ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)
A FIBERING MAP APPROACH TO A SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEM
KENNETH J. BROWN, TSUNG-FANG WU
Abstract. We prove the existence of at least two positive solutions for the semilinear elliptic boundary-value problem
−∆u(x) = λa(x)uq+ b(x)up for x ∈ Ω; u(x) = 0 for x ∈ ∂Ω on a bounded region Ω by using the Nehari manifold and the fibering maps associated with the Euler functional for the problem. We show how knowledge of the fibering maps for the problem leads to very easy existence proofs.
1. Introduction
We shall discuss the existence of positive solutions of the semilinear elliptic boundary-value problem
−∆u(x) = λa(x)uq+ b(x)up for x ∈ Ω; (1.1)
u(x) = 0 for x ∈ ∂Ω, (1.2)
where Ω is a bounded region with smooth boundary in RN, 0 < q < 1 < p < N +2 N −2,
λ > 0 and a, b : Ω → R are smooth functions which are somewhere positive but which may change sign on Ω. Equation (1.1), (1.2) has been recently studied in [3] by using the Mountain Pass Lemma and in [5] and [7] using the Nehari manifold.
In [4] and [2] it was shown that the Nehari manifold for an equation such as (1.1) is closely related to the fibering maps for the problem. In this paper we show how a fairly complete knowledge of all possible forms of the fibering maps provides a very simple and comparatively elementary means of establishing results similar to those proved in [5] and [7] on the existence of multiple solutions of (1.1), (1.2). In section 2 we recall the properties which we shall require of fibering maps and of the Nehari manifold. In section 3 we give a fairly complete description of the fibering maps associated with (1.1) and in section 4 we use this information to give a very simple variational proof of the existence of at least two positive solutions of (1.1), (1.2) for sufficiently small λ.
We shall throughout use the function space W01,2(Ω) with norm kuk =
Z
Ω
|∇u|2dx1/2
2000 Mathematics Subject Classification. 35J20, 36J65.
Key words and phrases. Semilinear elliptic boundary value problem; variational methods; Nehari manifold; fibering map.
c
2007 Texas State University - San Marcos.
Submitted Febraury 27, 2007. Published May 10, 2007.
and the standard Lp(Ω) spaces whose norms we denote by kukp.
2. Fibering Maps and the Nehari manifold The Euler functional associated with (1.1), (1.2) is
Jλ(u) = 1 2 Z Ω |∇u|2dx − λ q + 1 Z Ω a(x)|u|q+1dx − 1 p + 1 Z Ω b(x)|u|p+1dx for all u ∈ W01,2(Ω).
As Jλ is not bounded below on W01,2(Ω), it is useful to consider the functional
on the Nehari manifold
Mλ(Ω) = {u ∈ W 1,2
0 (Ω) : hJλ0(u), ui = 0}
where h, i denotes the usual duality. Thus u ∈ Mλ(Ω) if and only if
Z Ω |∇u|2dx − λ Z Ω a(x)|u|q+1dx − Z Ω b(x)|u|p+1dx = 0 (2.1) Clearly Mλ(Ω) is a much smaller set than W01,2(Ω) and, as we shall show, Jλ is
much better behaved on Mλ(Ω). In particular, on Mλ(Ω) we have that
Jλ(u) = ( 1 2 − 1 q + 1) Z Ω |∇u|2+ ( 1 q + 1− 1 p + 1) Z Ω b(x)|u|p+1 = (1 2 − 1 p + 1) Z Ω |∇u|2− λ( 1 q + 1− 1 p + 1) Z Ω a(x)|u|q+1 (2.2)
Theorem 2.1. Jλ is coercive and bounded below on Mλ(Ω).
Proof. It follows from (2.2) and the Sobolev embedding theorems that there exist positive constants c1, c2and c3 such that
Jλ(u) ≥ c1kuk2− c2
Z
Ω
|u|q+1dx ≥ c
1kuk2− c3kukq+1
and so Jλ is coercive and bounded below on Mλ(Ω).
