A fibering map approach to a potential operator
equation and its applications
∗
Kenneth J. Brown
School of Mathematical and Computer Sciences and the Maxwell Institute,
Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK
Tsung-fang Wu
Department of Applied Mathematics, National University of Kaohsiung,
Kaohsiung 811, Taiwan
Abstract
In this paper we study the existence of multiple solutions for operator equations involving homogeneous potential operators. With the help of the Nehari manifold and the fibering method, we prove that some such equations have at least two nonzero solutions. Furthermore, we apply this result to prove the existence of two positive solutions for some quasilinear elliptic problems involving sign-changing weight functions.
1
Introduction
In this paper, we consider multiplicity results for nonzero solutions of the operator
equation: ½
A (u) − B (u) − C (u) = 0,
u ∈ X, (E)
where X is a reflexive Banach space equipped with norm k·k and duality pair
h·, ·i between X and X∗ and where A, B, C : X → X∗ are homogeneous potential
operators. We shall assume throughout that A, B and C are homogeneous of degrees p − 1, q − 1 and r − 1 where 1 < q < p < r and so it is well-known that the corresponding functionals are given by
a(u) = 1
phA(u), ui; b(u) =
1
qhB(u), ui; c(u) =
1
rhC(u), ui.
For the definitions and properties of potential operators we refer the reader to Chabrowski [9, Chapter 1]. Moreover, we shall assume
(H1) u → hA(u), ui is weakly lower semicontinuous on X and there exists a con-tinuous function κ : [0, ∞) → [0, ∞) with κ(s) > 0 on (0, ∞) and lims→∞κ(s) =
∞ such that hA (u) , ui ≥ κ (kuk) kuk for all u ∈ X;
(H2) there exist u, v ∈ X such that hB (u) , ui > 0 and hC (v) , vi > 0; (H3) B : X → X∗ and C : X → X∗ are strongly continuous;
(H4) There exist positive numbers d1 and d2 with
dr−p1 dp−q2 < (p − q)p−q(r − p)r−p(r − q)q−r such that
hB (u) , ui ≤ d1[hA (u) , ui]q/p and hC (u) , ui ≤ d2[hA (u) , ui]r/p.
The study of (E) is motivated by a number of problems involving nonlinearities which are a combination of concave and convex terms e.g., quasilinear boundary value problems (see [1, 2, 4, 7, 8, 10, 11, 13, 17]), problems with nonlinear bound-ary conditions (see [6, 12, 16]) and quasilinear elliptic systems (see [3, 5, 18]). In this paper, we will use variational methods involving the Nehari manifold and fibering maps to prove the existence of at least two nonzero solutions of equation (E) and show how this provides corresponding existence results in a wide range of applications including these listed above.
This paper is organized as follows. In section 2, we discuss the Nehari manifold and fibering maps for equation (E). In section 3, we prove that equation (E) has at least two nontrivial solutions. In section 4, we apply this result to a wide range of applications with nonlinearities involving both concave and convex terms.
2
The Nehari manifold and fibering maps
A function u ∈ X is a solution of equation (E) if and only if
hA (u) − B (u) − C (u) , ϕi = 0 for all ϕ ∈ X.
Thus, the energy functional corresponding to equation (E) is defined by
J (u) = a (u) − b (u) − c (u) for u ∈ X,
i.e., u is a solution of (E) if and only if J0(u) = 0, i.e., u is a critical point of J.
As the energy functional J is not bounded below on X, it is useful to consider the functional on the Nehari manifold (see [14])
Thus, u ∈ N if and only if
hA (u) , ui − hB (u) , ui − hC (u) , ui = 0.
Moreover, we have the following result.
Lemma 2.1 The energy functional J is coercive and bounded below on N. Proof. If u ∈ N, then J(u) = µ 1 p− 1 r ¶ hA (u) , ui − µ 1 q − 1 r ¶ hB (u) , ui (1) and so by (H4) J(u) ≥ µ 1 p − 1 r ¶ hA (u) , ui − d1 µ 1 q − 1 r ¶
[hA (u) , ui]q/p.
Thus, by condition (H1), we have J is coercive and bounded below on N. ¤ The Nehari manifold N is closely linked to the behavior of functions of the form hu : t → J (tu) for t > 0. Such maps are known as fibering maps and were
introduced by Drabek-Pohozaev in [10] and are also discussed in Brown-Zhang [8]. If u ∈ X, we have hu(t) = tp p hA (u) , ui − tq q hB (u) , ui − tr r hC (u) , ui ; h0
u(t) = tp−1hA (u) , ui − tq−1hB (u) , ui − tr−1hC (u) , ui ;
h00
u(t) = (p − 1) tp−2hA (u) , ui − (q − 1) tq−2hB (u) , ui − (r − 1) tr−2hC (u) , ui .
