Volume 2011, Article ID 214289,19pages doi:10.1155/2011/214289
Research Article
Multiple Solutions of p-Laplacian with
Neumann and Robin Boundary Conditions for Both Resonance and Oscillation Problem
Jing Zhang and Xiaoping Xue
Department of Mathematics, Harbin Institute of Technology, Harbin 150025, China
Correspondence should be addressed to Jing Zhang,[email protected] Received 29 June 2010; Revised 7 November 2010; Accepted 18 January 2011
Academic Editor: Sandro Salsa
Copyrightq2011 J. Zhang and X. Xue. 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.
We discuss Neumann and Robin problems driven by thep-Laplacian with jumping nonlinearities.
Using sub-sup solution method, Fuc´ık spectrum, mountain pass theorem, degree theorem together with suitable truncation techniques, we show that the Neumann problem has infinitely many nonconstant solutions and the Robin problem has at least four nontrivial solutions. Furthermore, we study oscillating equations with Robin boundary and obtain infinitely many nontrivial solutions.
1. Introduction
Let Ωbe a bounded domain of Rn with smooth boundary∂Ω, we consider the following problems:
iNeumann problem:
−Δpuα|u|p−2ufx, u, inΩ,
∂u
∂ν 0, on∂Ω, p1
iiRobin problem:
−Δpuα|u|p−2ufx, u, inΩ,
|∇u|p−2∂u
∂ν bx|u|p−2u0, on∂Ω, p2
whereΔpudiv|∇u|p−2∇uis thep-Laplacian operator ofuwith 1< p <∞,α >0, bx∈L∞∂Ω,bx≥0, andbx/0 on∂Ω,fx,0 0 for a.e.x∈Ω, and∂u/∂ν denotes the outer normal derivative ofuwith respect to∂Ω. Our purpose is to show the multiplicity of solutions top1andp2.
It is known thatp1andp2are the Euler-Lagrange equations of the functionals
J1u 1 p
Ω|∇u|pdxα p
Ω|u|pdx−
ΩFx, udx, J2u 1
p
Ω|∇u|pdxα p
Ω|u|pdx1 p
∂Ωbx|u|pds−
ΩFx, udx,
1.1
respectively, defined on the Sobolev spaceW1,pΩ, whereFx, u u
0 fx, sds. The critical points of functionals correspond to the weak solutions of problems. In Li1 and Zhang et al.
2 , the authors study the existence and multiple solutions ofp1andp2using the critical points theory for the semilinear casep2. There also have been some papers dealing with the quasilinear casep /2 using the critical point theory, and some existence results of solutions have been generalized to this case in the work of Perera3 , Zhang et al.4 , and Zhang-Li5 . Most of these papers use the minimax arguments, and nontrivial solutions are obtained with the assumption that the nonlinearity is superlinear at 0. In this paper, we give the nontrivial solutions of p1 and p2 with a jumping nonlinearity when the asymptotic limits of the nonlinearity fall in the regions formed by the curves of the Fuc´ık spectrum. Our technique is based on mountain pass theorem, computing the critical groups and Fuc´ık spectrum.
Our general assumptions are the following.
f1There is constant C > 0 such thatfx, tsatisfies the following subcritical condi- tions:
fx, t≤C
|t|q1
for everyx∈Ω, t∈R, 1.2
withp−1< q < p∗−1, wherep∗np/n−pifn > p, andp∗∞ifn1,2, . . . , p.
f2∃sequence{ai}and{bi}, whereai, bi ∈R,i 1,2, . . ., which satisfyai >0,bi < 0 andai ∞,bi −∞asi → ∞. And at the same time{ai},{bi}satisfy
fx, ai αap−1i , fx, bi −α|bi|p−1, for everyx∈Ω 1.3
which means that{ai},{bi}are constant solution sequences ofp1.