The Nehari manifold is closely linked to the behaviour of the functions of the form φu : t → Jλ(tu) (t > 0). Such maps are known as fibering maps and were
introduced by Drabek and Pohozaev in [4] and are also discussed in Brown and Zhang [2]. If u ∈ W01,2(Ω), we have φu(t) = 1 2t 2 Z Ω |∇u|2− λ tq+1 q + 1 Z Ω a|u|q+1− t p+1 p + 1 Z Ω b|u|p+1 (2.3) φ0u(t) = t Z Ω |∇u|2− λ tq Z Ω a|u|q+1− tp Z Ω b|u|p+1 (2.4) φ00u(t) = Z Ω |∇u|2− λqtq−1 Z Ω a|u|q+1− ptp−1 Z Ω b|u|p+1 (2.5)
It is easy to see that u ∈ Mλ(Ω) if and only if φ0u(1) = 0 and, more generally,
that φ0
u(t) = 0 if and only if tu ∈ Mλ(Ω), i.e., elements in Mλ(Ω) correspond to
corresponding to local minima, local maxima and points of inflection and so we define
Mλ+(Ω) = {u ∈ Mλ(Ω) : φ00u(1) > 0},
Mλ−(Ω) = {u ∈ Mλ(Ω) : φ00u(1) < 0},
Mλ0(Ω) = {u ∈ Mλ(Ω) : φ00u(1) = 0},
and note that if u ∈ Mλ(Ω), i.e., φ0u(1) = 0, then
φ00u(1) = (1 − q) Z Ω |∇u|2dx − (p − q) Z b(x)|u|p+1dx = (1 − p) Z Ω |∇u|2dx − λ(q − p)Z a(x)|u|q+1dx . (2.6)
Also, as proved in Binding, Drabek and Huang [1] or in Brown and Zhang [2], we have the following lemma.
Lemma 2.2. Suppose that u0 is a local maximum or minimum for Jλ on Mλ(Ω).
Then, if u06∈ Mλ0(Ω), u0 is a critical point of Jλ.
3. Analysis of the Fibering Maps
In this section we give a fairly complete description of the fibering maps as-sociated with the problem. As we shall see the essential nature of the maps is determined by the signs of R a(x)|u|q+1dx and R
Ωb(x)|u|
p+1dx. We will find it
useful to consider the function mu(t) = t1−q Z Ω |∇u|2dx − tp−q Z Ω b(x)|u|p+1dx. Clearly, for t > 0, tu ∈ Mλ(Ω) if and only if t is a solution of
mu(t) = λ Z Ω a(x)|u|q+1dx. (3.1) Morever, m0u(t) = (1 − q)t−q Z Ω |∇u|2dx − (p − q)tp−q−1 Z Ω b(x)|u|p+1dx. (3.2) It is easy to see that mu is a strictly increasing function for t ≥ 0 whenever
R
Ωb(x)|u|
p+1dx ≤ 0 and m
u is initially increasing and eventually decreasing with
a single turning point as in Figure 1(b) whenR
Ωb(x)|u|
p+1dx > 0.
(a)
(b)
Suppose tu ∈ Mλ(Ω). It follows from (2.6) and (3.2) that φ00tu(1) = tq+2m0u(t)
and so tu ∈ Mλ+(Ω)(Mλ−(Ω)) provided m0u(t) > 0 (< 0).
We shall now describe the nature of the fibering maps for all possible signs of R Ωb(x)|u| p+1dx andR Ωa(x)|u| q+1dx. IfR Ωb(x)|u| p+1dx ≤ 0 andR Ωa(x)|u| q+1dx ≤
0, clearly φu is an increasing function of t and so has graph as shown in Figure
2(a); thus in this case no multiple of u lies in Mλ(Ω). If
R Ωb(x)|u| p+1dx ≤ 0 and R Ωa(x)|u| q+1dx > 0, then m
uhas graph as in Figure 1(a), and it is clear that there
is exactly one solution of (3.1). Thus there is a unique value t(u) > 0 such that t(u)u ∈ Mλ(Ω). Clearly m0u(t(u)) > 0 and so t(u)u ∈ M
+
λ(Ω). Thus the fibering
map φu has a unique critical point at t = t(u) which is a local minimum. Since
limt→∞φu(t) = ∞, it follows that φu has graph as shown in Figure 2(c).