It is easy to see that
h0 u(t) =
1
t (hA (tu) , tui − hB (tu) , tui − hC (tu) , tui)
and so, for u ∈ X\ {0} and t > 0, h0
u(t) = 0 if and only if tu ∈ N, i.e., positive
critical points of hu correspond to points on the Nehari manifold. In particular,
h0
u(1) = 0 if and only if u ∈ N. Thus it is natural to split N into three parts
corresponding to local minima, local maxima and points of inflection and so we define
N+ = {u ∈ N | h00u(1) > 0} ;
N0 = {u ∈ N | h00u(1) = 0} ;
N− = {u ∈ N | h00u(1) < 0} .
Lemma 2.2 Suppose that u0 is a local minimizer for J on N and that u0 ∈ N/ 0.
Then J0(u
0) = 0 in X∗.
Proof. The proof is essentially the same as that in Brown-Zhang [8, Theorem 2.3] (or see Binding-Drabek-Huang [4]). ¤ Lemma 2.3 (i) For any u ∈ N+, we have hB (u) , ui > 0;
(ii) for any u ∈ N0, we have hB (u) , ui > 0 and hC (u) , ui > 0;
(iii) For any u ∈ N−, we have hC (u) , ui > 0.
Proof. If u ∈ N we have
h00
u(1) = (p − 1) hA (u) , ui − (q − 1) hB (u) , ui − (r − 1) hC (u) , ui
= (p − r) hA (u) , ui − (q − r) hB (u) , ui (2) = (p − q) hA (u) , ui − (r − q) hC (u) , ui. (3) The result now follows immediately from (2) and (3) and the fact that hA (u) , ui >
0. ¤
Lemma 2.4 We have N0 = ∅.
Proof. Suppose otherwise, i.e., N0 6= ∅. Then, for u ∈ N0, by (2) and (H4) we
have
hA (u) , ui = r − q
r − phB (u) , ui ≤ d1 r − q
r − p[hA (u) , ui]
q p and so hA (u) , ui1p ≤ · d1 r − q r − p ¸ 1 p−q .
Similarly using (3) and (H4) we have
hA (u) , ui1p ≥ · 1 d2 p − q r − q ¸ 1 r−p .
Hence we must have
dr−p1 dp−q2 ≥ (p − q)p−q (r − p)r−p (r − q)q−r
which contradicts (H4). Hence N0 = ∅. ¤
In order to get a better understanding of the Nehari manifold and fibering maps we consider the function mu : R+→ R defined by
Clearly tu ∈ N if and only if mu(t) = hB (u) , ui. Moreover,
m0
u(t) = (p − q)tp−q−1hA (u) , ui − (r − q)tr−q−1hC (u) , ui (5)
and so it is easy to see that, if tu ∈ N, then tq−1m0
u(t) = h00u(t). Hence
tu ∈ N+ (N−) if and only if m0u(t) > 0 (< 0).
Suppose hC (u) , ui > 0. Then by (5), mu has a unique critical point at
t = tmax where tmax= µ (p − q) hA (u) , ui (r − q) hC (u) , ui ¶ 1 r−p > 0
and clearly muis strictly increasing on (0, tmax) and strictly decreasing on (tmax, ∞)
with limt→∞mu(t) = −∞. Moreover,
mu(tmax) = "µ p − q r − q ¶p−q r−p − µ p − q r − q ¶r−q r−p # hA (u) , uir−qr−p hC (u) , uip−qr−p = hA (u) , uiqp µ r − p r − q ¶ µ p − q r − q ¶p−q r−p Ã
[hA (u) , ui]rp
hC (u) , ui !p−q r−p ≥ 1 d1 hB (u) , ui µ r − p r − q ¶ µ p − q r − q ¶p−q r−p µ 1 d2 ¶p−q r−p = α hB (u) , ui where αr−p = dp−r 1 dq−p2 µ r − p r − q ¶r−pµ p − q r − q ¶p−q = dr−p1 dq−p2 (r − p)r−p(p − q)p−q(r − q)q−r > 1
by (H4). Hence mu(tmax) > hB (u) , ui. Thus, we have the following lemma.
Lemma 2.5 For each u ∈ X with hC (u) , ui > 0 we have (i) if hB (u) , ui ≤ 0, then there is unique t− > t
max such that t−u ∈ N− and hu
is increasing on (0, t−) and decreasing on (t−, ∞). Moreover,
J¡t−u¢= sup
t≥0 J (tu) . (6)
(ii) if hB (u) , ui > 0, then there are unique 0 < t+ < t
max < t− such that
t+u ∈ N
+, t−u ∈ N−, hu is decreasing on (0, t+), increasing on (t+, t−) and
decreasing on (t−, ∞). Moreover, J¡t+u¢= inf 0≤t≤tmax J (tu) ; J¡t−u¢= sup t≥t+ J (tu) . (7)
Proof. Fix u ∈ X with hC (u) , ui > 0.
(i) Suppose hB (u) , ui ≤ 0. Then mu(t) = hB (u) , ui has a unique solution
t− > t
max and m0u(t−) < 0. Hence hu has a unique turning point at t = t− and
h00
u(t−) < 0. Thus t−u ∈ N− and (6) holds.