Let a0 b0 0, fx, t < αtp−1 if t ∈ ai, ai1, whereiis an odd number, i ≥ 1;
fx, t> αtp−1ift∈ai, ai1, whereiis an even number,i≥0;fx, t<−α|t|p−1ift∈bi1, bi, whereiis an even number,i≥0;fx, t>−α|t|p−1ift∈bi1, bi, whereiis an odd number, i≥1, for everyx∈Ω.
f3For allt /ai, bi,f isC1;f−x, ai/fx, ai,f−x, bi/fx, bi, whereiis an even number,i ≥ 2, f−x, t,fx, tdenote the left and the right derivatives off at t, respectively.
f4Leta, b fx, ai−α, f−x, ai−αforiis an even number,i≥2. Fora, b∈R2, the problem
−Δpua
u−cp−1
−b
u−c−p−1
, inΩ,
∂u
∂ν 0, on∂Ω, 1.4
only has constant solutionc, whereu−c±x max{±u−c,0}andcis a constant.
Andfi− x, ai−α > λ2,fi x, ai−α > λ2foriis an even number,i≥2, where
fix, t
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
0, t <0, fx, t, 0≤t≤ai, fx, ai, t > ai,
1.5
andfi− x, ai, fi x, aidenote the left and the right derivatives offi atai, respectively, and λ2is the second of the eigenvalue problems with Neumann boundary value condition.
f5∃m > α, such thatfx, t m|t|p−2tis increasing int.
In particular, fromf2, we know thatp1has infinitely many constant solutions, a.e., {ai},{bi},i 0,1,2, . . .. In this paper, we mainly discuss whether it has many nonconstant solutions and what their locations are.
Then we have the main results of this paper.
Theorem 1.1. Assume that (f1)–(f5) hold. Thenp1 has infinitely many nonconstant solutions.
Moreover, if one chooses some order intervals which have two pairs of strict constant sub-sup solutions, thenp1has at least two nonconstant solutions in some order intervals.
Furthermore, if we assume that f−x,0/fx,0 under the same conditions as in Theorem 1.1, we can have at least one sign-changing solution which is of mountain pass type from the mountain pass theorem in order interval. When we discuss multiple solutions of p1, we notice that there may be infinitely many sign-changing solutions under stronger assumptions. In fact, if we give more assumptions,we can obtain infinitely many sign- changing solutions.
We assume the following.
FFx, t > λ2αε0/ptp,|t| ≥ M,Mis large enough, whereλ2 is the second eigenvalue of Neumann problem of−Δpandε0>0.
Corollary 1.2. Under the same conditions as inTheorem 1.1, (F) andf−x,0/fx,0, then one can get infinitely many sign-changing solutions for p1 which are of mountain pass type or not mountain pass type but with positive local degree.
For the Robin problem, if∃M1>0,M2 >0 such thatfx, M1 0,fx,−M2 0 for a.e.x∈Ω, then we give the following assumptions:
g1f ∈C1Ω×R1\ {0},f−x,0/fx,0, and min{fx,0, f−x,0}> λ1αfor a.e.
x ∈ Ω, wheref−x,0,fx,0denote the left and the right derivatives off at 0, respectively, andλ1is the first eigenvalue of Robin problem of−Δp;
g2leta, b fx,0−α, f−x,0−α. Fora, b∈R2, the problem
−Δpuaup−1−b u−p−1
, inΩ,
|∇u|p−2∂u
∂ν bx|u|p−2u0, on∂Ω, 1.6
only has trivial solution 0, whereu±x max{±u,0}.
In this case, we have the following.
Theorem 1.3. Assume that (f1), (f5), (g1), (g2) hold. Then one has at least four nontrivial solutions of problemp2.