Suppose nowR Ωb(x)|u| p+1dx > 0 andR Ωa(x)|u| q+1dx ≤ 0. Then m uhas graph
as shown in Figure 1(b) and it is clear that there is exactly one positive solution of (3.1). Thus there is again a unique value t(u) > 0 such that t(u)u ∈ Mλ(Ω) and
since m0
u(t(u)) < 0 in this case t(u)u ∈ Mλ−(Ω). Hence the fibering map φu has
a unique critical point which is a local maximum. Since limt→∞φu(t) = −∞, it
follows that φu has graph as shown in Figure 2(b).
Finally we consider the caseR
Ωb(x)|u|
p+1dx > 0 andR
Ωa(x)|u|
q+1dx > 0 where
the situation is more complicated. As in the previous case muhas a graph as shown
in Figure 1(b). If λ > 0 is sufficiently large, (3.1) has no solution and so φu has
no critical points - in this case φu is a decreasing function. Hence no multiple
of u lies in Mλ(Ω). If, on the other hand, λ > 0 is sufficiently small, there are
exactly two solutions t1(u) < t2(u) of (3.1) with m0u(t1(u)) > 0 and m0u(t2(u)) < 0.
Thus there are exactly two multiples of u ∈ Mλ(Ω), namely t1(u)u ∈ Mλ+(Ω)
and t2(u)u ∈ Mλ−(Ω). It follows that φu has exactly two critical points - a local
minimum at t = t1(u) and a local maximum at t = t2(u); moreover φuis decreasing
in (0, t1), increasing in (t1, t2) and decreasing in (t2, ∞) as in Figure 2(d).
The following result ensures that when λ is sufficiently small the graph of φu
must be as shown in Figure 2(d) for all non-zero u.
Lemma 3.1. There exists λ1 > 0 such that, when λ < λ1, φu takes on positive
values for all non-zero u ∈ W01,2(Ω).
Proof. If RΩb(x)|u|p+1dx ≤ 0, then φu(t) > 0 for t sufficiently large. Suppose
u ∈ W01,2(Ω) and R Ωb(x)|u| p+1dx > 0. Let hu(t) = t2 2 Z Ω |∇u|2dx − t p+1 p + 1 Z Ω b(x)|u|p+1dx. Then elementary calculus shows that hu takes on a maximum value of
p − 1 2(p + 1) n (R Ω|∇u| 2dx)p+1 (R Ωb(x)|u|p+1dx)2 op−11 when t = tmax= R Ω|∇u| 2dx R Ωb(x)|u|p+1dx p−11 . However (R Ω|∇u| 2dx)p+1 (R Ω|u|p+1dx)2 ≥ 1 Sp+12(p+1)
where Sp+1denotes the Sobolev constant of the embedding of W01,2(Ω) into Lp+1(Ω).
Hence hu(tmax) ≥ p − 1 2(p + 1) 1 kb+k2 ∞S 2(p+1) p+1 p−11 = δ
u u u u
φ
φ
φ
φ
t
t
t
t
(d)
(c)
(a)
(b)
Figure 2. Possible forms of fibering maps where δ is independent of u.
We shall now show that there exists λ1> 0 such that φu(tmax) > 0, i.e.,
hu(tmax) − λ (tmax)q+1 q + 1 Z Ω a(x)|u|q+1dx > 0 for all u ∈ W01,2(Ω) − {0} provided λ < λ1. We have
(tmax)q+1 q + 1 Z Ω a(x)|u|q+1dx ≤ 1 q + 1kak∞S q+1 q+1 R Ω|∇u| 2dx R Ωb(x)|u|p+1dx q+1p−1Z Ω |∇u|2dx q+1 2 = 1 q + 1kak∞S q+1 q+1 n R Ω|∇u| 2dxp+1 R Ωb(x)|u| p+1dx2 o2(p−1)q+1 = 1 q + 1kak∞S q+1 q+1 2(p + 1) p − 1 q+12 h u(tmax) q+1 2 = c hu(tmax) q+1 2
where c is independent of u. Hence φu(tmax) ≥ hu(tmax) − λchu(tmax)
q+1 2 = hu(tmax) q+1 2 hu(tmax) 1−q 2 − λc
and so, since hu(tmax) ≥ δ for all u ∈ W 1,2
0 (Ω) − {0}, it follows that φu(tmax) > 0
for all non-zero u provided λ < δ1−q2 /2c = λ1. This completes the proof.