(ii) Suppose hB (u) , ui > 0. Since mu(tmax) > hB (u) , ui, the equation mu(t) =
hB (u) , ui has exactly two solutions t+ < t
max < t− such that m0u(t+) > 0 and
m0
u(t−) < 0. Hence there are exactly two multiples of u lying in N, viz, t+u ∈ N+
and t−u ∈ N
−. Thus, hu has turning points at t = t+ and t = t−with h00u(t+) > 0
and h00
u(t−) < 0. Thus, hu is decreasing on (0, t+), increasing on (t+, t−) and
decreasing on (t+, ∞). Hence (7) must hold. ¤
Similarly we define the function mu : R+ → R by
m (t) = tp−rhA (u) , ui − tq−rhB (u) , ui for t > 0.
If hB (u) , ui > 0, it is clear that m (t) → −∞ as t → 0+ and m (t) → 0+ as
t → ∞. Moreover, the function attains its maximum at tmax = µ (r − q) hB (u) , ui (r − p) hA (u) , ui ¶ 1 p−q .
Using arguments similar to those used in the proof of the previous lemma we have the following.
Lemma 2.6 For each u ∈ X with hB (u) , ui > 0 we have (i) if hC (u) , ui ≤ 0, then there is unique t+ < t
max such that t+u ∈ N+, hu is
decreasing on (0, t+) and increasing on (t+, 0). Moreover,
J¡t+u¢= inf
t≥0J (tu) ;
(ii) if hC (u) , ui > 0, then there are unique 0 < t+ < t
max < t− such that
t+u ∈ N
+, t−u ∈ N− and hu is decreasing on (0, t+), increasing on (t+, t−) and
decreasing on (t−, ∞). Moreover, J¡t+u¢= inf 0≤t≤tmax J (tu) ; J¡t−u¢= sup t≥t+ J (tu) .
Finally we remark that it follows from (H2) and Lemmas 2.5 and 2.6 that N+
and N− are nonempty.
3
Existence of nonzero solutions
By Lemma 2.4, we may write N = N+∪ N− and by Lemma 2.1 may define
α+ = inf u∈N+
J (u) and α− = inf u∈N−
For u ∈ N+, we have by (2) hB (u) , ui < r − p r − q hA (u) , ui . (8) Hence by (1) J(u) = µ 1 p− 1 r ¶ hA (u) , ui − µ 1 q − 1 r ¶ hB (u) , ui > µ 1 p− 1 r ¶ hA (u) , ui − µ 1 q − 1 r ¶ r − p r − q hA (u) , ui = r − p rp hA (u) , ui − r − p qr hA (u) , ui < 0
and so α+ < 0. Furthermore, we have the following result.
Theorem 3.1 The functional J has a minimizer u+
0 in N+ such that J ¡ u+ 0 ¢ = α+.
Proof. Let {un} be a minimizing sequence for J on N+. Then by Lemma 2.1
J is coercive on N and so {un} is bounded in X. Hence by (H3) there exist a
subsequence {un} and u+0 ∈ X such that
un * u+0 weakly in X, hB (un) , uni → B¡u+0¢, u+0® as n → ∞, hC (un) , uni → C¡u+0¢, u+0® as n → ∞. Since J (un) = r − p pr hA (un) , uni − r − q qr hB (un) , uni and so r − q qr hB (un) , uni = r − p pr hA (un) , uni − J (un) ,
letting n → ∞, we see that B¡u+ 0
¢
, u+ 0
®
> 0. Hence by Lemma 2.6 there is a
unique t+0 such that t+0u+0 ∈ N+. Now we prove that
hA (un) , uni →
A¡u+0¢, u+0® as n → ∞. Supposing the contrary, then
A¡u+0¢, u+0®< lim inf hA (un) , uni . (9) Since h0 un(t) = t p−1hA (u n) , uni − tq−1hB (un) , uni − tr−1hC (un) , uni (10)
and h0u+ 0 (t) = t p−1A¡u+ 0 ¢ , u+0®− tq−1B¡u+0¢, u0+®− tr−1C¡u+0¢, u+0® and h0 u+
0(t0) = 0, it follows that that h 0
un(t0) > 0 for n sufficiently large. Since
{un} ⊆ N+, h0un(1) = 0 and by Lemma 2.6 h
0
un(t) < 0 for all t ∈ (0, 1). Hence
t0 > 1. But t+0u+0 ∈ N+ and so by Lemma 2.6 and (9)
J¡t+ 0u+0 ¢ < J¡u+ 0 ¢ < lim n→∞J (un) = α+.
This is impossible and so
hA (un) , uni → A¡u+ 0 ¢ , u+ 0 ® as n → ∞. Hence h0 u+0(1) = limn→∞h 0 un(1) = 0 and h 00 u+0(1) = limn→∞h 00 un(1) ≥ 0 and so u+0 ∈ N+. Moreover, J (un) → J ¡ u+ 0 ¢ = α+ as n → ∞ and so u+ 0 is a minimizer for J on N+. ¤
Next, we establish the existence of a local minimum for J on N−.
