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EXISTENCE OF RADIAL POSITIVE SOLUTIONS FOR A QUSILINEAR NON-POSITONE PROBLEM IN A BALL†
EXISTENCE OF RADIAL POSITIVE SOLUTIONS FOR A QUSILINEAR NON-POSITONE PROBLEM IN A BALL†
Journal of Applied Mathematics & Informatics. 2015. Sep, 33(5_6): 749-757
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : February 10, 2015
  • Accepted : April 06, 2015
  • Published : September 30, 2015
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WEIHUI WANG
ZUODONG YANG

Abstract
In this paper, we prove existence of radial positive solutions for the following boundary value problem where λ > 0, Ω denotes a ball in R N ; f has more than one zero and f (0) < 0(the nonpositone case). AMS Mathematics Subject Classification : 65H05, 65F10.
Keywords
1. Introduction
Let us consider the the existence of radial positive solutions of the problem
PPT Slide
Lager Image
where λ > 0, Ω denotes a ball in R N ; f (0) < 0, f has more than one zero and is not strictly increasing entirely on [0,∞). △ pu = div(|∇ u | p −2 u ) (1 < p N ) is the p -Laplacian operator of u .
The problem (1.1) arises in the theory of quasiregular and quasiconformal mappings or in the study of non-Newtonian fluids. In the latter case, the quantity p is a characteristic of the medium.Media with p > 2 are called dilatant fluids and these with p < 2 are called pseudoplastics(see [18 , 19] ). If p = 2, they are Newtonian fluid. When p ≠ 2, the problem becomes more complicated certain nice properties inherent to the case p = 2 seem to be lost or at least difficult to verify. The main differences between p = 2 and p ≠ 2 can be founded in [9 , 11] .
In recent years, the asymptotic behavior, existence and uniqueness of the positive solutions for the quasilinear eigenvalue problems:
PPT Slide
Lager Image
where λ > 0; p > 1;Ω ∈ R N , N ≥ 2 have been considered by a number of authors, see [5 - 15 , 20 - 24 , 26 - 28] and the references therein. In [11] , Guo and Webb proved existence and uniqueness results of (1.2) for λ large when f ≥ 0, ( f ( x )= x p −1 )′ < 0 for x > 0 and f satisfies some p-sublinearity conditions at 0 and ∞, generalizing a result in [11] where Ω is a ball. When p = 2, uniqueness results for semilinear equations were obtained in [29 , 30] where the assumption ( f ( x )= x )′ < 0 is required only for large x . Similar results for systems were discussed in [31] . Related results for the superlinear case when f ≥ 0 can be found in [26 , 32] . When p = 2, f (0) < 0, f ( s ) has only one zero and Ω being a unit ball or an annulus in R N the related results have been obtained by Castro and Shivaji [2] , Arcoya and Zertiti [1] . The case when f (0) < 0 and p = 2 was treated in [33] , in which uniqueness of positive solution to single equation of (1.1) for λ large was established for sublinear f . See also [34] where this result was extended to the case when Ω is any bounded domain with convex outer boundary.
In this paper, we study this problem for p ≠ 2, f (0) < 0 and Ω being a unit ball in R N . It extends and complements previous results in the literature [1] .
The paper is organized as follows. In section 2, we recall some facts that will be needed in the paper and give the main results. In section 3, we give the proofs of the main results in this paper.
2. Main results
We consider radial solution of (1.1), then, the existence of radial positive solutions of (1.1) is equivalent to the existence of positive solutions of the problem
PPT Slide
Lager Image
where Ω is the unit ball of R N and λ > 0. Here f : [0,+∞) → R satisfies the following assumptions:
(H1) f C 1 ([0,+∞), R ) such that f ≥ 0 on [ β ,+∞), where β is the greatest zero of f ;
(H2) f (0) < 0;
(H3)
PPT Slide
Lager Image
where p −1 < q < p −1,
PPT Slide
Lager Image
p = ∞ if p N ;
(H4) For some k ∈ (0, 1),
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Remark 2.1. We note that in hypothesis (H1), there is no restriction on the function f ( u ) for 0 < u < β .
