Electronic Journal of Differential Equations, Vol. 2003(2003), No. 07, pp. 1–5.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
Multiple positive solutions for a class of
quasilinear elliptic boundary-value problems
∗
Kanishka Perera
Abstract
Using variational arguments we prove some nonexistence and multiplicity results for positive solutions of a class of elliptic boundary-value
problems involving the p-Laplacian and a parameter.
1
Introduction
In a recent paper, Maya and Shivaji [4] studied the existence, multiplicity, and
non-existence of positive classical solutions of the semilinear elliptic boundaryvalue problem
−∆u = λf (u) in Ω,
(1.1)
u = 0 on ∂Ω
where Ω is a smooth bounded domain in Rn , n ≥ 1, λ > 0 is a parameter, and
f is a C 1 function such that
f (0) = 0,
lim
t→∞
f (t)
= 0.
t
(1.2)
Assuming
(f1 ) f 0 (0) < 0,
(f2 ) ∃β > 0 such that f (t) < 0 for 0 < t < β and f (t) > 0 for t > β,
(f3 ) f is eventually increasing,
they showed using sub-super solutions arguments and recent results from semipositone problems that there are λ and λ such that (1.1) has no positive solution
for λ < λ and at least two positive solutions for λ ≥ λ.
In the present paper we consider the corresponding quasilinear problem
−∆p u = λf (x, u) in Ω,
u = 0 on ∂Ω
∗ Mathematics Subject Classifications: 35J20, 35J65.
Key words: p-linear p-Laplacian problems, positive solutions, non-existence,
multiplicity, variational methods.
c
2003
Southwest Texas State University.
Submitted May 29, 2002. Published January 23, 2003.
1
(1.3)
2
Multiple positive solutions
EJDE–2003/07
where ∆p u = div |∇u|p−2 ∇u is the p-Laplacian, 1 < p < ∞, λ > 0, and f is
a Carathéodory function on Ω × [0, ∞) satisfying
f (x, 0) = 0,
|f (x, t)| ≤ Ctp−1
(1.4)
for some constant C > 0. Note that when p = 2 and f is C 1 and satisfies (1.2),
the existence of the limits limt→0 f (t)/t = f 0 (0) and limt→∞ f (t)/t imply (1.4).
Using variational methods, we shall prove the following theorems.
Theorem 1.1. There is a λ such that (1.3) has no positive solution for λ < λ.
Rt
Theorem 1.2. Set F (x, t) = 0 f (x, s)ds, and assume
(F1 ) ∃δ > 0 such that F (x, t) ≤ 0 for 0 ≤ t ≤ δ,
(F2 ) ∃t0 > 0 such that F (x, t0 ) > 0,
(F3 ) lim
t→∞
F (x, t)
≤ 0 uniformly in x.
tp
Then there is a λ such that (1.3) has at least two positive solutions u1 > u2 for
λ ≥ λ.
Note that we have substantially relaxed the assumptions in [4] and therefore
our results seem to be new even in the semilinear case p = 2. More specifically,
we have let f depend on x and dropped the assumption of differentiability in
t, and replaced (f1 ), (f2 ), and (f3 ) with the much weaker assumptions (F1 ) and
(F2 ) on the primitive F . We emphasize that (F1 ) follows from (f1 ), while (f2 )
and (f3 ) together imply (F2 ), and that we make no monotonicity assumptions.
The limit in (F3 ) equals 0 in the p-sublinear case
lim
t→∞
f (x, t)
= 0 uniformly in x,
tp−1
(1.5)
in particular, in the special case considered in [4].
2
Proofs of Theorems 1.1 and 1.2
Recall that the first Dirichlet eigenvalue of −∆p is positive and is given by
R
|∇u|p
λ1 = min RΩ
(2.1)
u6=0
|u|p
Ω
(see Lindqvist [3]). If (1.3) has a positive solution u, multiplying (1.3) by u,
integrating by parts, and using (1.4) gives
Z
Z
Z
|∇u|p = λ
f (x, u)u ≤ Cλ
up ,
(2.2)
Ω
Ω
and hence λ ≥ λ1 /C by (2.1), proving Theorem 1.1.
Ω
EJDE–2003/07
Kanishka Perera
3
We will prove Theorem 1.2 using critical point theory. Set f (x, t) = 0 for
t < 0, and consider the C 1 functional
Z
Φλ (u) =
|∇u|p − λpF (x, u), u ∈ W01,p (Ω).
(2.3)
Ω
If u is a critical point of Φλ , denoting by u− the negative part of u,
Z
0 = (Φ0λ (u), u− ) =
|∇u|p−2 ∇u · ∇u− − λf (x, u)u− = ku− kp
(2.4)
Ω
shows that u ≥ 0. Furthermore, u ∈ L∞ (Ω) ∩ C 1 (Ω) by Anane [1] and
di Benedetto [2], so it follows from the Harnack inequality (Theorem 1.1 of
Trudinger [6]) that either u > 0 or u ≡ 0. Thus, nontrivial critical points of Φλ
are positive solutions of (1.3).
By (F3 ) and (1.4), there is a constant Cλ > 0 such that
λpF (x, t) ≤
λ1 p
|t| + Cλ
2
(2.5)
and hence
Z
|∇u|p −
Φλ (u) ≥
Ω
λ1 p
1
|u| − Cλ ≥ kukp − Cλ µ(Ω)
2
2
(2.6)
where µ denotes the Lebesgue measure in Rn , so Φλ is bounded from below and
coercive. This yields a global minimizer u1 since Φλ is weakly lower semicontinuous.
