Boundary Value Problems
Volume 2010, Article ID 139126,22pages doi:10.1155/2010/139126
Research Article
Sign-Changing and Extremal Constant-Sign
Solutions of Nonlinear Elliptic Neumann Boundary Value Problems
Patrick Winkert
Institut f ¨ur Mathematik, Technische Universit¨at Berlin, Straße des 17.
Juni 136, 10623 Berlin, Germany
Correspondence should be addressed to Patrick Winkert,winkert@math.tu-berlin.de Received 23 November 2009; Accepted 15 June 2010
Academic Editor: Pavel Dr´abek
Copyrightq2010 Patrick Winkert. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Our aim is the study of a class of nonlinear elliptic problems under Neumann conditions involving thep-Laplacian. We prove the existence of at least three nontrivial solutions, which means that we get two extremal constant-sign solutions and one sign-changing solution by using truncation techniques and comparison principles for nonlinear elliptic differential inequalities. We also apply the properties of the Fu˘cik spectrum of the p-Laplacian and, in particular, we make use of variational and topological tools, for example, critical point theory, Mountain-Pass Theorem, and the Second Deformation Lemma.
1. Introduction
LetΩ⊂ RN be a bounded domain with Lipschitz boundary∂Ω. We consider the following nonlinear elliptic boundary value problem. Findu∈W1,pΩ\ {0}and constantsa∈R, b∈R such that
−Δpufx, u− |u|p−2u inΩ,
|∇u|p−2∂u
∂ν aup−1−b u−p−1
gx, u on∂Ω, 1.1
where −Δpu −div|∇u|p−2∇u,1 < p < ∞, is the negativep-Laplacian, ∂u/∂ν denotes the outer normal derivative of u, andu max{u,0} as well as u− max{−u,0} are the positive and negative parts ofu, respectively. The nonlinearities f : Ω×R → R and g :
∂Ω×R → Rare some Carath´eodory functions which are bounded on bounded sets. For
reasons of simplification, we drop the notation for the trace operatorγ :W1,pΩ → Lp∂Ω which is used on the functions defined on the boundary∂Ω.
The motivation of our study is a recent paper of the author in 1in which problem 1.1 was treated in case a b. We extend this approach and prove the existence of multiple solutions for the more general problem 1.1. To be precise, the existence of a smallest positive solution, a greatest negative solution, as well as a sign-changing solution of problem1.1is proved by using variational and topological tools, for example, critical point theory, Mountain-Pass Theorem, and the Second Deformation Lemma. Additionally, the Fu˘cik spectrum for thep-Laplacian takes an important part in our treatments.
Neumann boundary value problems in the form of1.1arise in different areas of pure and applied mathematics, for example, in the theory of quasiregular and quasiconformal mappings in Riemannian manifolds with boundary see 2, 3, in the study of optimal constants for the Sobolev trace embedding see 4–7, or at non-Newtonian fluids, flow through porus media, nonlinear elasticity, reaction diffusion problems, glaciology, and so on see 8–11.
The existence of multiple solutions for Neumann problems like those in the form of 1.1has been studied by a number of authors, such as, for example, the authors of 12–15, and homogeneous Neumann boundary value problems were considered in 16,17and 15, respectively. Analogous results for the Dirichlet problem have been recently obtained in 18–
21. Further references can also be found in the bibliography of 1.
In our consideration, the nonlinearities f and g only need to be Carath´eodory functions which are bounded on bounded sets whereby their growth does not need to be necessarily polynomial. The novelty of our paper is the fact that we do not need differentiability, polynomial growth, or some integral conditions on the mappingsfandg.
First, we have to make an analysis of the associated spectrum of 1.1. The Fu˘cik spectrum for thep-Laplacian with a nonlinear boundary condition is defined as the setΣp
ofa, b∈R×Rsuch that
−Δpu−|u|p−2u inΩ,
|∇u|p−2∂u
∂ν aup−1−b u−p−1
on∂Ω, 1.2
has a nontrivial solution. In view of the identity
|u|p−2u|u|p−2
u−u−
up−1− u−p−1
, 1.3
we see at once that forabλproblem1.2reduces to the Steklov eigenvalue problem
−Δpu−|u|p−2u inΩ,
|∇u|p−2∂u
∂ν λ|u|p−2u on∂Ω. 1.4
We say thatλis an eigenvalue if1.4has nontrivial solutions. The first eigenvalueλ1 >0 is isolated and simple and has a first eigenfunctionϕ1which is strictly positive inΩ see 22.