Theorem 3.2 J has a minimizer u−0 in N− such that J
¡
u−0¢= α−.
Proof. Let {un} be a minimizing sequence for J on N−. Then as in the previous
proof there exists a subsequence {un} and u−0 ∈ X such that
un * u−0 weakly in X, hB (un) , uni → B¡u−0¢, u−0® as n → ∞, hC (un) , uni → C¡u−0¢, u−0® as n → ∞. Moreover, since un ∈ N−, (p − q) hA (un) , uni − (r − q) hC (un) , uni < 0 and so £ C¡u− 0 ¢ , u− 0 ®¤r−p r ≥ p − q r − q µ 1 d2 ¶p r > 0. (11) In particular, u−
0 6= 0. Now we prove that
hA (un) , uni →
A¡u−0¢, u−0® as n → ∞. Suppose otherwise, then
A¡u− 0 ¢ , u− 0 ® < lim inf hA (un) , uni .
By Lemma 2.5, there is a unique t−
0 such that t−0u−0 ∈ N−. Then h0u−
0 (t −
0) = 0 and
so it follows from (10) that h0 un(t
−
0) > 0 for n sufficiently large. Since un ∈ N−,
h0
un(1) = 0 and it is clear from Lemma 2.5 and Lemma 2.6 that hun is increasing on (t−
0, 1). Hence hun(t
−
0) < hun(1), i.e., J(t
−
0un) < J(un). Thus, using (9),
J¡t−0u−0¢< lim inf J¡t−0un
¢
≤ lim inf J (un) = α−,
which is a contradiction. Hence
hA (un) , uni → A¡u− 0 ¢ , u− 0 ® as n → ∞.
The proof can now be completed in the same way as that used in Theorem 3.1. ¤
Furthermore, we have the following result.
Theorem 3.3 The equation (E) has at least two nonzero solutions. Proof. By Theorems 3.1, 3.2 there exist u+
0 ∈ N+ and u−0 ∈ N− such that
J¡u+0¢ = α+ and J
¡
u−0¢ = α−. Since N+ ∩ N− = ∅, this implies that u+0 and
u−
0 are distinct. Moreover, by Lemma 2.2 u+0 and u−0 are nonzero solutions of
equation (E) . ¤
4
Applications
In this section we discuss a variety of applications of Theorem 3.3. (I) We study the p–Laplacian elliptic equation:
½
−∆pu = λf (x) |u|q−2u + µg (x) |u|r−2u in Ω,
u = 0 on Ω, (Dλ,µ)
where 1 < q < p < r < p∗ (p∗ = pN
N −p if N > p, p∗ = ∞ if N ≤ p), Ω ⊂ RN
is a bounded domain with smooth boundary, λ and µ are parameters in R+,
and the weight functions f, g ∈ C¡Ω¢ satisfy f+ = max {f, 0} 6≡ 0 and g+ =
max {g, 0} 6≡ 0. We shall apply Theorem 3.3 with potential operators A, B and
C from W01,p(Ω) into W−1,p0 (Ω) given by hA (u) , vi = Z Ω |∇u|p−2∇u∇v dx, hB (u) , vi = λ Z Ω f |u|q−2uv dx
and
hC (u) , vi = µ
Z
Ω
g |u|r−2uv dx
for u and v in W01,p(Ω) , where 1
p + p10 = 1. Thus, the corresponding energy functional of equation (Dλ,µ) is defined by
J (u) = 1 phA (u) , ui − 1 qhB (u) , ui − 1 rhC (u) , ui for u ∈ W 1,p 0 (Ω)
and the Nehari manifold is defined by
N = ©u ∈ W01,p(Ω) \ {0} | hA (u) , ui = hB (u) , ui + hC (u) , uiª.
Clearly hA (u) , ui = Z Ω |∇u|pdx = kukpW1,p 0 and so A satisfies (H1).
Since f+ 6≡ 0 and g+ 6≡ 0, B and C satisfy (H2). Moreover, by standard
compact embedding theorems B and C satisfy (H3) and
hB (u) , ui ≤ λ°°f+°°∞Sqq[hA (u) , ui]q/p and
hC (u) , ui ≤ µ°°g+°°∞Srr[hA (u) , ui]r/p
where Sl is the best Sobolev constant for the embedding of W01,p(Ω) in Ll(Ω) for
1 ≤ l < p∗. Thus if we choose d
1 = λ kf+k∞Sqq and d2 = µ kg+k∞Srr, then it is
clear that (H4) is satisfied provided that λµ is chosen to be sufficiently small. Theorem 4.1 There exists Λ0 > 0 such that for 0 < λµ < Λ0 equation (Dλ,µ)
has at least two positive solutions.
