We consider semilinear elliptic equations with Dirichlet boundary conditions in a Lipschitz, possibly unbounded, domain. Under suitable assumptions on the nonlinearity, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here, uniform means that the estimate is independent of the domain. Besides of its simplicity, the main advantage of our approach is that we can remove a restrictive assumption on the nonlinearity that was imposed in a recent paper. Moreover, we can remove a non-degeneracy condition that was assumed in the latter paper

Sourdis, Christos

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On a uniform estimate for positive solutions of semilinear elliptic
equations
Christos Sourdis
Original Citation:
Sourdis, Christos
(2012)
On a uniform estimate for positive solutions of semilinear elliptic equations.
(Unpublished)
This version is available at: http://preprints.acmac.uoc.gr/137/
Available in ACMAC’s PrePrint Repository: July 2012
ACMAC’s PrePrint Repository aim is to enable open access to the scholarly output of ACMAC.
http://preprints.acmac.uoc.gr/ON A UNIFORM ESTIMATE FOR POSITIVE SOLUTIONS OF
SEMILINEAR ELLIPTIC EQUATIONS
CHRISTOS SOURDIS
Abstract. We consider the semilinear elliptic equation ∆u = W￿(u) with
Dirichlet boundary conditions in a Lipschitz, possibly unbounded, domain
Ω ⊂ Rn. Under suitable assumptions on the potential W, we deduce a condi-
tion on the size of the domain that implies the existence of a positive solution
satisfying a uniform pointwise estimate. Here uniform means that the esti-
mate is independent of Ω. Besides of its simplicity, the main advantage of our
approach is that we can remove a restrictive monotonicity assumption on W
that was imposed in the recent paper [12]. Moreover, we can remove a non-
degeneracy condition on the global minimum of W that was assumed in the
latter reference.
1. Introduction and statement of the main result
Recently, the authors of [12] considered the semilinear elliptic problem