Furthermore, one can show thatϕ1belongs toL∞Ω cf., 23, Lemma 5.6 and Theorem 4.3
or 24, Theorem 4.1, and along with the results of Lieberman in 25, Theorem 2it holds thatϕ1 ∈ C1,αΩ. This fact combined with ϕ1x > 0 inΩyieldsϕ1 ∈ intC1Ω, where intC1Ωdenotes the interior of the positive coneC1Ω {u∈C1Ω:ux≥ 0, ∀x∈ Ω}in the Banach spaceC1Ω, given by
int C1
Ω
u∈C1 Ω
:ux>0, ∀x∈Ω
. 1.5
Let us recall some properties of the Fu˘cik spectrum. Ifλis an eigenvalue of1.4, then the pointλ, λbelongs toΣp. Since the first eigenfunction of1.4is positive,Σpclearly contains the two linesR× {λ1} and{λ1} ×R. A first nontrivial curveC inΣp through λ2, λ2was constructed and variationally characterized by a mountain-pass procedure by Mart´ınez and Rossi 26. This yields the existence of a continuous path in {u ∈ W1,pΩ : Ia,bu <
0, u Lp∂Ω 1}joining−ϕ1andϕ1provided thata, bis above the curveC. The functional Ia,bonW1,pΩis given by
Ia,bu
Ω
|∇u|p|u|p dx−
∂Ω
aupb u−p
dσ. 1.6
Due to the fact thatλ2belongs toC, there exists a variational characterization of the second eigenvalue of1.4meaning thatλ2can be represented as
λ2 inf
g∈Π max
u∈g −1,1 Ω
|∇u|p|u|p
dx, 1.7
where
Π
g ∈C −1,1, S|g−1 −ϕ1, g1 ϕ1 , S
u∈W1,pΩ:
∂Ω|u|pdσ1
. 1.8
The proof of this result is given in 26.
An important part in our considerations takes the following Neumann boundary value problem defined by
−Δpu−ς|u|p−2u1 in Ω,
|∇u|p−2∂u
∂ν 1 on∂Ω, 1.9
whereς >1 is a constant. As pointed out in 1, there exists a unique solutione∈intC1Ω of problem 1.9 which is required for the construction of sub- and supersolutions of problem1.1.
2. Notations and Hypotheses
Now, we impose the following conditions on the nonlinearitiesfandgin problem1.1. The mapsf :Ω×R → Randg :∂Ω×R → Rare Carath´eodory functions, which means that they are measurable in the first argument and continuous in the second one. Furthermore, we suppose the following assumptions.
H1 f1
slim→0
fx, s
|s|p−2s
0, uniformly with respect to a.a.x∈Ω. 2.1
f2
|s| → ∞lim
fx, s
|s|p−2s
−∞, uniformly with respect to a.a. x∈Ω. 2.2
f3fis bounded on bounded sets.
f4There existsδf >0 such that fx, s
|s|p−2s ≥0, for all 0<|s| ≤δf for a.a.x∈Ω. 2.3 H2 g1
slim→0
gx, s
|s|p−2s
0, uniformly with respect to a.a. x∈∂Ω. 2.4
g2
|s| → ∞lim
gx, s
|s|p−2s
−∞, uniformly with respect to a.a. x∈∂Ω. 2.5
g3gis bounded on bounded sets.
g4gsatisfies the condition
gx1, s1−gx2, s2≤L
|x1−x2|α|s1−s2|α
, 2.6
for all pairsx1, s1,x2, s2in∂Ω× −M0, M0, whereM0is a positive constant and α∈0,1.
H3Let a, b ∈ R2 be above the first nontrivial curve C of the Fu˘cik spectrum constructed in 26 seeFigure 1.
Note that H2g4 implies that the function x, s → a|s|p−1 − b|s|p−1 gx, s fulfills a condition as in H2g4, too. Moreover, we see at once that u 0 is a trivial solution of problem 1.1 because of the conditions H1f1 and H2g1, which guarantees that fx,0 gx,0 0. It should be noted that hypothesisH3includes thata, b > λ1see 26 orFigure 1.
Example 2.1. Let the functionsf:Ω×R → Randg:∂Ω×R → Rbe given by
fx, s
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
|s|p−2s1 s1e−s ifs≤ −1, sgns|s|p
2 |s−1coss1|s1 if −1≤s≤1, sp−1e1−s− |x|s−1sp−1es ifs≥1,
gx, s
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
|s|p−2s
s1es1
ifs≤ −1,
|s|p−1ses2−1√
|x| if −1≤s≤1,
sp−1cos1−s 1−ses ifs≥1.
2.7
Then all conditions inH1f1–f4andH2g1–g4are fulfilled.
Definition 2.2. A functionu∈W1,pΩis called a weak solution of1.1if the following holds:
Ω|∇u|p−2∇u∇ϕ dx
Ω
fx, u− |u|p−2u ϕ dx ∂Ω
aup−1−b u−p−1
gx, u
ϕ dσ, ∀ϕ∈W1,pΩ.