Furthermore, we give the following stronger assumption:
FFx, t>λ2αε0/pC t p,|t| ≥M,Fx, u u
0 fx, sds,u∈E2, whereE2 {u ∈W1,pΩ:u kϕ1tϕ2},C C2/2bxL∞∂Ω. HereCis the imbedding constant of Sobolev Trace Theoremsee6 ,Mis large enough,ε0is small enough, λ2is the second of the eigenvalue problems with Robin boundary value condition, andϕ1,ϕ2are the first and the second eigenfunction, respectively.
Then we have the following.
Corollary 1.4. Assume thatf is satisfied as inTheorem 1.3and (F), then one can have infinitely many sign-changing solutions forp2which are of mountain pass type or not mountain pass type but with positive local degree.
In the oscillating problems of Robin boundary, a.e.,f2holds. We make the following assumption.
F
ΩFx, tϕ1dx ≥ λ1 αε0/p Ct p
Ωϕp1dx, |t| ≥ M, where ϕ1 is the first eigenvalue of the Robin problem and
Ωϕp1dx1.
Then we have the following.
Theorem 1.5. Assume that f is satisfied as in Theorem 1.3 and (f2), (F), one can get infinitely many nontrivial solutions of problemp2. Some of them are minimum points; others are mountain pass points.
2. Preliminaries
Now we recall the notion of critical groups of an isolated critical pointuof aC1functionalJ briefly. LetU⊂Mbe an isolated neighborhood ofusuch that there are no critical points ofJ inU\ {u};Mis a Banach space. The critical groups ofuare defined as
CqJ, u HqJc∩U,Jc\ {u}∩U;G, q0,1,2, . . ., 2.1
wherec JuandJc {u ∈ M|Ju ≤ c}is a level set ofJ andHqX, Y;Gare singular relative homology groups with a Abelian coefficient groupG,Y ⊂X,q0,1,2, . . .. They are independent of the choices ofU, hence are well defined. UseHqX;Gto stand for theqth singular cohomology group with an Abelian coefficient groupG; from now on we denote it byHqX. Assume that J ∈ C2M, R, and a critical point uis called nondegenerate if the HessianJuat this point has a bounded inverse. Letube a nondegenerate critical point of J; we call the dimension of the negative space corresponding to the spectral decomposition ofJu, that is, the dimension of the subspace of negative eigenvectors ofJu, the Morse index ofu, and denote it by indJu. IfC1J, u/0, then we call an isolated critical pointu ofJas a mountain pass point. For the details, we refer to7 .
We have the following basic facts on the critical groups for an isolated critical point ofJ.
aLetube is an isolated minimum point ofJ, thenCqJ, u δq0G.
bIfJ ∈C2M, Randuis a nondegenerate critical point ofJwith Morse indexj, then CqJ, u δqjG.
Definition 2.1. If any sequence{uk} ⊂Mwhich satisfiesJuk → candJuk → 0k → ∞ has a convergent subsequence, one says thatJsatisfies theP Sccondition. IfJsatisfiesP Sc condition for allc∈R, one says thatJsatisfies theP Scondition.
Lemma 2.2 see 8 . Assume that u and u are, respectively, lower and upper solutions for the problem
−Δpugx, u, in Ω,
|∇u|p−2∂u
∂ν bx|u|p−2u0, on∂Ω,
2.2
withu≤ua.e. inΩ, wheregx, sis a Carath´eodory function onΩ×Rwith the property that, for anys0 > 0, there exists a constantAsuch that|gx, s| ≤ Afor a.e.x ∈ Ωand alls ∈ −s0, s0 . Consider the associated functional
Φu: 1 p
Ω|∇u|p−
ΩGx, u, 2.3
whereGx, u:s
0gx, tdtand the intervalM:{u∈W1,PΩ:u≤u≤ua.e.inΩ}. Then the infimum ofΦonMis achieved at someu, and such auis a solution of the above problem.
