, some biological experiments, etc. [–]. In the case
where g(s) is singular at s = . They obtained similar results as that of []. Moreover,
Boccardo and Orsina in [] discussed how the summability of f and the values of α affected
the existence, regularity and nonexistence of solutions. For more results, the interested
readers may refer to [, ]. When p ∈ (, ∞), p = , Giacomoni, Schindler and Takáč in
[] applied lower and uppersolution method and the mountain pass theorem to prove
that the problem
⎧⎨ – div(∇up–∇u) = λu–δ + uq, in ,
⎩ u = ,
on ∂ ,
where δ ∈ (, ), q ∈ (p – , p* – ), has multiple weak solutions. And then, the authors in
[] not only improved the results in [] but also obtained that the solution was not in
W,p( ) if α > pp–– . However, we need to point out that all the papers mentioned discussed
the existence of solutions by means of upperlower solution techniques. In this paper, we
apply the method of regularization and Schauder’s fixed point theorem as well as a
necessary compactness argument to overcome some difficulties arising from the nonlinearity
of the differential operator, the singularity of nonlinear terms and the summability of the
weighted function f (x) and then prove the existence of positive solutions in W,p( ) for
suitable m and α when f (x) ∈ Lm( ) and α ≥ , which implies that the summability of the
weighted function f (x) determines whether or not problem (.) has a solution in W,p( ).
2 Main results
In this section, we apply the method of regularization and Schauder’s fixed point theorem
to prove the existence of solutions. In order to prove the main results of this section, we
consider the following auxiliary problem:
⎧⎨ – div(∇unp–∇un) = (unfn+(xn))α , x ∈ ,
⎩ un = ,
x ∈ ∂ ,
where fn = min{f (x), n}.
Definition . A function u ∈ W,p( ) is called a solution of problem (.) if the following
identity holds:
∇up–∇u∇ϕ dx =
∀ϕ ∈ C∞( ).
Since the proof of the following lemmas are similar to that in [], we only give a sketch
of the proof.
Proof Let n ∈ N be fixed. For any w ∈ Lp( ), we get that the following problem has a
unique solution v ∈ W,p( ) ∩ L∞( ) by applying the variational method to
⎧⎨ – div(∇vp–∇v) = (wfn+(xn))α , x ∈ ,
⎩ v = ,
x ∈ ∂ .
We may refer to [, ] for the existence and uniqueness of the solution for problem (.).
So, for any w ∈ Lp( ), we may define the mapping : Lp( ) → Lp( ) as (w) = v. In fact,
multiplying the first identity in (.) by v, and integrating over , we have
∇vp dx =
Applying the embedding theorem W ,p( ) → L( ), we obtain
which implies that
Lemma . The sequence {un} is increasing with respect to n. un > in for any
and there exists a positive constant C (independent of n) such that for all n ∈ N *,
un ≥ C
> for every x ∈
≤ fn ≤ fn+,
∇unp–∇un – ∇un+p–∇un+ ∇(un – un+)+ ≥ ,
α
(un – un+)+ ≤ , for every α > ,
≤
∇unp–∇un – ∇un+p–∇un+ ∇(un – un+)+ dx ≤ .
This inequality yields (un – un+)+ = a.e. in , that is, un ≤ un+ for every n ∈ N *. Since
the sequence un is increasing with respect to n, we only need to prove that u satisfies
inequality (.). According to Lemma ., we know that there exists a positive constant C
(only depending on  , N , p) such that u L∞( ) ≤ C f L∞( ) ≤ C, then
– div ∇up–∇u = (u f+ )α ≥ (C +f )α .
Noting that (C +f)α ≥ , (C +f)α ≡ , the strong maximum principle implies that u > in ,
i.e., inequality (.) holds.
Theorem . Suppose that f is a nonnegative function in L( ) and α = , then problem
(.) has a solution in W,p( ).
Proof We consider the existence of solutions in the case when f (x) ∈ L( ). Multiplying
the first identity in problem (.) by un and integrating over , we get
∇unp dx =
ufnnu+nn dx ≤
fn dx ≤
p
Then we know that there exist u ∈ W ,p( ) and V ∈ L p– ( , RN ) such that
⎧⎪⎪⎨ uunn → uu wa.eea.kinly i n, W ,p( ) and strongly in Lp( ),
⎪⎪⎩ ∇unp–∇un V weakly in L pp– ( , RN ).
For every ϕ ∈ C∞( ), we get from inequality (.) that
≤
as un satisfies the following identity:
Combining with (.)(.), we have that
V ∇ϕ dx =
∀ϕ ∈ C∞( ).
∇unp–∇un∇ϕ dx =
∀ϕ ∈ C∞( ).