Remark 2.2. If f satisfies (H1), any nonnegative solution u of (2.1) is positive in Ω, radial symmetric and radially decreasing, that is
PPT Slide
Lager Image
By a modification of the method given in [1] , we obtain the following results.
Theorem 2.1. Let assumptions (H1)-(H4) be satisfied. Then there exists a positive real number λ 0 such that if λ ∈ [0, λ 0 ], problem (1.1) has at least one radial positive solution which is decreasing on [0,1].
The proof of the theorem is based on the following preliminaries and four Lemmas.
Lemma 2.2. Let u ( r ) be a solution of (2.1) in ( r 1 , r 2 ) ⊂ (0,∞) and let a be an arbitrary constant, then for each r ∈ ( r 1 , r 2 ) we have
PPT Slide
Lager Image
Remark 2.3. The identity of Pohozaev type was obtained by Ni and Serrin [6] .
By a modification of the method given in [1] , we first introduce the notations and the following preliminaries. Let F be defined as
PPT Slide
Lager Image
and θ denotes the greatest zero of F .
From (H4), we have
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
Given d R, λ R , we define
PPT Slide
Lager Image
By Lemma 2.2, we show the following Pohozaev identity on ( r 0 , r 1 )
PPT Slide
Lager Image
Moreover, for d γ , there exists t 0 such that
PPT Slide
Lager Image
Next from (H1), we obtain that f is nondecreasing on [ kd, d ] ⊂ ( β ,+∞), and from (2.1) we have
PPT Slide
Lager Image
PPT Slide
Lager Image
Integrating on [0, t 0 ], which implies
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Hence, taking r 0 = 0, r 1 = t 0 in (2.4), and using (2.5)-(2.6), we find
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Lemma 2.3. There exists λ 1 > 0 such that if λ ∈ (0, λ 1 ), then u ( r, γ, λ ) ≥ β, for r ∈ [0, 1].
Proof . Let r = sup{0 ≤ r ≤ 1 : u ( r, γ, λ ) ≥ β }. For u is decreasing on [0, r ], then β u ( r, γ, λ ). ≤ u (0, r, λ ) = γ , ∀ r ∈ [0, r ]
Moreover, since f ≥ 0 on [ β ,+∞) and
PPT Slide
Lager Image
we obtain
PPT Slide
Lager Image
Then for
PPT Slide
Lager Image
we have
PPT Slide
Lager Image
Next, by using the mean value theorem and (2.8), there exists
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
Assume that r < 1, we have
PPT Slide
Lager Image
which contradicts the definition of r . Then, the lemma is proved for r = 1. □
Lemma 2.4. There exists λ 2 > 0 such that for λ ∈ (0, λ 2 )
PPT Slide
Lager Image
Proof . From Lemma 2.2, we have the following Pohozaev identity on ( r , t 0 )
PPT Slide
Lager Image
Extending f by f ( x ) = f (0) < 0, for x ∈ (−∞, 0], then there exists B < 0 such that
PPT Slide
Lager Image
For sufficiently large γ , from (H4), we deduce
PPT Slide
Lager Image
By (2.7) and (2.9), we get
PPT Slide
Lager Image
Then, there exists λ2 such that
PPT Slide
Lager Image
Hence, for all λ ∈ (0, λ 2 ) and r ∈ [0, 1], H ( t ) > 0, ∀ d γ . This also implies that u ( r, d, λ ) 2 + u′ ( r, d, λ ) 2 > 0, for all t ∈ [0, 1] and all d γ . □.
Lemma 2.5. For r ∈ [0, 1], there exists d γ such that u ( r, d, λ ) < 0.
Proof . By contradiction, let d λ , we assume that u ( r, d, λ ) ≥ 0 for ∀ r ∈ [0, 1].
Let
PPT Slide
Lager Image
is decreasing on (0, r )}. Define ω be the solution of the following equation:
PPT Slide
Lager Image
where δ is chosen such that the first zero of ω is
PPT Slide
Lager Image
and ω satisfies
PPT Slide
Lager Image
r ∈ (0; 1).