In what follows, we setX W1,pΩwhich is is uniformly convex1 < p < ∞and equipped with the normu
Ω|∇u|pdxmα
Ω|u|pdx1/p. LetEbe a Hilbert space and PE⊂Ea closed convex cone such thatXis densely embedded inE. Assume thatPX∩PE, P has nonempty interior ˙P and any order interval is bounded. It is well known that P S condition implies the compactness of the critical set at each levelc ∈ R, on the case of the above condition. Then we assume the following:
J1J ∈C2E, Rand satisfiesP Scondition inEand deformation property inX;
J2∇J id−KE, whereKE : E → E is compact. KEX ⊂ X and the restriction KKE|X :X → Xis of classC1and strongly preserving, that is,uv⇔u−v∈P;˙ J3Jis bounded from below on any order interval inX.
Lemma 2.3Mountain pass theorem in half-order intervals, sup-solutions casesee9 . Suppose that J satisfies (J1)–(J3).v1 < v2 is a pair of strict supersolution of∇J 0.v0 < v1 is a subsolution of∇J 0. Suppose thatv0, v1 andv0, v2 are admissible invariant sets forJ. IfJ has a local strict minimizerwinv0, v2 \v0, v1 . ThenJhas mountain pass pointsu0inv0, v2 \ v0, v1 .
Lemma 2.4Mountain pass theorem in order intervalssee10 . Suppose thatJsatisfies (J1)–
(J3) and{v1, v2},{ω1, ω2}are two pairs of strict sub-sup solutions of∇J 0 inXwith v1 < ω2, v1, v2 ∩ω1, ω2 ∅. ThenJ has a mountain pass pointu0,u0∈v1, ω2 \v1, v2 ∪ω1, ω2 . More precisely, letv0be the maximal minimizer ofJinv1, v2 andω0the minimal minimizer ofJin ω1, ω2 . Thenv0u0 ω0. Moreover,C1J, u0, the critical group ofJatu0, is nontrivial.
Remark 2.5. aLemma 2.4still holds ifJ ∈C1E, R,Kis of classC0see10 .
b For X W1,pΩ, we define gpt : |t|p−2t. From assumption f5, there exists m > αsuch thatfx, u−α|u|p−2umgpuis strictly increasing inu. The assumption is not essential but is assumed for simplicity. If suchmdoes not exist then we can approximatef by a sequence of functions so thatmas above exists, and obtain the solutions by passing to limits. Form > α, we need the operator
Am:X −→X, Amu
−Δpmgp·−1
x, u mgpu
. 2.4
From11 , we know thatAmis compact, that is, it is continuous and maps bounded subsets ofXinto relatively compact subsets ofX. Since−Δpumgpuis a positive operator,
K:
−Δpumgpu−1
fx, u−α|u|p−2umgpu
2.5
is strongly orderpreserving. From the above discussion, we have the mountain pass theorem in order intervals ofJ1andJ2.
Next, let us recall some notions and known results on Fuc´ık spectrum.
The Fuc´ık spectrum ofp-Laplacian onW1,pΩis defined as the setΣpof those points a, b∈R2for which the problem
−Δpuaup−1−b u−p−1
, u∈W1,pΩ 2.6
has nontrivial solutions. Hereu±x max{±u,0}.
For the semilinear casep 2, it is known thatΣ2 consists, at least locally, of curves emanating from the pointsλl, λlwhere{λl}l∈Nare the distinct eigenvalues of−Δ see, e.g., 12 . It was shown in Schechter 13 thatΣ2 contains continuous and strictly decreasing curvesCl1,Cl2 through λl, λl such that the points in the squareQl λl−1, λl12 that are either below the lower curveCl1or above the upper curveCl2are free ofΣ2, while the points on the curves are in Σ2 when they do not coincide. The points in the region between the curves may or may not belong toΣ2.