2.8
Definition 2.3. A functionu∈W1,pΩis called a subsolution of1.1if the following holds:
Ω
∇up−2∇u∇ϕ dx≤
Ω
f
x, u
−up−2u ϕ dx ∂Ω
a
up−1
−b u−p−1
g x, u
ϕ dσ, ∀ϕ∈W1,pΩ. 2.9
Definition 2.4. A functionu∈W1,pΩis called a supersolution of1.1if the following holds:
Ω|∇u|p−2∇u∇ϕ dx≥
Ω
fx, u− |u|p−2u ϕ dx ∂Ω
a up−1
−b u−p−1
gx, u
ϕ dσ, ∀ϕ∈W1,pΩ.
2.10
(λ1,λ1)
(λ2,λ2) (a, b)
C b
a Figure 1: Fu˘cik spectrum
We recall that W1,pΩ : {ϕ ∈ W1,pΩ : ϕ ≥ 0} denotes all nonnegative functions of W1,pΩ. Furthermore, for functionsu, v, w ∈ W1,pΩsatisfyingv ≤ u ≤ w, we have the relationγv≤ γu ≤γw, whereγ :W1,pΩ → Lp∂Ωstands for the well-known trace operator.
3. Extremal Constant-Sign Solutions
For the rest of the paper we denote byϕ1∈intC1Ωthe first eigenfunction of the Steklov eigenvalue problem1.4corresponding to its first eigenvalueλ1. Furthermore, the function e ∈ intC1Ωstands for the unique solution of the auxiliary Neumann boundary value problem defined in1.9. Our first lemma reads as follows.
Lemma 3.1. Let conditions (H1)-(H2) be satisfied and let a, b > λ1. Then there exist constants ϑa, ϑb > 0 such that ϑae and −ϑbe are a positive supersolution and a negative subsolution, respectively, of problem1.1.
Proof. Settinguϑaewith a positive constantϑato be specified and considering the auxiliary problem1.9, we obtain
Ω|∇ϑae|p−2∇ϑae∇ϕ dx−ς
Ωϑaep−1ϕ dx
Ωϑp−1a ϕ dx ∂Ωϑap−1ϕ dσ, ∀ϕ∈W1,pΩ.
3.1
In order to satisfyDefinition 2.4foru ϑae, we have to show that the following inequality holds true meaning:
Ω
ϑp−1a −cϑ aep−1−fx, ϑae ϕ dx
∂Ω
ϑap−1−aϑaep−1−gx, ϑae
ϕ dσ ≥0, 3.2
wherecς−1 withc >0. ConditionH1f2implies the existence ofsς>0 such that fx, s
sp−1 <−c, for a.a. x∈Ωand alls > sς, 3.3 and due toH1f3, we have
−fx, s−cs p−1≤fx, scs p−1≤cς, for a.a. x∈Ωand all s∈ 0, sς
. 3.4
Hence, we get
fx, s≤ −csp−1cς, for a.a. x∈Ωand all s≥0. 3.5
Because of hypothesisH2g2, there existssa>0 such that gx, s
sp−1 <−a, for a.a. x∈∂Ωand alls > sa, 3.6 and thanks to conditionH2g3, we find a constantca>0 such that
−gx, s−asp−1≤gx, sasp−1≤ca, for a.a. x∈∂Ωand all s∈ 0, sa. 3.7
Finally, we have
gx, s≤ −asp−1ca, for a.a. x∈∂Ωand alls≥0. 3.8
Using the inequality in3.5to the first integral in3.2yields
Ω
ϑp−1a −cϑ aep−1−fx, ϑae ϕ dx≥
Ω
ϑap−1−cϑaep−1cϑ aep−1−cς ϕ dx
Ω
ϑap−1−cς
ϕ dx,
3.9
which proves its nonnegativity ifϑa ≥c1/p−1ς . Applying3.8to the second integral in3.2 ensures that
∂Ω
ϑp−1a −aϑaep−1−gx, ϑae ϕ dx≥
∂Ω
ϑp−1a −aϑaep−1aϑaep−1−ca ϕ dx
≥ ∂Ω
ϑp−1a −ca ϕ dx.
3.10
We take ϑa : max{c1/p−1ς , c1/p−1a } to verify that both integrals in 3.2 are nonnegative.
Hence, the functionu ϑaeis in fact a positive supersolution of problem1.1. In a similar way one proves that u −ϑbe is a negative subsolution, where we apply the following estimates:
fx, s≥ −csp−1−cς, for a.a. x∈Ωand alls≤0,
gx, s≥ −bsp−1−cb, for a.a. x∈∂Ωand all s≤0. 3.11
This completes the proof.
The next two lemmas show that constant multipliers of ϕ1 may be sub- and supersolution of1.1. More precisely, we have the following result.