It follows from the above lemma that when λ < λ1,
R Ωa(x)|u| q+1dx > 0 and R Ωb(x)|u| p+1dx > 0 then φ
u must have exactly two critical points as discussed in
Thus when λ < λ1 we have obtained a complete knowledge of the number of
critical points of φu, of the intervals on which φu is increasing and decreasing
and of the multiples of u which lie in Mλ(Ω) for every possible choice of signs of
R
Ωb(x)|u|
p+1dx andR
Ωa(x)|u|
q+1dx. In particular we have the following result.
Corollary 3.2. M0
λ(Ω) = ∅ when 0 < λ < λ1.
Corollary 3.3. If λ < λ1, then there exists δ1 > 0 such that Jλ(u) ≥ δ1 for all
u ∈ Mλ−(Ω).
Proof. Consider u ∈ Mλ−(Ω). Then φuhas a positive global maximum at t = 1 and
R b(x)|u|p+1dx > 0. Thus Jλ(u) = φu(1) ≥ φu(tmax) ≥ hu(tmax) q+1 2 hu(tmax) 1−q 2 − λc ≥ δq+12 (δ 1−q 2 − λc)
and the left hand side is uniformly bounded away from 0 provided that λ < λ1.
4. Existence of Positive Solutions
In this section using the properties of fibering maps we shall give simple proofs of the existence of two positive solutions, one in Mλ+(Ω) and one in Mλ−(Ω). Theorem 4.1. If λ < λ1, there exists a minimizer of Jλ on Mλ+(Ω).
Proof. Since Jλ is bounded below on Mλ(Ω) and so on Mλ+(Ω), there exists a
minimizing sequence {un} ⊆ Mλ+(Ω) such that
lim
n→∞Jλ(un) =u∈Minf+ λ(Ω)
Jλ(u).
Since Jλ is coercive, {un} is bounded in W01,2(Ω). Thus we may assume, without
loss of generality, that un* u0in W01,2(Ω) and un → u0in Lr(Ω) for 1 < r < N −22N .
If we choose u ∈ W01,2(Ω) such that R
Ωa(x)|u|
q+1dx > 0, then the graph of
the fibering map φu must be of one of the forms shown in Figure 2(c) or (d)
and so there exists t1(u) such that t1(u)u ∈ Mλ+(Ω) and Jλ(t1(u)u) < 0. Hence,
infu∈M+ λ(Ω)Jλ(u) < 0. By (2.2), Jλ(un) = ( 1 2− 1 p + 1) Z Ω |∇un|2dx − λ ( 1 q + 1 − 1 p + 1) Z Ω a(x)|un|q+1dx and so λ ( 1 q + 1 − 1 p + 1) Z Ω a(x)|un|q+1dx = ( 1 2− 1 p + 1) Z Ω |∇un|2dx − Jλ(un).
Letting n → ∞, we see thatR
Ωa(x)|u0|
q+1dx > 0.
Suppose un6→ u0in W01,2(Ω). We shall obtain a contradiction by discussing the
fibering map φu0. Since
R
Ωa(x)|u0|
q+1dx > 0, the graph of φ
u0 must be either
of the form shown in Figure 2(c) or (d). Hence there exists t0 > 0 such that
t0u0∈ Mλ+(Ω) and φu0 is decreasing on (0, t0) with φ 0 u0(t0) = 0. Since un6→ u0 in W 1,2 0 (Ω), R Ω|∇u0| 2dx < lim inf n→∞ R Ω|∇un| 2dx. Thus, as φ0un(t) = t Z Ω |∇un|2dx − λtq Z Ω a(x)|un|q+1dx − tp Z Ω b(x)|un|p+1dx
and φ0u0(t) = t Z Ω |∇u0|2dx − λtq Z Ω a(x)|u0|q+1dx − tp Z Ω b(x)|u0|p+1dx,
it follows that φ0un(t0) > 0 for n sufficiently large. Since {un} ⊆ Mλ+(Ω), by
considering the possible fibering maps it is easy to see that φ0un(t) < 0 for 0 < t < 1
and φ0u
n(1) = 0 for all n. Hence we must have t0> 1. But t0u0∈ M + λ(Ω) and so Jλ(t0u0) < Jλ(u0) < lim n→∞Jλ(un) =u∈Minf+ λ(Ω) Jλ(u)
and this is a contradiction. Hence un → u0 in W 1,2 0 (Ω) and so Jλ(u0) = lim n→∞Jλ(un) =u∈Minf+ λ(Ω) Jλ(u).