Proof. By Theorem 3.3 there exist u+
0 ∈ N+ and u−0 ∈ N− such that J
¡ u+ 0 ¢ = α+ and J ¡ u− 0 ¢ = α−. Moreover, J ¡ u± 0 ¢ = Jλ ¡¯ ¯u± 0 ¯ ¯¢ and ¯¯u± 0 ¯ ¯ ∈ N± and so we may assume u±
0 ≥ 0. By Lemma 2.2 u±0 are critical points for J on W01,p(Ω) and
hence are weak solutions (and so by standard regularity results classical solutions) of equation (Dλ,µ) . Moreover, by the Harnack inequality due to Trudinger [15],
we obtain that u±
0 are positive solutions of (Dλ,µ). ¤
(II) We consider the multiplicity of positive solutions for the following quasilinear equation with nonlinear boundary condition:
½
−∆pu + |u|p−2u = g (x) |u|r−2u in Ω, ∂u
∂ν = λf (x) |u|
where 1 < q < p < r < p∗, q < p
∗ (p∗ = p(N −1)N −p if N > p, p∗ = ∞ if N ≤ p),
λ ∈ R−{0}, Ω is a bounded domain in RN with smooth boundary, ∂
∂ν is the outer
normal derivative and the weight functions f, g satisfy the following conditions: (F ) f ∈ C (∂Ω) with f± = max {±f, 0} 6≡ 0;
(G) g ∈ C¡Ω¢ with g+ = max {g, 0} 6≡ 0.
We shall apply Theorem 3.3 with potential operators A, B and C from W1,p(Ω)
into W−1,p0 (Ω) given by hA (u) , vi = Z Ω |∇u|p−2∇u∇v dx + Z Ω |u|p−2uv dx, hB (u) , vi = λ Z ∂Ω f |u|q−2uv dσ and hC (u) , vi = Z Ω g |u|r−2uv dx
for u and v in W1,p(Ω) , where 1
p+p10 = 1 and dσ is the measure on the boundary. As before (H1), (H2) and (H3) are satisfied. Moreover, by the Sobolev em-bedding theorems,
hB (u) , ui ≤ |λ| kf k∞Sqq[hA (u) , ui]q/p and
hC (u) , ui ≤°°g+°° ∞S
r
r[hA (u) , ui]r/p
where Sq be the best Sobolev trace constant for the embedding of W1,p(Ω) in
Lq(∂Ω) and S
r be the best Sobolev constant for the embedding of W1,p(Ω) in
Lr(Ω) . Thus, if we choose d
1 = |λ| kf k∞S q
q and d2 = kg+k∞Srr, then (H4) is
satisfied provided λ is sufficiently small. Hence arguing exactly as in the previous example we can prove
Theorem 4.2 There exists Λ0 > 0 such that for 0 < |λ| < Λ0, equation (Γλ) has
at least two positive solutions.
(III) In a very similar way we can consider the nonlinear boundary value problem ½
∆pu − |u|p−2u = 0 in Ω, ∂u
∂ν = λf (x) |u|
q−2u + g (x) |u|r−2u on ∂Ω, (Θλ)
where 1 < q < p < r < p∗, λ ∈ R − {0}, Ω is a bounded domain in RN with
smooth boundary, ∂
∂ν is the outer normal derivative and the weight functions
f, g ∈ C (∂Ω) satisfy f± 6≡ 0 and g+ 6≡ 0. We now apply Theorem 3.3, with
potential operators A, B and C from W1,p(Ω) into W−1,p0
(Ω) given by
hA (u) , vi =
Z
Ω
hB (u) , vi = λ Z ∂Ω f |u|q−2uv dσ and hC (u) , vi = Z ∂Ω g |u|r−2uv dσ.
In this case by the Sobolev embedding theorem
hB (u) , ui ≤ |λ| kf k∞Sqq[hA (u) , ui]q/p and
hC (u) , ui ≤°°g+°°∞Srr[hA (u) , ui]r/p If we set d1 = |λ| kf k∞S
q
q and d2 = kg+k∞S r
r, then (H4) is satisfied provided
λ is sufficiently small. Thus arguing exactly as in the previous examples we can
prove
Theorem 4.3 There exists Λ0 > 0 such that for 0 < |λ| < Λ0, equation (Θλ)
has at least two positive solutions.
(IV ) We now consider multiplicity results for positive solutions of the following quasilinear elliptic system:
−∆pu = λf (x) |u|q−2u +α+βα h (x) |u|α−2u |v|β in Ω, −∆pv = µg (x) |v|q−2v +α+ββ h (x) |u|α|v|β−2v in Ω, u = v = 0 on ∂Ω, (Lλ,µ)
where Ω is a bounded domain in RN with smooth boundary, α > 1, β > 1 satisfy
1 < q < p < α + β < p∗, λ and µ are parameters such that λ ∈ R − {0} and
µ ∈ R − {0}, and the weight functions f, g, h ∈ C¡Ω¢ satisfy
(U1) the intersections of the set {x ∈ Ω : h(x) > 0} with each of the sets
{x ∈ Ω : f (x) > 0}, {x ∈ Ω : f (x) < 0}, {x ∈ Ω : g(x) > 0}, {x ∈ Ω : g(x) < 0}
have positive measures.