As shown in Lindqvist14 that the first eigenvalueλ1of−Δpis positive, simple and admits a strictly positive eigenfunction ϕ1, soΣp contains the two lines λ1×RandR×λ1. This generalized notion of spectrum was introduced for the semilinear casep2 in the 1970s by Fuc´ık 12 in connection with jumping nonlinearities. A first nontrivial curveC2 in Σp
throughλ2, λ2that is continuous, strictly decreasing, and asymptotic toλ1×RandR×λ1
at infinity was constructed and variationally characterized by a mountain-pass procedure in Cuesta et al.15 .
Consider the problem
−Δpua
u−cp−1
−b
u−c−p−1
, inΩ,
∂u
∂ν 0, on∂Ω,
2.7
−Δpuaup−1−b u−p−1
, inΩ,
|∇u|p−2∂u
∂νbx|u|p−2u0, on∂Ω,
2.8
from the variational point of view; solutions of2.7and2.8are the critical points of the functional
I1u I1u, a, b
Ω
|∇u|p−a
u−cp
−b
u−c−p dx,
I2u I2u, a, b
Ω
|∇u|p−aup−b u−p
dx 1 p
∂Ωbx|u|pds,
2.9
respectively, wherecis a constant.
Ifa, bdoes not belong toΣp,cis the constant solution of2.7, that is,cis an isolated critical point of I1; 0 is the trivial solution of2.8, that is, 0 is an isolated critical point of I2, then from the definition of critical group, we have theCqI1, candCqI2,0defined,q 0,1,2, . . .. Now, we give some results relative to the computation of the critical groups which are the results of Dancer and Perera16 . Let C11 −∞, λ1 ×λ1∪λ1 ×−∞, λ1and C12 λ1×λ1,∞∪λ1,∞×λ1.
Lemma 2.6. iIfa, blies belowC11, thenCqI, c δq0Z.
iiIfa, blies betweenC11andC12, thenCqI, c 0 for allq.
iiiIfa, blies betweenC12andC2, thenCqI, c δq1Z.
ivIfa, bdoes not belong toΣp, but lies aboveC2, thenCqI, c 0 forq0,1.
Denote
Isu
Ω|∇u|p−su−cp, u∈X, 2.10 andIsis the restriction ofIsto theC1manifold
S
u∈X:
Ω|u−c|p1
. 2.11
As noted in16 , the critical groups ofI are related to the homology groups of sublevel sets ofIa−b. We have that
I|SIa−b−b, 2.12
so the sublevel sets
Id{u∈X:Iu≤d}, Isd
u∈S:Is≤d
2.13
are related by
Id∩SIa−bdb. 2.14
Lemma 2.7. Ifa, bdoes not belong toΣp, then
CqI, c∼
⎧⎨
⎩
δq0Z, if Ia−bb ∅, Hq−1
Ia−bb
, otherwise, 2.15
whereHq denote reduced homology groups. It also holds with Ia−bb replaced by b {u ∈ S : Ia−bu> b}.
3. The Proof of the Main Results
Let
fix, t
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
0, t <0, fx, t, 0≤t≤ai, fx, ai, t > ai.
Fix, t t
0
fix, sds
J1iu 1 p
Ω|∇u|pdxα p
Ω|u|pdx−
ΩFix, udx.
3.1
It is well known that critical points of J1i correspond to weak solutions of the following equation:
−Δpuα|u|p−2ufix, u, inΩ,
∂u
∂ν 0, on∂Ω. p
We have thatfix, t∈C0R, RandJ1i∈C1E, R. We can discuss similar case forbi. Next, we give the relation of the solutions of p and the solutions of p1, that is, Lemma 3.2below. In order to proveLemma 3.2, we firstly give the comparison principle.
Let
Lp:−Δpax|u|p−2u, λ1,pa inf
Ω
|∇u|pax|u|p
dx, u∈W01,pΩ,
Ω|u|pdx1
.
3.2
Lemma 3.1 comparison principle see17 . Assumea ∈ L∞Ω,λ1,pa > 0. The Lpu ∈ L∞Ωwithu|∂Ω∈C1α∂Ω, andLpu≤0 withu∈W1,pΩ∩L∞Ω, thenu≤0.