Lemma 3.2. Assume that (H1)-(H2) are satisfied. Ifa > λ1, then forε >0 sufficiently small and any b∈Rthe functionεϕ1is a positive subsolution of problem1.1.
Proof. The Steklov eigenvalue problem1.4implies for allϕ∈W1,pΩ,
Ω
∇
εϕ1p−2∇ εϕ1
∇ϕ dx−
Ω
εϕ1p−1 ϕ dx
∂Ωλ1 εϕ1p−1
ϕ dσ. 3.12
Definition 2.3is satisfied foruεϕ1provided that the inequality
Ω−f x, εϕ1
ϕ dx
∂Ω
λ1−a εϕ1
p−1
−g x, εϕ1
ϕ dσ ≤0 3.13
is valid for all ϕ ∈ W1,pΩ. With regard to hypothesis H1f4, we obtain, for ε ∈ 0, δf/ ϕ1 ∞,
Ω−f x, εϕ1
ϕ dx
Ω−f x, εϕ1
εϕ1
p−1 εϕ1p−1
ϕ dx≤0, 3.14
where · ∞denotes the usual supremum norm. Thanks to conditionH2g1, there exists
a numberδa>0 such that gx, s
|s|p−1 < a−λ1, for a.a. x∈∂Ωand all 0<|s| ≤δa. 3.15
In caseε∈0, δa/ ϕ1 ∞,we get
∂Ω
λ1−a εϕ1
p−1
−g x, εϕ1
ϕ dσ ≤
∂Ω
λ1−ag x, εϕ εϕ1p−1
εϕ1
p−1 ϕ dσ
<
∂Ωλ1−aa−λ1 εϕ1p−1
ϕ dσ 0.
3.16
Selecting 0< ε≤min{δf/ ϕ1 ∞, δλ/ ϕ1 ∞}guarantees thatuεϕ1is a positive subsolution.
The following lemma on the existence of a negative supersolution can be proved in a similar way.
Lemma 3.3. Assume that (H1)-(H2) are satisfied. Ifb > λ1, then forε >0 sufficiently small and any a∈Rthe function−εϕ1is a negative supersolution of problem1.1.
Concerning Lemmas3.1–3.3, we obtain a positive pair εϕ1, ϑaeand a negative pair −ϑbe,−εϕ1of sub- and supersolutions of problem1.1provided thatε > 0 is sufficiently small.
In the next step we are going to prove the regularity of solutions of problem 1.1 belonging to the order intervals 0, ϑaeand −ϑbe,0, respectively. We also point out that uu0 is both a subsolution and a supersolution because of the hypothesesH1f1and H2g1.
Lemma 3.4. Assume (H1)-(H2) and leta, b > λ1. Ifu∈ 0, ϑae(resp.,u∈ −ϑbe,0) is a solution of problem1.1satisfyingu /≡0 inΩ, then it holds thatu∈intC1Ω(resp.,u∈ −intC1Ω).
Proof. We just show the first case; the other case acts in the same way. Letube a solution of 1.1 satisfying 0 ≤ u ≤ ϑae. We directly obtain the L∞-boundedness, and, hence, the regularity results of Lieberman in 25, Theorem 2imply thatu ∈ C1,αΩwith α ∈ 0,1.
Due to assumptionsH1f1,H1f3,H2g1, andH2g3, we obtain the existence of constantscf, cg>0 satisfying
fx, s≤cfsp−1, for a.a. x∈Ωand all 0≤s≤ϑa e ∞, gx, s≤cgsp−1, for a.a. x∈∂Ωand all 0≤s≤ϑa e ∞.
3.17
Applying3.17to1.1provides
Δpu≤cu p−1, a.e.inΩ, 3.18 where c is a positive constant. We set βs csp−1 for all s > 0 and use V´azquez’s strong maximum principle cf., 27which is possible because
01/sβs1/pds ∞.
Hence, it holds that u > 0 in Ω. Finally, we suppose the existence of x0 ∈ ∂Ω satisfying ux0 0. Applying again the maximum principle yields∂u/∂νx0<0. However, because of gx0, ux0 gx0,0 0 in combination with the Neumann condition in 1.1, we get ∂u/∂νx0 0. This is a contradiction and, hence, u > 0 in Ω, which proves that u∈intC1Ω.
The main result in this section about the existence of extremal constant-sign solutions is given in the following theorem.
Theorem 3.5. Assume (H1)-(H2). For everya > λ1 and b ∈ R, there exists a smallest positive solutionuua∈intC1Ωof 1.1in the order interval 0, ϑaewith the constantϑaas in Lemma 3.1. For everyb > λ1anda∈R,there exists a greatest solutionu−u−b∈ −intC1Ω in the order interval −ϑbe,0with the constantϑbas inLemma 3.1.