From (H3), there exists d 0 γ such that
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
Let v = ,
PPT Slide
Lager Image
Then, we obtain
PPT Slide
Lager Image
Suppose u ( r, d, λ ) ≥ d 0 for all
PPT Slide
Lager Image
from (2.12),
PPT Slide
Lager Image
On the other hand, from the quality of
PPT Slide
Lager Image
we know that
PPT Slide
Lager Image
then
PPT Slide
Lager Image
From (2.13)-(2.15), we obtain
PPT Slide
Lager Image
On the other hand, since
PPT Slide
Lager Image
PPT Slide
Lager Image
which is contradiction with (2.16).
Hence, there exists
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
And since d 0 γ > β , there exists
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
Now, we consider t 0 defined in (2.5), also
PPT Slide
Lager Image
On [0, t 0 ], from (H1), F is nondecreasing on [ β ,+∞) and u ( r, d, λ ) ≥ kd β r ∈ (0, t 0 ]. We have
PPT Slide
Lager Image
On the other hand, since
PPT Slide
Lager Image
then
PPT Slide
Lager Image
Hence, by (2.10) we get
PPT Slide
Lager Image
From (H4), (2.18), (2.19)
PPT Slide
Lager Image
Therefore, there exists d 1 d 0 such that for d d 1 , we get
PPT Slide
Lager Image
By (2.17), (2.18)
PPT Slide
Lager Image
Which implies
PPT Slide
Lager Image
The mean value theorem gives us
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
hence
PPT Slide
Lager Image
and since
PPT Slide
Lager Image
there exists T ∈ (0, 1) such that u ( T, d, λ ) < 0, which contradicts with the assuming, the lemma is proved. □
3. Proof of the Main Results
The proof of Theorem 2.1. Let λ 0 = min{ λ 1 , λ 2 }, for ∀ λ ∈ (0, λ 0 ). Define
PPT Slide
Lager Image
From Lemma 2.3, we obtain that the set { d γ : u ( r, d, λ ) ≥ 0, ∀ r ∈ (0, 1]} is nonempty. From Lemma 2.5 implies that
PPT Slide
Lager Image
Then we claim that
PPT Slide
Lager Image
is the solution of problem (1.1). Moreover, the solution
PPT Slide
Lager Image
satisfies the following properties:
  • (i)for allr∈ [0, 1);
  • (ii)
  • (iii)
  • (iv) u is decreasing in [0,1].
For (i). By contradiction, if there exists 0 ≤ R 1 < 1 such that
PPT Slide
Lager Image
from Lemma 2.4,
PPT Slide
Lager Image
then we can suppose
PPT Slide
Lager Image
Hence from
PPT Slide
Lager Image
and
PPT Slide
Lager Image
we find there exists R 2 ∈ ( R 1 , 1) such that
PPT Slide
Lager Image
which contradicts with the definition of
PPT Slide
Lager Image
So
PPT Slide
Lager Image
for all r ∈ [0, 1).
For (ii). By contradiction,we assume
PPT Slide
Lager Image
then from (i) there exists η such that
PPT Slide
Lager Image
for ∀ r ∈ (0, 1), moreover, there exists δ > 0 such that
PPT Slide
Lager Image
for ∀ t ∈ (0, 1], which is a contradiction with the definition of
PPT Slide
Lager Image
(ii) is proved.
For (iii). From (2.1),
PPT Slide
Lager Image
Taking into account Lemma 2.3 and (H1), we have for ∀ λ ∈ (0, λ 0 ), u ( r, γ, λ ) ≥ β and f ( s ) > 0, for ∀ s ∈ ( β ,+∞), which implies for ∀ λ ∈ (0, λ 1 ),
PPT Slide
Lager Image
PPT Slide
Lager Image
So
PPT Slide
Lager Image
(iv) is also proved.
BIO
Weihui Wang received master degree from Nanjing Normal University. Her research interests include elliptic equations.
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210023, China.
Zuodong Yang received Ph.D. from Beijing University of Areonautics and Astronautics. His research interests are elliptic and parabolic equations.
School of Teacher Education, Nanjing Normal University, Jiangsu Nanjing 210097, China; Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210023, China.
e-mail: zdyang-jin@263.net
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