Lemma 3.2. Ifuixis a solution of p, then uix is also a solution of p1and satisfies 0 ≤ uix≤ai,i1,2, . . ..
Proof. Suppose that the conclusion is false. Now, consider the domainUi {x∈Ω|uix>
ai}, then we have
−Δpufix, u−α|u|p−2u≤0, inUi,
uai, on∂Ui, 3.3
where−Δpu fix, u−α|u|p−2u fx, ai−α|u|p−2u ≤ fx, ai−αap−1i 0, x ∈Ui by the definition offix, u. By the comparison principle, we can conclude thatuix≤0 inUi. It is a contradiction, so we have thatUi∅, that is,uix≤ai.
Similarly, we considerVi {x∈Ω|uix<0}, by the comparison principle; we also get the contradiction, so we have thatVi∅, that is,uix≥0. From the above discussion, we
have that 0≤uix≤ai,i1,2, . . .andfix, u fx, ui, souixis a solution ofp1. This completes the proof of the lemma.
Remark 3.3. From the above discussion, by applyingLemma 3.2, we know that solutions of pare also the solutions ofp1if we want to proveTheorem 1.1, we only need to prove that phas infinitely many nonconstant solutions under the assumptions as inTheorem 1.1and phas two nonconstant solutions in every order interval.
Theorem 3.4. There are infinitely many nonconstant solutions ofp. Moreover, if there exists some order intervals which have two pairs of strict constant sub-sup solutions, then there are at least two nonconstant solutions in these order intervals.
Proof. We treat the case ofai; the other case ofbiis proved by a similar argument.
Iff2holds, then
−Δpai 0fix, ai−αap−1i , for a.e. x∈Ω, 3.4
so{ai}are all positive constant solutions ofp. Assuming thatiis large enough andiis an even number, we also infer that{a2k−1}are local minimums,k1,2, . . . , i/2. So we getu2k−1 andu2k−1 a strict subsolution and sup-solution pair forp, satisfyingu2k−1 < a2k−1 < u2k−1 for eachk,k1,2, . . . , i/2.
Now, we study the order intervalu1, u3 inXwhich includes two suborder intervals u1, u1 andu3, u3 ,a2∈u1, u3 .
We infer that J1iu satisfies deformation properties and is bounded from below on u1, u3 and so we get a mountain pass pointu1 ∈u1, u3 \u1, u1 ∪u3, u3 according to mountain pass theorem in order interval, we have thatC1J1i, u1is nontrivial.
From assumptionf3, we know that the left and the right derivatives offiata2are different; we consider the problem
−Δpufix, u−α|u|p−2u, inΩ,
∂u
∂ν 0, on∂Ω, 3.5
wherefi∈CΩ×Rand asu → a2we have fix, u−α|u|p−2u
fix, a2−α
u−a2p−1
−
fi− x, a2−α
u−a2−p−1 ◦
|u−a2|p−1 .
3.6
We takea fi x, a2−α,bfi− x, a2−α, then from assumptionf4and the definition of Σp, we know thata, bdoes not belong toΣp. So, we have the following.
1Ifa, bdoes not belong toΣp, but lies aboveC2, then
CqJ1i, a2 0 forq0,1 3.7
byLemma 2.6iv. In this case,C1J1i, a2 0, soCqJ1i, a2 CqJ1i, u1, and we haveu1/a2.
2Denote
Ja−bu
Ω|∇u|p−a−bu−a2p, u∈X, 3.8 andJa−bis the restriction ofJa−bto theC1manifold
S
u∈X:
Ω|u−a2|p1
, 3.9
whereafi x, a2−α,bfi− x, a2−αas shown above.