Proof. Leta > λ1. Lemmas3.1and3.2guarantee thatuεϕ1 ∈intC1Ωis a subsolution of problem1.1andu ϑae ∈intC1Ωis a supersolution of problem1.1. Moreover, we chooseε > 0 sufficiently small such that εϕ1 ≤ ϑae. Applying the method of sub- and supersolutionsee 28corresponding to the order interval εϕ1, ϑaeprovides the existence of a smallest positive solutionuε uελof problem1.1fulfillingεϕ1 ≤uε ≤ϑae. In view of Lemma 3.4, we haveuε ∈ intC1Ω. Hence, for every positive integer nsufficiently large, there exists a smallest solutionun ∈intC1Ωof problem1.1in the order interval 1/nϕ1, ϑae. We obtain
un↓u pointwise, 3.19
with some functionu:Ω → Rsatisfying 0≤u≤ϑae.
Claim 1. uis a solution of problem1.1.
Asun ∈ 1/nϕ1, ϑaeandγun∈ γ1/nϕ1, γϑae, we obtain the boundedness ofun inLpΩ andLp∂Ω, respectively. Definition 2.2holds, in particular, foru un and ϕun,which results in
∇un p LpΩ≤
Ω
fx, unundx un p
LpΩa un p Lp∂Ω
Ω
gx, unundσ
≤a1 un LpΩ un p
LpΩa un p
Lp∂Ωa2 un Lp∂Ω
≤a3,
3.20
with some positive constantsai, i1, . . . ,3 independent ofn. Consequently,unis bounded in W1,pΩ,and due to the reflexivity ofW1,pΩ, 1< p <∞,we obtain the existence of a weakly
convergent subsequence of un. Because of the compact embedding W1,pΩ → LpΩ, the monotony ofun, and the compactness of the trace operatorγ, we get for the entire sequence un
un u inW1,pΩ,
un−→u inLpΩand for a.a. x∈Ω, un−→u inLp∂Ωand for a.a. x∈∂Ω.
3.21
Sinceunsolves problem1.1, one obtains, for allϕ∈W1,pΩ,
Ω|∇un|p−2∇un∇ϕ dx
Ω
fx, un−up−1n ϕ dx
∂Ω
aup−1n gx, un
ϕ dσ. 3.22
Settingϕun−u∈W1,pΩin3.22results in
Ω|∇un|p−2∇un∇un−udx Ω
fx, un−up−1n
un−udx
∂Ω
aup−1n gx, un
un−udσ.
3.23
Using3.21and the hypothesesH1f3as well asH2g3yields
lim sup
n→ ∞ Ω|∇un|p−2∇un∇un−udx≤0, 3.24
which provides, by theS-property of−ΔponW1,pΩalong with3.21,
un−→u inW1,pΩ. 3.25
The uniform boundedness of the sequenceunin conjunction with the strong convergence in3.25and conditionsH1f3as well asH2g3admit us to pass to the limit in3.22.
This shows thatuis a solution of problem1.1.
Claim 2. One hasu∈intC1Ω.
In order to applyLemma 3.4, we have to prove thatu/≡0. Let us assume that this assertion is not valid meaning thatu ≡0. From3.19it follows that
unx↓0 ∀x∈Ω. 3.26
We set
un un
un W1,pΩ ∀n. 3.27
It is clear that the sequenceun is bounded in W1,pΩ,which ensures the existence of a weakly convergent subsequence ofun, denoted again byun, such that
unu inW1,pΩ,
un−→u inLpΩand for a.a. x∈Ω,
un−→u inLp∂Ωand for a.a. x∈∂Ω,
3.28
with some functionu:Ω → Rbelonging toW1,pΩ. In addition, we may suppose that there are functionsz1∈LpΩ, z2∈Lp∂Ωsuch that
|unx| ≤z1x for a.a. x∈Ω,
|unx| ≤z2x for a.a. x∈∂Ω. 3.29
With the aid of3.22, we obtain forunthe following variational equation:
Ω|∇un|p−2∇un∇ϕ dx
Ω
fx, un
up−1n up−1n −up−1n
ϕ dx
∂Ωaup−1n ϕ dσ ∂Ω
gx, un
up−1n up−1n ϕ dσ, ∀ϕ∈W1,pΩ.
3.30
We selectϕun−u∈W1,pΩin the last equality to get
Ω|∇un|p−2∇un∇un−udx
Ω
fx, un
up−1n up−1n −up−1n
un−udx
∂Ωaup−1n un−udσ
∂Ω
gx, un
up−1n up−1n un−udσ.