Lemma 2.1. There is a λ such that inf Φλ < 0, and hence u1 6= 0, for λ ≥ λ.
Proof. Taking a sufficiently large compact subset Ω0 of Ω and a function u0 ∈
W01,p (Ω) such that u0 (x) = t0 on Ω0 and 0 ≤ u0 (x) ≤ t0 on Ω \ Ω0 , where t0 is
as in (F2 ), we have
Z
Z
F (x, u0 ) ≥
F (x, t0 ) − Ctp0 µ(Ω \ Ω0 ) > 0
(2.7)
Ω
Ω0
and hence Φλ (u0 ) < 0 for λ large enough.
Now fix λ ≥ λ, let

f (x, t),
t ≤ u1 (x),
fe(x, t) =
f (x, u (x)), t > u (x),
1
1
Then consider
Z
e λ (u) =
Φ
Ω
Z
and Fe(x, t) =
|∇u|p − λpFe(x, u).
t
fe(x, s)ds.
(2.8)
0
(2.9)
4
Multiple positive solutions
EJDE–2003/07
e λ , then u ≥ 0 as before, and
If u is a critical point of Φ
e 0λ (u) − Φ0λ (u1 ), (u − u1 )+
0= Φ
Z
=
|∇u|p−2 ∇u − |∇u1 |p−2 ∇u1 · ∇(u − u1 )+
Ω
− λ fe(x, u) − f (x, u1 ) (u − u1 )+
Z
=
|∇u|p−2 ∇u − |∇u1 |p−2 ∇u1 · (∇u − ∇u1 )
Zu>u1
≥
|∇u|p−1 − |∇u1 |p−1 (|∇u| − |∇u1 |) ≥ 0
(2.10)
u>u1
implies that u ≤ u1 , so u is a solution of (1.3) in the order interval [0, u1 ]. We
e λ (u2 ) > 0 via the mountain-pass lemma,
will obtain a critical point u2 with Φ
e λ (0) = 0 > Φ
e λ (u1 ).
which would complete the proof since Φ
e λ.
Lemma 2.2. The origin is a strict local minimizer of Φ
Proof. Setting Ωu = x ∈ Ω : u(x) > min u1 (x), δ , by (2.8) and (F1 ),
Fe(x, u(x)) ≤ 0 on Ω \ Ωu , so
Z
e λ (u) ≥ kukp − λp
Φ
Fe(x, u).
(2.11)
Ωu
By (1.4), Hölder’s inequality, and Sobolev imbedding,
Z
Z
p
Fe(x, u) ≤ C
up ≤ Cµ(Ωu )1− q kukp
Ωu
(2.12)
Ωu
where q = np/(n − p) if p < n and q > p if p ≥ n, so it suffices to show that
µ(Ωu ) → 0 as kuk → 0.
Given ε > 0, take a compact subset Ωε of Ω such that µ(Ω \ Ωε ) < ε and let
Ωu,ε = Ωu ∩ Ωε . Then
Z
p
kukp ≥
up ≥ cp µ(Ωu,ε )
(2.13)
Ωu,ε
where c = min min u1 (Ωε ), δ > 0, so µ(Ωu,ε ) → 0. But, since Ωu ⊂ Ωu,ε ∪
(Ω \ Ωε ),
µ(Ωu ) < µ(Ωu,ε ) + ε,
(2.14)
and ε is arbitrary.
e λ is also coAn argument similar to the one we used for Φλ shows that Φ
e
ercive, so every Palais-Smale sequence of Φλ is bounded and hence contains a
convergent subsequence as usual. Now the mountain-pass lemma gives a critical
e λ at the level
point u2 of Φ
c := inf
max
γ∈Γ u∈γ([0,1])
e λ (u) > 0
Φ
(2.15)
where Γ = γ ∈ C([0, 1], W01,p (Ω)) : γ(0) = 0, γ(1) = u1 is the class of paths
joining the origin to u1 (see, e.g., Rabinowitz [5]).
EJDE–2003/07
Kanishka Perera
5
References
[1] A. Anane. Etude des valeurs propres et de la résonnance pour l’opérateur
p-Laplacien. PhD thesis, Université Libre de Bruxelles, 1987. C. R. Acad.
Sci. Paris Sér. I Math., 305(16):725–728, 1987.
[2] E. di Benedetto. C 1+α local regularity of weak solutions of degenerate elliptic
equations. Nonlinear Anal., 7(8):827–850, 1983.
[3] P. Lindqvist. On the equation div(|∇u|p−2 ∇u) + λ|u|p−2 u = 0. Proc.
Amer. Math. Soc., 109(1):157–164, 1990. Addendum: Proc. Amer. Math.
Soc., 116(2):583–584, 1992.
[4] C. Maya and R. Shivaji. Multiple positive solutions for a class of semilinear
elliptic boundary value problems. Nonlinear Anal., 38(4, Ser. A: Theory
Methods):497–504, 1999.
[5] P. H. Rabinowitz. Minimax methods in critical point theory with applications to differential equations. Published for the Conference Board of the
Mathematical Sciences, Washington, DC, 1986.
[6] N. Trudinger. On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math., 20:721–747, 1967.
Kanishka Perera
Department of Mathematical Sciences
Florida Institute of Technology
Melbourne, FL 32901, USA
e-mail: kperera@fit.edu http://my.fit.edu/∼kperera/