∆u = W￿(u),x ∈ Ω,
u =0 ,x ∈ ∂Ω,
(1.1)
where Ω ⊂ Rn, n ≥ 1, is a domain with nonempty Lipschitz boundary (see for
instance [10]), under the following assumptions on the C2 function W : R → R:
(a): There exists a constant µ>0 such that
0=W(µ) <W(t),t ∈ [0,∞),t ￿= µ,
W(−t) ≥ W(t),t ∈ [0,∞);
(b): W￿(t) ≤ 0,t ∈ (0,µ);
(c): W￿￿(µ) > 0.
For a typical example of such a potential, see (1.8) below. We stress that, in the
case where the domain is unbounded, the boundary conditions in (1.1) do not refer
to u(x) → 0 as |x|→∞with x ∈ Ω.
For x ∈ Rn, ρ>0, we let
Bρ(x)={y ∈ Rn : |y − x| <ρ },B ρ = Bρ(0),
A + B = {x + y : x ∈ A, y ∈ B}, A,B ⊂ Rn,
and denote d(x,E) the Euclidean distance of the point x ∈ Rn from the set E ⊂ Rn,
and |E| the n-dimensional Lebesgue measure of E (see [10]).
The main result of [12] was the following:
Theorem 1.1. Assume Ω and W as above. There are positive constants R∗,k,K,
depending only on W and n, such that if Ω contains a closed ball of radius R∗, then
problem (1.1) has a solution u ∈ C2(Ω) ∩ C(¯ Ω) verifying
0 <u (x) <µ , x∈ Ω, (1.2)
1POSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 2
µ − u(x) ≤ Ke−kd(x,∂Ω),x ∈ Ω. (1.3)
In addition, there are r∗ ∈ (0,R ∗) and a∗ ∈ (0,µ), depending only on W and n,
such that
µ − a∗ <u (x),x ∈ ΩR∗ + Br∗, (1.4)
where
ΩR∗ = {x ∈ Ω:d(x,∂Ω) >R ∗}.
The approach of [12] to the proof of Theorem 1.1 is variational, involving the
construction of various judicious radial comparison functions on BR∗, see also [1].
We note that, once (1.4) is established, the exponential decay estimate (1.3) can also
be deduced as in Lemma 4.2 in [11], making use of the non-degeneracy condition
(c) (it holds that W￿￿(u) > 0,u∈ [µ − a∗,µ]). Moreover, an examination of the
proof of Lemma 2.1 in [12] shows that assumption (a) above can be relaxed to
(a’): There exists a constant µ>0 such that
0=W(µ) <W(t),t ∈ [0,µ),W (t) ≥ 0,t ≥ µ,
W(−t) ≥ W(t),t ∈ [0,∞).
The main purpose of this note is to show that relation (1.4) can be established
in a simple manner without assuming the monotonicity condition (b). We will
accomplish this by making use of some variational arguments that can be traced
back to [7], in the context of semilinear elliptic singular perturbation problems. In
passing, we remark that a similar monotonicity assumption to (b) also appears in
[1], in the context of variational systems of the form (1.1), where W : Rn → Rn (see
also Remark 2.4 below). Moreover, we remove the non-degeneracy condition (c).
Our result is
Theorem 1.2. Assume that Ω is as above and W ∈ C2 satisﬁes (a’). There exist
positive constants R￿ and a ∈ (0,µ), depending only on W and n, such that if Ω
contains some ball BR￿(x0), then problem (1.1) has a solution u ∈ C2(Ω) ∩ C(¯ Ω)
verifying (1.2), and
µ − a<u (x),x ∈ B R￿
2 (x0). (1.5)
In our opinion, Theorems 1.1 and 1.2 are important for the following reasons. If
we additionally assume that W is even, namely
W(−t)=W(t),t ∈ R, (1.6)
by means of these theorems, we can derive the existence of various sign-changing
entire solutions for the problem
∆u = W￿(u),x ∈ Rn, (1.7)
by ﬁrst proving existence of a positive solution is a suitable “fundamental” domain
ΩF, as above, and consecutive odd reﬂections to cover the entire space. We refer
the interested reader to the introduction of [12]. In the case where
W(t)=
1
4
(t2 − 1)2,t ∈ R, (1.8)
then (1.7) becomes the well known Allen-Cahn equation (see for instance [18]).
Assuming (1.6), then (1.1) has always the trivial solution. In this regard, the
purpose of estimate (1.5) is twofold: In the case where ΩF is bounded, it ensures
that the solution of (1.1), provided by Theorem 1.2, is nontrivial. The situation of
unbounded domains ΩF can be treated by exhausting them by an inﬁnite sequencePOSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 3
{Ωn} of bounded ones, each containing the same ball BR￿(x0), and a standard
compactness argument, making use of (1.2) together with elliptic estimates. The
fact that estimate (1.5) is independent of the domain is needed to rule out the
possibility of subsequences of the (chosen) solutions un of (1.1)n on Ωn converging,
uniformly in compact subsets of Ω, to the trivial solution of (1.1) on ΩF. Another
approach can be found in [6].
2. Proof of the main result
We will need the following lemma, which is motivated from Lemma 2 in [15], and
can be traced back to [7].
Lemma 2.1. Assume that W ∈ C2 satisﬁes condition (a’). There exist positive
constants R￿ and a ∈ (0,µ), depending only on W and n, such that any global
minimizer of the energy functional
J(v;BR)=
￿
BR
￿
1
2
|∇v|2 + W(v)
￿
dx in W
1,2
0 (BR),
satisﬁes
0 <u (x) < µ, x ∈ BR, (2.1)
and
µ − a<u (x),x ∈ B R
2 , (2.2)
provided that R ≥ R￿.
Proof. Under our assumptions on W, it is standard to show the existence of a global
minimizer u ∈ W
1,2
0 (BR) satisfying 0 ≤ u(x) ≤ µ a.e. in BR, see [12]. (The second
bound in the latter inequality can also be derived from Lemma 2.3 in [7] or Lemma
1 in [15]). Moreover, this minimizer is a smooth solution, in C2(BR) ∩ C( ¯ BR), of
∆u = W￿(u)i nBR; u = 0 on ∂BR. (2.3)
By the strong maximum principle, see for example Lemma 3.4 in [14], we deduce that
u(x) < µ, x ∈ BR, and that either u is identically equal to zero or u(x) > 0,x∈ BR
(recall that assumption (a’) implies that W￿(0) ≤ 0 and W￿(µ) = 0). Observe that
u depends on R but we have chosen not to make this apparent in the notations.
Next, adapting an argument in Section 4 in [18], we will show that u is nontrivial,
provided R is suﬃciently large. It is easy to cook up a test function, and use it as a
competitor, to show that there exists a constant C1, depending only on n and W,
such that
J(u;BR) ≤ C1Rn−1, say for R ≥ 2. (2.4)
(Plainly construct a function which interpolates smoothly from µ to 0 in a layer of
size 1 around the boundary of BR and which is identically equal to µ elsewhere).
Notice that the energy of the trivial solution is
J(0,B R)=
￿
BR
W(0)dx = C2Rn,
where C2 > 0 depends only on n, W. From the relation
C2Rn = J(0;BR) ≤ C1Rn−1,R ≥ 2,
we infer that u is certainly not identically equal to zero for
R ≥ C1C
−1
2 +2 .POSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 4
We thus conclude that (2.1) holds. (In the above calculation, we relied on the fact
that (a’) implies that W(0) > 0; in this regard, see Remark 2.2 below).
Since u is strictly positive in the ball BR, by (2.3) and the method of moving
planes [5, 13], we infer that u is radially symmetric and
ur(r) < 0,r ∈ (0,R), (2.5)
(with the obvious notation). We note that, since u is a global minimizer, the same
conclusion can be asserted as in [17]. Relation (2.4) and the positivity of W clearly
imply that ￿
¯ BR\B R
2
W(u)dx ≤ C1Rn−1,R ≥ C1C
−1
2 +2 .
So, by the mean value theorem, there exists a ξ ∈
￿R
2 ,R
￿
such that
W (u(ξ))
￿
￿ ￿ ¯ BR\B R
2
￿
￿ ￿ ≤ C1Rn−1,
i.e.,
W (u(ξ)) ≤ C3R−1,
where the positive constat C3 > 0 depends only on n and W. Now, observe that
assumption (a’) implies that
there exist δ, a > 0 such that W−1(0,δ) ∩ (0,µ)=( µ − a,µ). (2.6)
Thus, choosing
R￿ > max
￿
C3
δ
,C 1C
−1
2 +2
￿
,
then assumption (c’), (2.1), and the above relation, yield that
u(ξ) >µ− a.
Recalling that ξ ∈
￿R
2 ,R
￿
, by relation (2.5), we infer that the desired estimate (2.2)
holds.
The proof of the lemma is complete. ￿
Remark 2.1. If we additionally assume that W￿￿(0) < 0, then there exists an
explicitly computable (in terms of the principal eigenvalue of the Dirichlet Laplacian
in BR), critical radius Rc ∈ (0,R ￿) such that (2.3) admits a nontrivial positive
solution which is a global minimizer of J(·;BR) in W
1,2
0 (BR), as long as R>R c.
If we further assume that
W￿(t) ≥ W￿￿(0)t, t ≥ 0,
then (2.3) for R ∈ (0,R c) has no positive solution. These assertions can be proven
by adapting Lemma 2.1 in [9].
Remark 2.2. In the case where W(0) = 0 and Ω is Lipschitz, bounded, and star-
shaped with respect to some point, then Pohozaev’s identity easily implies that there
does not exist a nontrivial solution of (1.1) such that W(u(x)) ≥ 0,x ∈ Ω (see for
instance relation (11) in [3]).
We can now proceed to the
Proof of Theorem 1.2: Once Lemma 2.1 is established, the proof of Theorem
1.2 proceeds in a rather standard way. We will adapt an argument from the proof ofPOSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS 5
Theorem 11 in [8], and prove existence of the desired solution to (1.1) by the method
of upper solutions. We use ¯ u ≡ µ as an upper solution (recall that W￿(µ) = 0), and
u =