3.31
Making use of3.17in combination with3.29results in fx, unx
up−1n x up−1n x|unx−ux| ≤ cfz1xp−1z1x |ux|, 3.32 and, respectively,
gx, unx
up−1n x up−1n x|unx−ux| ≤ cgz2xp−1z2x |ux|. 3.33
We see at once that the right-hand sides of 3.32and 3.33belong to L1Ω andL1∂Ω, respectively, which allows us to apply Lebesgue’s dominated convergence theorem. This fact and the convergence properties in3.28show that
nlim→ ∞ Ω
fx, un
up−1n up−1n un−udx 0,
nlim→ ∞ ∂Ω
gx, un
up−1n up−1n un−udσ 0.
3.34
From3.28,3.31, and3.34we infer that
lim sup
n→ ∞ Ω|∇un|p−2∇un∇un−udx 0, 3.35 and theS-property of−Δpcorresponding toW1,pΩimplies that
un−→u inW1,pΩ. 3.36
Remark that u W1,pΩ 1, which means thatu /≡0. Applying3.26and3.36along with conditionsH1f1,H2g1to3.30provides
Ω|∇u|p−2∇u∇ϕ dx−
Ωup−1ϕ dx
∂Ωaup−1ϕ dσ, ∀ϕ∈W1,pΩ. 3.37 The equation above is the weak formulation of the Steklov eigenvalue problem in1.4where
u≥ 0 is the eigenfunction with respect to the eigenvaluea > λ1. Asu ≥0 is nonnegative in Ω, we get a contradiction to the results of Mart´ınez and Rossi in 22, Lemma 2.4becauseu must change sign on∂Ω. Hence,u/≡0. ApplyingLemma 3.4yieldsu∈intC1Ω. Claim 3. u∈intC1Ωis the smallest positive solution of1.1in 0, ϑae.
Letu ∈ W1,pΩbe a positive solution of1.1satisfying 0 ≤ u ≤ ϑae.Lemma 3.4 immediately implies that u ∈ intC1Ω. Then there exists an integernsufficiently large such thatu∈ 1/nϕ1, ϑae. However, we already know thatun is the smallest solution of 1.1in 1/nϕ1, ϑaewhich yieldsun≤u. Passing to the limit proves thatu≤u. Hence,u must be the smallest positive solution of1.1. The existence of the greatest negative solution of1.1within −ϑbe,0can be proved similarly. This completes the proof of the theorem.
4. Variational Characterization of Extremal Solutions
Theorem 3.5ensures the existence of extremal positive and negative solutions of 1.1 for all a > λ1 and b > λ1 denoted by u ua ∈ intC1Ω and u− u−b ∈
−intC1Ω, respectively. Now, we introduce truncation functionsT, T−, T0 :Ω×R → R
andT∂Ω, T−∂Ω, T0∂Ω:∂Ω×R → Ras follows:
Tx, s
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
0 ifs≤0
s if 0< s < ux ux ifs≥ux
, T∂Ωx, s
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
0 ifs≤0
s if 0< s < ux ux ifs≥ux
T−x, s
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
u−x ifs≤u−x s ifu−x< s <0 0 ifs≥0
, T−∂Ωx, s
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
u−x ifs≤u−x s ifu−x< s <0 0 ifs≥0
T0x, s
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
u−x if s≤u−x
s if u−x< s < ux ux if s≥ux
, T0∂Ωx, s
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
u−x ifs≤u−x
s ifu−x< s < ux ux ifs≥ux
4.1
For u ∈ W1,pΩ the truncation operators on∂Ω apply to the corresponding traces γu.
We just write for simplificationT∂Ωx, u, T∂Ωx, u, T∂Ωx, uwithout γ. Furthermore, the truncation operators are continuous, uniformly bounded, and Lipschitz continuous with respect to the second argument. By means of these truncations, we define the following associated functionals given by
Eu 1 p
∇u pLpΩ u pLpΩ
−
Ω ux 0
fx, Tx, sds dx
− ∂Ω ux 0
aT∂Ωx, sp−1g
x, T∂Ωx, s ds dσ,
E−u 1 p
∇u pLpΩ u pLpΩ
−
Ω ux 0
fx, T−x, sds dx
∂Ω ux 0
bT−∂Ωx, sp−1−g
x, T−∂Ωx, s ds dσ,
E0u 1 p
∇u pLpΩ u pLpΩ
−
Ω ux 0
fx, T0x, sds dx
− ∂Ω ux 0
aT∂Ωx, sp−1−bT−∂Ωx, sp−1g
x, T0∂Ωx, s ds dσ,
4.2
which are well defined and belong toC1W1,pΩ. Due to the truncations, one can easily show that these functionals are coercive and weakly lower semicontinuous, which implies that their global minimizers exist. Moreover, they also satisfy the Palais-Smale condition.