Fromf4, we know thata, bdoes not belong toΣp, and ifJa−bu> b, a.e.Ja−bb ∅, then
CqJ1i, a2 δq0Z 3.10
byLemma 2.7. In this case,C1J1i, a2 0, soCqJ1i, a2 CqJ1i, u1, and we haveu1/a2. Similarly, applying the mountain pass theorem in order interval to u3, u5 which contain two sub-order intervalsu3, u3 and u5, u5 , we get a mountain pass pointu2 and prove thatCqJ1i, a4 CqJ1i, u2, sou2/a4from Lemmas2.6and2.7.
We let the procedure go on. Soi/2−1 mountain pass points are available which are nonconstant solutions ofp, whereiis large enough andiis an even number. Then we have infinitely many nonconstant positive solutions ofpby the arbitrary ofi.
We can discuss the similar case forbi and get infinitely many nonconstant negative solutions.
Now, we discuss the solutions inu1, u3 more deeply. Since u1 is a mountain pass point, for the Leray-Schauder degree ofid−Ki, we have the computing formular
deg
id−Ki, Bu1, r,0
−1, 3.11
where r > 0 is small enough, Ki −Δp mαgp·−1fi∗|X : X → X is of class C0 and strongly preserving,fi∗x, u fix, u mgpu seeRemark 2.5b. Then according to Poincar´e-Hopf formular forC1case and the computation ofCqJ1i, a2, we have
indexJ1i, a2 −1l. 3.12
Furthermore, for minimum pointsa1,a3,
CqJ1i, a1∼δq0G, CqJ1i, a3∼δq0G. 3.13
From the additivity of Leray-Schauder degree and Theorem 1.1 in10 , we can get
1deg
id−Ki, u1, u3
,0
deg
id−Ki, u1, u1
,0 deg
id−Ki, u3, u3
,0 deg
id−Ki, Ba2, r,0 deg
id−Ki, Bu1, r,0 11 −1l −1.
3.14
So we have−1l 1. It is impossible. From the above discussion, we conclude that there must exist another critical pointu∗1∈u1, u3 , which satisfiesu∗1/u1and is nonconstant.
Similarly, we can discuss the order intervalu3, u5 , and we get another critical point u∗2/u2. We let the procedure go on.
This completes the proof ofTheorem 3.4.
Thus, we prove that the conclusion ofTheorem 1.1holds.
The Proof of Corollaries1.2and1.4.
Proof. See Theorem 3.5 of Li1 .
Proof ofTheorem 1.3. From the variational point of view, solutions ofp2are the critical points of the functional
J2u 1 p
Ω|∇u|pdxα p
Ω|u|pdx 1 p
∂Ωbx|u|pds−
ΩFx, udx, 3.15
defined onX:W1,pΩ, whereFx, u u
0fx, sds.
We show thatJ2belongs toC1X, R. In fact, we set
J21u 1 p
Ω|∇u|pdxα p
Ω|u|pdx−
ΩFx, udx, J22u 1 p
∂Ωbx|u|pds. 3.16 Under the conditionf1, it is well known thatJ21is aC1-functional. Next, we considerJ22. If we letu, v∈X, 0<|t|<1,
J22utv−J22u
t
∂Ωbx|u|p−2uv ds
q≥2
Cqp p tq−1
∂Ωbx|u|p−q|v|qds
−→
∂Ωbx|u|p−2uv ds, t−→0.
3.17
So we have thatJ22has a Gateaux derivative andJ22 u, v
∂Ωbx|u|p−2uv ds.