We pose Problem (Lλ,µ) in the framework of the Sobolev space W = W01,p(Ω)×
W01,p(Ω) with the standard norm
k(u, v)kW = µZ Ω |∇u|pdx + Z Ω |∇v|p dx ¶1 p .
Moreover, a pair of functions (u, v) ∈ W is said to be weak solution of problem (Lλ,µ) if Z Ω |∇u|p−2∇u∇ϕ1dx + Z Ω |∇v|p−2∇v∇ϕ2dx − λ Z Ω f |u|q−2uϕ1dx − µ Z Ω g |v|q−2vϕ2dx − α α + β Z Ω h |u|α−2u |v|βϕ1dx − β α + β Z Ω h |u|α|v|β−2vϕ2dx = 0
for all (ϕ1, ϕ2) ∈ W. We shall apply Theorem 3.3, with potential operators A, B
and C from W into W∗ given by
hA (u, v) , (ϕ1, ϕ2)i = Z Ω |∇u|p−2∇u∇ϕ1dx + Z Ω |∇v|p−2∇v∇ϕ2dx, hB (u, v) , (ϕ1, ϕ2)i = λ Z Ω f |u|q−2uϕ1dx + µ Z Ω g |v|q−2vϕ2dx and hC (u, v) , (ϕ1, ϕ2)i = α α + β Z Ω h |u|α−2u |v|βϕ1dx + β α + β Z Ω h |u|α|v|β−2vϕ2dx
for (u, v) and (ϕ1, ϕ2) in W. Thus, the corresponding energy functional of equation
(Lλ,µ) is defined by
J (u, v) = 1
phA (u, v) , (u, v)i −
1
qhB (u, v) , (u, v) −
1
α + βhC (u, v) , (u, v)i
for (u, v) ∈ W. Moreover, the Nehari manifold is defined by
N = {u ∈ W \ {(0, 0)} | hA (u, v) , (u, v)i = hB (u, v) , (u, v)i + hC (u, v) , (u, v)i} . It is easy to see that (H1)and (H2) are satisfied and by standard compact em-bedding theorems B and C satisfy (H3) and moreover
hB (u, v) , (u, v)i = λ
Z Ω f |u|qdx + µ Z Ω g|v|qdx ≤ |λ|kf k∞ Z Ω |u|qdx + |µ|kgk∞ Z Ω |v|qdx ≤ Sq q(|λ|kf k∞+ |µ|kgk∞) k (u, v) kq
≤ Sqq(|λ|kf k∞+ |µ|kgk∞) [hA (u, v) , (u, v)i]q/p
and
hC (u, v) , (u, v)i =
Z Ω h|u|α|v|βdx ≤ Sα+β α+β ° °h+°°
∞[hA (u, v) , (u, v)i] (α+β)/p
where we have used the fact that |u|α|v|β ≤ |u|α+β + |v|α+β. Thus, if we set
d1 = Sqq(|λ|kf k∞+ |µ|kgk∞) and d2 = Sα+βα+βkh+k∞, it is clear that (H4) is
satisfied provided that |λ| + |µ| is sufficiently small. Theorem 4.4 There exists Λ0 > 0 such that for
|λ| + |µ| < Λ0
Proof. By Theorem 3.3 there exist ¡u+ 0, v0+ ¢ ∈ N+ and ¡ u− 0, v−0 ¢ ∈ N− such that J¡u+ 0, v+0 ¢ = α+ and J ¡ u− 0, v0− ¢ = α−. Moreover, J ¡ u± 0, v±0 ¢ = J¡¯¯u± 0 ¯ ¯ ,¯¯v± 0 ¯ ¯¢ and ¡¯¯u±0¯¯ ,¯¯v0±¯¯¢ ∈ N±. Thus, we may assume that u±0 ≥ 0 and v0± ≥ 0. By
Lemma 2.2 ¡u±0, v0±¢ are critical points for J on W01,p(Ω) and hence are weak solutions of equation (Lλ,µ) . Moreover, by (11), u−0 6= 0, v0− 6= 0. Finally, we
prove that u+
0 6≡ 0, v+0 6≡ 0. We assume, without loss of generality, that v0+ ≡ 0.