Lemma 4.1. Letuandu−be the extremal constant-sign solutions of 1.1. Then the following hold.
iA critical pointv∈W1,pΩofEis a nonnegative solution of1.1satisfying 0≤v≤u. iiA critical pointv∈W1,pΩofE−is a nonpositive solution of 1.1satisfyingu−≤v≤0.
iiiA critical pointv∈W1,pΩofE0is a solution of 1.1satisfyingu− ≤v≤u. Proof. Letvbe a critical point ofE0meaningE0v 0. We have for allϕ∈W1,pΩ
Ω|∇v|p−2∇v∇ϕ dx
Ω
fx, T0x, v− |v|p−2v ϕ dx
∂ΩaT∂Ωx, vp−1ϕ dσ ∂Ω
−bT−∂Ωx, vp−1g
x, T0∂Ωx, v ϕ dσ.
4.3
Asuis a positive solution of1.1, it satisfies
Ω|∇u|p−2∇u∇ϕ dx
Ω
fx, u−up−1 ϕ dx ∂Ω
aup−1 gx, u
ϕ dσ, ∀ϕ∈W1,pΩ.
4.4
Subtracting4.4from4.3and settingϕ v−u∈W1,pΩprovide
Ω
|∇v|p−2∇v− |∇u|p−2∇u
∇v−udx
Ω
|v|p−2v−up−1
v−udx Ω
fx, T0x, v−fx, u
v−udx ∂Ω
aT∂Ωx, vp−1−bT−∂Ωx, vp−1−aup−1
v−udσ ∂Ω
g
x, T0∂Ωx, v
−gx, u
v−udσ.
4.5
Based on the definition of the truncation operators, we see that the right-hand side of the equality above is equal to zero. On the other hand, the integrals on the left-hand side are strictly positive in casev > z,which is a contradiction. Thus, we getv−u0 and, hence, v≤u. The proof forv≥u−acts in a similar way which shows thatT0x, v v, T∂Ωx, v v, and T−∂Ωx, v v−, and therefore,vis a solution of 1.1 satisfyingu− ≤ v ≤ u. The statements iniandiican be shown in the same way.
An important tool in our considerations is the relation between local C1Ω- minimizers and localW1,pΩ-minimizers forC1-functionals. The fact is that every localC1- minimizer ofE0 is a localW1,pΩ-minimizer ofE0which was proved in similar form in 1, Proposition 5.3. This result reads as follows.
Proposition 4.2. Ifz0∈W1,pΩis a localC1Ω-minimizer ofE0meaning that there existsr1>0 such that
E0z0≤E0z0h ∀h∈C1 Ω
with h C1Ω≤r1, 4.6
thenz0is a local minimizer ofE0inW1,pΩmeaning that there existsr2>0 such that
E0z0≤E0z0h ∀h∈W1,pΩwith h W1,pΩ≤r2. 4.7 We also refer to a recent papersee 29in which the proposition above was extended to the more general case of nonsmooth functionals. With the aid ofProposition 4.2, we can formulate the next lemma about the existence of local and global minimizers with respect to the functionalsE, E−, andE0.
Lemma 4.3. Leta > λ1 andb > λ1. Then the extremal positive solutionu of 1.1is the unique global minimizer of the functionalE, and the extremal negative solutionu− of 1.1is the unique global minimizer of the functionalE−. In addition, bothuandu−are local minimizers of the functional E0.
Proof. AsE :W1,pΩ → Ris coercive and weakly sequentially lower semicontinuous, its global minimizerv ∈ W1,pΩexists meaning thatv is a critical point ofE. Concerning Lemma 4.1, we know thatvis a nonnegative solution of1.1satisfying 0≤v≤u. Due to conditionH2g1, there exists a numberδa>0 such that
gx, s≤a−λ1sp−1, ∀s: 0< s≤δa. 4.8 Choosingε <min{δf/ ϕ1 ∞, δa/ ϕ1 ∞}and applying assumptionH1f4, inequality4.8 along with the Steklov eigenvalue problem in1.4implies that
E εϕ1
−
Ω εϕ1x 0
fx, sds dxλ1−a
p εpϕ1p
Lp∂Ω−
∂Ω εϕ1x 0
gx, sds dσ
< λ1−a p εpϕ1
Lp∂Ω
∂Ω εϕ1x 0
a−λ1sp−1ds dσ 0.
4.9
From the calculations above, we see at once thatEv < 0, which means thatv/0. This allows us to apply Lemma 3.4getting v ∈ intC1Ω. Since u is the smallest positive solution of1.1in 0, ϑaefulfilling 0≤v≤u, it must hold thatvu, which proves that uis the unique global minimizer ofE. The same considerations show thatu−is the unique global minimizer ofE−. In order to complete the proof, we are going to show thatuandu− are local minimizers of the functionalE0as well. The extremal positive solutionubelongs to intC1Ω, which means that there is a neighborhoodVuofuin the spaceC1Ωsatisfying Vu ⊂C1Ω. Therefore,E E0onVuproves thatu is a local minimizer ofE0onC1Ω.