Next, we shall prove that V = ∇up–∇u a.e. in . It is easy to see that both (.) and (.)
hold for all ϕ ∈ W ,p( ) with compact support. Thus in (.) we choose ϕ = (un – ξ )ζ ,
where ζ ∈ C∞( ), ζ ≥ , and ξ ∈ W ,p( ), to obtain
which implies that
ζ ∇ξ p–∇ξ – V ∇(u – ξ ) dx ≤ .
Let u – ξ = εψ in (.), where ψ is an arbitrary function in W ,p( ) and ε > is a constant,
we get that
ζ ε ∇u – ε∇ψ p–(∇u – ε∇ψ ) – V ∇ψ dx ≤ ,
ζ ∇u – ε∇ψ p–(∇u – ε∇ψ ) – V ∇ψ dx ≤ .
dx =
which yields that V = ∇up–∇u a.e. in . This proves that u is a weak solution of problem
(.) when f (x) ∈ L( ).
The first question is what happens to the solution if the inhomogeneous function f (x)
is not in L( ) but a nonnegative bounded Radon measure μ. Since a nonnegative Radon
measure μ may always be approximated by a sequence fn of L∞( ) functions, we want to
know whether the approximate solutions may converge to a nontrivial function in W,p( )
or whether the approximate solutions converge. The existence of solutions in this case is
still unknown, but we have the following result.
Theorem . Suppose that μ is a nonnegative Radon measure concentrated on a Borel
set E of zero pcapacity, and that gn is a bounded sequence of nonnegative L( ) functions
which converges to μ in the narrow topology of measures. Let un be the solution of problem
(.) with the nonhomogeneous function fn = gn(x). Then
unp dx = .
Proof By the conclusion of Theorem ., we get that the solution un of problem (.) with
fn = gn is bounded in W,p( ). Since the set E has zero pcapacity, by [, Lemma .], for
any real number σ > , there exists a function σ ∈ C∞( ) satisfying
≤
≤
Define T(un) = min{un, }. Choosing T(un)( – σ ) as a test function in (.) with a
nonhomogeneous function gn, we obtain that
dx ≤
Using un W,p( ) ≤ C, we assume that un is any subsequence such that un u in W ,p( )
and un → u in Lp( ). We show that the two limits in the theorem hold for any such
subsequence. This completes the proof. Note that
By (.)(.) and weak lower semicontinuity, we have
≤
∇T(u) p dx ≤ ,
The above theorem shows that problem (.) has a solution in W,p( ) when f ∈ L( )
and α = . But if f is only a Radon measure, the solution may not exist. At least, the solution
can not be approximated by the solution of problem (.). The second question we are
interested in is whether this problem has a solution in W,p( ) when f ∈ Lm( ) (m > )
and α > . We have the following.
Theorem . Let f be a nonnegative function in Lm( ) (f ≡ ) (m > ). If < α < – m ,
then problem (.) has a solution u ∈ W,p( ) satisfying
∇up–∇u∇ϕ dx =
∀ϕ ∈ C∞( ).
In order to prove this theorem, we need the following lemma.
u–r dx < ∞,
∀r < .
ur dx < ∞
if and only if r > –.
min{f (x),} ≤ , and Lemma . in [], we know that there exists < β < such
Proof By (u+)α
that u ∈ C,β ( ) and u C,β ≤ C, which implies that the gradient of u exists everywhere,
then the Hopf lemma in [] shows that ∂u∂ν(x) > , in , where ν is the outward unit normal
vector of ∂ at x. Moreover, following the lines of proof of the lemma in [], we get
Proof of Theorem . Multiplying the first identity in problem (.) by un, integrating over
, and applying Hölder’s inequality and Lemma ., we get
∇unp dx =
≤ f Lm u–α Lm ≤ c f Lm ,
p
From (.), we know that there exist u ∈ W,p( ) and V ∈ L p– ( , RN ) such that
weakly in W,p( ) and strongly in Lp( ),
For every ϕ ∈ C∞( ), from Lemma ., we get that
≤
Then applying Lebesgue’s dominated convergence theorem, we have
since un satisfies the following identity:
∇un
∇un∇ϕ dx =
∀ϕ ∈ C∞( ).
V ∇ϕ dx =
∀ϕ ∈ C∞( ).
Following the lines of proof of Theorem ., we get that problem (.) has a solution in
,p
W ( ).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors collaborated in all the steps concerning the research and achievements presented in the final manuscript.
Acknowledgements
The work was supported by the Natural Science Foundation of China (11271154) and by the 985 program of Jilin
University. We are very grateful to the anonymous referees for their valuable suggestions that improved the article.
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Cite this article as: Chu and Gao: Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms . Boundary Value Problems 2013 , 2013 : 229