Then, since α+ < 0, we must have u+0 6≡ 0. Then, as u+0 is a solution of
−∆pu = λf |u|q−2u in Ω; u = 0 on ∂Ω, we have λ Z Ω f |u+0|qdx = Z Ω |∇u+0|pdx > 0. (12) Moreover by (U1) we may choose w ∈ W01,p(Ω) \ {0} such that
Z Ω |∇w|pdx = µ Z Ω g |w|q dx (13) and Z Ω h¯¯u+0¯¯α|w|β dx ≥ 0. (14) Now B¡u+ 0, w ¢ ,¡u+ 0, w ¢® = λ Z Ω f¯¯u+ 0 ¯ ¯q dx + µ Z Ω g |w|q dx > 0
and so by Lemma 2.6 there is a unique t+ > 0 such that 0 < t+ < t
max and ¡ t+u+ 0, t+w ¢ ∈ N+ where tmax = Ã (α + β − q) hB¡u+0, w¢,¡u+0, , w¢i (α + β − p) hA¡u+ 0, w ¢ ,¡u+ 0, w ¢ i ! 1 p−q . By (12) and (13) hA¡u+ 0, w ¢ ,¡u+ 0, w ¢ i = hB¡u+ 0, w ¢ ,¡u+ 0, w ¢ i and so tmax > 1. By Lemma 2.6 J¡t+u+ 0, t+w ¢ = inf 0≤t≤tmax J¡tu+ 0, tw ¢ and so, by (14), J¡t+u+ 0, t+w ¢ ≤ J¡u+ 0, w ¢ < J¡u+ 0, 0 ¢ = α+
which is a contradiction. Moreover, by the Harnack inequality due to Trudinger [15], we may conclude that ¡u+0, v+0¢ and ¡u−0, v0−¢ are positive solutions of
(V ) Similarly we can consider multiplicity results for positive solutions of the following quasilinear elliptic system with nonlinear boundary conditions:
−∆pu + |u|p−2u = α+βα h (x) |u|α−2u |v|β in Ω, −∆pv + |v|p−2v = α+ββ h (x) |u|α|v|β−2v in Ω, ∂u ∂n = λf (x) |u| q−2u, ∂v ∂n = µg (x) |v| q−2v on ∂Ω, (Γλ,µ)
where Ω is a bounded domain in RN with smooth boundary, α > 1, β > 1 satisfy
1 < q < p < α + β < p∗, q < p
∗, λ and µ are parameters in R − {0}, and the
weight functions f, g ∈ C (∂Ω) and h ∈ C¡Ω¢ satisfy the following condition: (U2) f±= max {±f, 0} 6≡ 0, g± = max {±g, 0} 6≡ 0 and h+= max {h, 0} 6≡ 0
We consider problem (Γλ,µ) in the framework of the Sobolev space W =
W1,p(Ω) × W1,p(Ω) with the standard norm
k(u, v)kW = µZ Ω (|∇u|p+ |u|p) dx + Z Ω (|∇v|p+ |v|p) dx ¶1 p .
Moreover, a pair of functions (u, v) ∈ W is said to be weak solution of problem (Γλ,µ) if
Z
Ω
¡
|∇u|p−2∇u∇ϕ1+ |u|p−2uϕ1
¢ dx + Z Ω ¡ |∇v|p−2∇v∇ϕ2+ |v|p−2vϕ2 ¢ dx − λ Z ∂Ω f |u|q−2uϕ1dσ − µ Z ∂Ω g |v|q−2vϕ2dσ − α α + β Z Ω h |u|α−2u |v|βϕ1dx− β α + β Z Ω h |u|α|v|β−2vϕ2dx = 0 ∀ (ϕ1, ϕ2) ∈ W.
where dσ is the measure on the boundary. We shall apply Theorem 3.3, with potential operators A, B and C from W into W∗ given by
hA (u, v) , (ϕ1, ϕ2)i =
Z
Ω
|∇u|p−2∇u∇ϕ1+|u|p−2uϕ1dx+
Z Ω |∇v|p−2∇v∇ϕ2+|v|p−2vϕ2dx, hB (u, v) , (ϕ1, ϕ2)i = λ Z ∂Ω f |u|q−2uϕ1dσ + µ Z ∂Ω g |v|q−2vϕ2dσ and hC (u, v) , (ϕ1, ϕ2)i = α α + β Z Ω h |u|α−2u |v|βϕ1dx + β α + β Z Ω h |u|α|v|β−2vϕ2dx
for (u, v) and (ϕ1, ϕ2) in W.
Again (H1), (H2) and (H3) are satisfied and by standard compact embedding theorems
and
hC (u, v) , (u, v)i ≤ Sα+βα+β kh+k
∞[hA (u, v) , (u, v)i](α+β)/p.
Thus, if we set d1 = S q
q(|λ| kf k∞+ |µ| kgk∞) and d2 = Sα+βα+β kh+k∞, it is clear
(H4) is satisfied provided |λ| + |µ| is sufficiently small. Hence, arguing as in the previous example, we have
Theorem 4.5 There exists Λ0 > 0 such that for
|λ| + |µ| < Λ0
problem (Γλ,µ) has at least two positive solutions.
(V I) Finally we consider multiplicity results for positive solutions of the following quasilinear elliptic system with boundary condition containing both convex and concave nonlinearities. ∆pu − |u|p−2u = 0 in Ω, ∆pv − |v|p−2v = 0 in Ω, ∂u ∂n = λf (x) |u| q−2u + α α+βh (x) |u| α−2u |v|β on ∂Ω, ∂v ∂n = µg (x) |v| q−2v + β α+βh (x) |u| α|v|β−2v on ∂Ω, (Θλ,µ)
where Ω is a bounded domain in RN with smooth boundary, α > 1, β > 1 satisfy
1 < q < p < α + β < p∗, λ and µ are parameters in R − {0}, and the weight
functions f, g, h ∈ C (∂Ω) satisfy the following condition:
(U3) the intersections of the set {x ∈ ∂Ω : h(x) > 0} with each of the sets {x ∈
∂Ω : f (x) > 0}, {x ∈ ∂Ω : f (x) < 0}, {x ∈ ∂Ω : g(x) > 0}, {x ∈ ∂Ω : g(x) < 0}
have positive measures.