ApplyingProposition 4.2yields thatuis also a localW1,pΩ-minimizer ofE0. Similarly, we see thatu−is a local minimizer ofE0, which completes the proof.
Lemma 4.4. The functionalE0 : W1,pΩ → Rhas a global minimizerv0 which is a nontrivial solution of 1.1satisfyingu−≤v0≤u.
Proof. As we know, the functionalE0 : W1,pΩ → Ris coercive and weakly sequentially lower semicontinuous. Hence, it has a global minimizerv0. More precisely,v0 is a critical point ofE0which is a solution of1.1satisfyingu−≤v0 ≤useeLemma 4.1. The fact that E0u Eu < 0see the proof ofLemma 4.3proves thatv0is nontrivial meaning that v0/0.
5. Existence of Sign-Changing Solutions
The main result in this section about the existence of a nontrivial solution of problem1.1 reads as follows.
Theorem 5.1. Under hypotheses (H1)–(H3), problem1.1has a nontrivial sign-changing solution u0∈C1Ω.
Proof. In view ofLemma 4.4, the existence of a global minimizerv0∈W1,pΩofE0satisfying v0/0 has been proved. This means that v0 is a nontrivial solution of 1.1 belonging to u−, u. Ifv0/u−andv0/u, thenu0:v0must be a sign-changing solution becauseu−is the greatest negative solution anduis the smallest positive solution of1.1, which proves the theorem in this case. We still have to show the theorem in case that eitherv0u−orv0 u. Let us only consider the casev0 ubecause the casev0 u− can be proved similarly. The functionu−is a local minimizer ofE0. Without loss of generality, we suppose thatu−is a strict local minimizer; otherwise, we would obtain infinitely many critical pointsvofE0which are sign-changing solutions due tou− ≤v≤uand the extremality of the solutionsu−, u. Under these assumptions, there exists aρ∈0, u−u− W1,pΩsuch that
E0u≤E0u−<inf
E0u:u∈∂Bρu−
, 5.1
where∂Bρ {u ∈ W1,pΩ : u−u− W1,pΩ ρ}. Now, we may apply the Mountain-Pass Theorem toE0cf., 30thanks to5.1along with the fact thatE0satisfies the Palais-Smale condition. This yields the existence ofu0∈W1,pΩsatisfyingE0u0 0 and
inf
E0u:u∈∂Bρu−
≤E0u0 inf
π∈Π max
t∈ −1,1E0πt, 5.2
where
Π π ∈C
−1,1, W1,pΩ
:π−1 u−, π1 u
. 5.3
It is clear that5.1and 5.2 imply thatu0/u− and u0/u. Hence,u0 is a sign-changing solution provided thatu0/0. We have to show thatE0u0/0,which is fulfilled if there exists a pathπ∈Πsuch that
E0πt /0, ∀t∈ −1,1. 5.4
LetSW1,pΩ∩∂B1Lp∂Ω, where∂BL1p∂Ω{u∈Lp∂Ω: u Lp∂Ω1}, andSC S∩C1Ω be equipped with the topologies induced byW1,pΩandC1Ω, respectively. Furthermore, we set
Π0
π∈C −1,1, S:π−1 −ϕ1, π1 ϕ1
, Π0,C
π∈C −1,1, SC:π−1 −ϕ1, π1 ϕ1
. 5.5
Because of the results of Mart´ınez and Rossi in 26, there exists a continuous pathπ ∈Π0
satisfyingt→πt∈ {u∈W1,pΩ:Ia,bu<0, u Lp∂Ω1}provided thata, bis above the curveCof hypothesisH3. Recall that the functionalIa,bis given by
Ia,bu
Ω
|∇u|p|u|p dx−
∂Ω
aupb u−p
dσ. 5.6
This implies the existence ofμ >0 such that
Ia,bπt≤ −μ <0, ∀t∈ −1,1. 5.7
It is well known that SC is dense in S, which implies the density of Π0,C in Π0. Thus, a continuous pathπ0 ∈Π0,Cexists such that
Ia,bπt−Ia,bπ0t< μ
2, ∀t∈ −1,1. 5.8
The boundedness of the setπ0 −1,1ΩinRensures the existence ofM >0 such that
|π0tx| ≤M ∀x∈Ω, ∀t∈ −1,1. 5.9
Theorem 3.5 yields that u,−u− ∈ intC1Ω. Thus, for every u ∈ π0 −1,1 and any bounded neighborhoodVuofuinC1Ω,there exist positive numbershuandjusatisfying
u−hv∈int C1
Ω
, −u−jv∈int C1
Ω
, 5.10
for allh : 0 ≤ h ≤ hu, for allj : 0 ≤ j ≤ ju, and for allv ∈ Vu. Using5.10along with a compactness argument implies the existence ofε0>0 such that
u−x≤επ0tx≤ux, 5.11