Thus u0 is a minimizer for Jλon Mλ+(Ω).
Theorem 4.2. If λ < λ1, there exists a minimizer of Jλ on Mλ−(Ω).
Proof. By Corollary 3.3 we have Jλ(u) ≥ δ1 > 0 for all u ∈ Mλ−(Ω) and so
infu∈M− λ(Ω)
Jλ(u) ≥ δ1. Hence there exists a minimizing sequence {un} ⊆ Mλ−(Ω)
such that
lim
n→∞Jλ(un) =u∈Minf− λ(Ω)
Jλ(u) > 0.
As in the previous proof, since Jλ is coercive, {un} is bounded in W 1,2
0 (Ω) and we
may assume, without loss of generality, that un * u0 in W01,2(Ω) and un→ u0 in
Lr(Ω) for 1 < r < 2N N −2. By (2.2) Jλ(un) = ( 1 2 − 1 q + 1) Z Ω |∇un|2dx + ( 1 q + 1− 1 p + 1) Z Ω b(x)|un|p+1dx
and, since limn→∞Jλ(un) > 0 and
lim n→∞ Z Ω b(x)|un|p+1dx = Z Ω b(x)|u0(x)|p+1dx,
we must have that R
Ωb(x)|u0(x)|
p+1dx > 0. Hence the fibering map φ
u0 must
have graph as shown in Figure 2(b) or (d) and so there exists ˆt > 0 such that ˆ
tu0∈ Mλ−(Ω).
Suppose un6→ u0 in W 1,2
0 (Ω). Using the facts that
Z
Ω
|∇u0|2dx < lim inf n→∞
Z
Ω
and that, since un∈ Mλ−(Ω), J (un) ≥ J (sun) for all s ≥ 0, we have J (ˆtu0) = 1 2ˆt 2 Z Ω |∇u0|2dx − λˆtq+1 q + 1 Z Ω a(x)|u0|q+1dx − ˆ tp+1 p + 1 Z Ω b(x)|u0|p+1dx < lim n→∞ h1 2 ˆt 2 Z Ω |∇un|2dx − λˆtq+1 q + 1 Z Ω a(x)|un|q+1dx − ˆt p+1 p + 1 Z Ω b(x)|un|p+1dx i = lim n→∞J (ˆtun) ≤ lim n→∞J (un) =u∈Minf− λ(Ω) Jλ(u)
which is a contradiction. Hence un → u0in W01,2(Ω) and the proof can be completed
as in the previous theorem.
Corollary 4.3. Equation (1.1), (1.2) has at least two positive solutions whenever 0 < λ < λ1.
Proof. By Theorems 4.1 and 4.2 there exist u+ ∈ M
λ(Ω) and u− ∈ Mλ−(Ω)
such that J (u+) = infu∈M+
λ(Ω)J (u) and J (u
−) = inf
u∈Mλ−(Ω)J (u). Moreover
J (u±) = J (|u±|) and |u±| ∈ M±
λ(Ω) and so we may assume u± ≥ 0. By Lemma
2.2 u± are critical points of J on W01,2(Ω) and hence are weak solutions (and so by standard regularity results classical solutions) of (1.1), (1.2). Finally, by the Harnack inequality due to Trudinger [6], we obtain that u± are positive solutions
of (1.1), (1.2).
Acknowledgement. We would like to thank the referee for bringing [5] to our attention and for making the important observation that the fact that Mλ0(Ω) = ∅ follows from sufficient knowledge of the fibering maps.
References
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[4] P. Drabek and S. I. Pohozaev; Positive solutions for the p-Laplacian: application of the fibering method, Proc. Royal Soc. Edinburgh Sect A, 127, (1997), 703-726.
[5] Y. Il’yasov; On non-local existence results for elliptic operators with convex-concave nonlin-earities, Nonlinear Analysis, 61, 2005 211-236.
[6] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Applied Math. 20 (1967) 721-747.
[7] T.-F. Wu, Multiplicity results for a semilinear elliptic equation involving sign-changing weight function, to appear in Rocky Mountain J. Math.
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