uR￿(x),x ∈ BR￿(x0),
0,x ∈ Ω\BR￿(x0),
where uR￿ is as in Lemma 2.1 but centered at x0, as a lower solution (here we used
that W￿(0) ≤ 0 and Proposition 1 in [4] to make sure that u is a lower solution). Note
that u(x) < ¯ u(x),x ∈ Ω. In the case where Ω is bounded, it follows immediately
from the method of monotone iterations, see Theorem 2.3.1 in [19], that there exists
a solution u of (1.1) such that
u(x) <u (x) < ¯ u(x),x ∈ Ω.
The same property also holds in the case where Ω is unbounded, by exhausting
it with a sequence of bounded domains, see Theorem 2.10 in [16] (also recall our
discussion following the statement of Theorem 1.2). The validity of estimates (1.2)
and (1.5) follows at once, in view of (2.1), (2.2).
The proof of the theorem is complete. ￿
Remark 2.3. The same assertions of Theorem 1.2 continue to hold if assumption
(a’) is replaced by the weaker one
(a”): There exists a constant µ>0 such that
0=W(µ) ≤ W(t),t ∈ [0,∞),
there exist δ, a > 0 such that W−1(0,δ) ∩ [0,µ) ⊆ (µ − a,µ),
W(−t) ≥ W(t),t ∈ [0,∞),
which allows W to have zeros in any interval containing µ.
Remark 2.4. We recently found the paper [2], where it is stated that G. Fusco, in
work in progress, has been able to remove the corresponding monotonicity assump-
tion to (b) from the vector-valued Allen-Cahn type equation that was treated in [1].
After the ﬁrst version of the current paper was completed, I was informed by G.
Fusco that himself, F. Leonetti and C. Pignotti are working in a paper where, using
the same technique developed for the vector case, they also extend the result in [12]
to general potentials.
Acknowledgment. The research leading to these results has received funding
from the European Union’s Seventh Framework Programme (FP7-REGPOT-2009-
1) under grant agreement no 245749.
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Department of Applied Mathematics, University of Crete, GR–714 09 Heraklion,
Crete, Greece.
E-mail address: csourdis@tem.uoc.gr

On a uniform estimate for positive solutions of semilinear elliptic equations

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On a uniform estimate for positive solutions of semilinear elliptic equations

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