As in the previous example we consider the problem (Θλ,µ) in the framework
of the Sobolev space W = W1,p(Ω) × W1,p(Ω) with the standard norm. Thus a
pair of functions (u, v) ∈ W is said to be weak solution of problem (Θλ,µ) if
Z Ω ¯ ¯(∇u|p−2∇u∇ϕ 1+ |u|p−2uϕ1 ¢ dx + Z Ω ¡ |∇v|p−2∇v∇ϕ2+ |v|p−2vϕ2 ¢ dx − λ Z ∂Ω f |u|q−2uϕ1dσ − µ Z ∂Ω g |v|q−2vϕ2dσ − α α + β Z ∂Ω h |u|α−2u |v|βϕ1dσ− β α + β Z ∂Ω h |u|α|v|β−2vϕ2dσ = 0 ∀ (ϕ1, ϕ2) ∈ W
where dσ is the measure on the boundary. We shall apply Theorem 3.3, with potential operators A, B and C from W into W∗ given by
hA (u, v) , (ϕ1, ϕ2)i =
Z
Ω
|∇u|p−2∇u∇ϕ1+|u|p−2uϕ1dx+
Z
Ω
hB (u, v) , (ϕ1, ϕ2)i = λ Z ∂Ω f |u|q−2uϕ1dσ + µ Z ∂Ω g |v|q−2vϕ2dσ and hC (u, v) , (ϕ1, ϕ2)i = α α + β Z ∂Ω h |u|α−2u |v|βϕ1dσ+ β α + β Z ∂Ω h |u|α|v|β−2vϕ2dσ.
Again (H1), (H2) and (H3) are satisfied and by standard compact embedding theorems
hB (u, v) , (u, v)i ≤ Sqq(|λ| kf k∞+ |µ| kgk∞) [hA (u, v) , (u, v)i]q/p
and
hC (u, v) , (u, v)i ≤ Sα+βα+β kh+k
∞[hA (u, v) , (u, v)i](α+β)/p.
If we set d1 = Sqq(|λ| kf k∞+ |µ| kgk∞) and d2 = Sα+βα+β kh+k∞, then it is clear
that (H4) is satisfied provided |λ| + |µ| is sufficiently small. Arguing as before we have
Theorem 4.6 There exists Λ0 > 0 such that for
|λ| + |µ| < Λ0
problem (Θλ,µ) has at least two positive solutions.
References
[1] A. Ambrosetti, G. J. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996) 219–242.
[2] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519–543.
[3] K. Adriouch and A. El Hamidi, The Nehari manifold for systems of nonlinear elliptic equations, Nonlinear Anal. 64 (2006) 2149–2164.
[4] P. A. Binding, P. Drabek and Y. X. Huang, On Nuumann boundary value problems for some quasilinear elliptic equations, Electr. J. Diff. Eqns. 5 (1997) 1–11.
[5] Y. Bozhkov and E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method, J. Differential Equations 190 (2003) 239–267.
[6] K. J. Brown and T. F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function, J. Math. Anal. Appl., in press.
[7] K. J. Brown and T. F. Wu, A fibrering map approach to a semilinear elliptic boundary value problem, Electr. J. Diff. Eqns. 69 (2007) 1–9.
[8] K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations 193 (2003) 481–499.
[9] J. Chabrowski, Variational Methods for Potential Operator Equations, Wal-ter de GruyWal-ter, 1997.
[10] P. Drabek and S. I. Pohozaev, Positive solutions for the p–Laplacian: ap-plication of the fibering method, Proc. Roy. Soc. Edinburgh 127A (1997) 703–726.
[11] D. G. de Figueiredo, J. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal. 199 (2003) 452–467.
[12] J. Garcia-Azorero, I. Peral and J. D. Rossi, A convex-concave problem with a nonlinear boundary condition, J. Differential Equations 198 (2004) 91–128. [13] Y. Il’yasov, On nonlocal existence results for elliptic equations with
convex-concave nonlinearities, Nonlinear Anal. 61 (2005) 211–236.
[14] Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Am. Math. Soc. 95 (1960) 101–123.
[15] N. S. Trudinger, On Harnack type inequalities and their application to quasi-linear elliptic equations, Comm. Pure Appl. Math. 20 (1967) 721–747. [16] T. F. Wu, A semilinear elliptic problem involving nonlinear boundary
con-dition and sign-changing potential, Electr. J. Diff. Eqns. 131 (2006) 1–15. [17] T. F. Wu, Multiplicity results for a semilinear elliptic equation involving
sign-changing weight function, Rocky Mountain Journal of Math., in press. [18] T. F. Wu, The Nehari manifold for a semilinear elliptic system involving