Letun → uinX; now, by H ¨older’s and Sobolev’s inequalities we can estimate J22 un−J22 u, v
∂Ωbx
|un|p−2un− |u|p−2u v ds
≤ bL∞∂Ω
∂Ω
|un|p−2un− |u|p−2u vds
≤
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
cbL∞∂Ω
∂Ω|un||u|p−2|un−u||v|ds ifp≥2, cbL∞∂Ω
∂Ω|un−u|p−1|v|ds ifp <2,
≤
⎧⎪
⎨
⎪⎩
cbL∞∂ΩTun−uLp∂ΩTvLp∂Ω ifp≥2, cbL∞∂ΩTun−up−1Lp∂ΩTvLp∂Ω ifp <2,
3.18
but, whenp≥2,pp/p−1≤p, we haveuLp ≤ uLp, then we have
J22 un−J22 u, v≤
⎧⎪
⎨
⎪⎩
cbL∞∂ΩTun−uLp∂ΩTvLp∂Ω ifp≥2, cbL∞∂ΩTun−up−1Lp∂ΩTvLp∂Ω ifp <2,
≤
⎧⎪
⎨
⎪⎩
cbL∞∂Ωun−uW1,pΩvW1,pΩ ifp≥2, cbL∞∂Ωun−up−1W1,pΩvW1,pΩ ifp <2,
3.19
where 1/p1/p1,T :W1,pΩ → Lp∂Ωis trace operator, andTuLp∂Ω≤CuW1,pΩ
for allu∈W1,pΩwith the constantCdepending onΩby Sobolev Trace Theoremsee6 . To get3.18, we have used the following well-known inequalities:
|u|p−2u− |v|p−2v≤
⎧⎨
⎩
c|u||v|p−2|u−v| ifp≥2,
c|u−v|p−1 ifp <2, 3.20
which hold for a convenientc >0,u, v∈Rn. So
J22 un−J22 u≤
⎧⎨
⎩
cbL∞∂Ωun−uW1,pΩ ifp≥2,
cbL∞∂Ωun−up−1W1,pΩ ifp <2. −→0, n−→ ∞. 3.21
SoJ22 uis continuous andJ2∈C1X, R.
Consider the truncated functions
fx, t
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
0, t≤ −M2, fx, t, −M2 ≤t≤M1, 0, t≥M1
3.22
and the corresponding functional
Ju 1 p
Ω|∇u|pdxα p
Ω|u|pdx 1 p
∂Ωbx|u|pds−
Ω
Fx, udx, 3.23
Fx, t t
0fx, sds.
From Perera 18 we have that J satisfies P S condition. From the deformation theorem, we know thatJsatisfies deformation property whenJsatisfiesP Scondition. By a similar discussion as inTheorem 1.1, we only need to discuss the critical points ofJ.
Now, we construct the sub-sup solutions ofp2. It is easy to see thatM1is a constant sup-solution ofp2and −M2 is a constant subsolution. Moreover, we consider εϕ1 for all ε >0 small enough. From14 we know thatϕ1x>0,x∈Ω. In fact, withu:εϕ1, byg1 we have
−Δpuαup−2u−f x, u
εp−1ϕp−11 x
⎡
⎣λ1α−f x, εϕ1 εp−1ϕp−11
⎤
⎦≤0. 3.24
Furthermore, ϕ1 ∈ W1,pΩ∩L∞Ω satisfies −Δpϕ1 λϕp−11 in the weak sense, then the regularity theory for thep-Laplaciane.g.,19 impliesϕ1 ∈C1,αΩfor someααn, p∈ 0,1. Moreoverϕ1≥0. In addition, by the strong maximum principle of20 andϕ1/0, then ϕ1 >0 inΩand∂ϕ1/∂ν <0 on∂Ω. So ifbxis small enough at some pointx0 ∈∂Ω, we can have|∇u|p−2∂u/∂νbx|u|p−2u |x0≤0. From the above discussion, we have a sub-solution ofp2, a.e.,εϕ1x0. Withεϕ1:εϕ1x0.
By a similar argument we can find that−M2,−εϕ1 is a pair of strict sub-sup solutions.
Now we study the order interval −M2, M1 in X which includes two suborder intervals −M2,−εϕ1 and εϕ1, M1 . By Lemma 2.2, there exists weak solutions of p2 relative minimum points u2,u3 in −M2,−εϕ1 and εϕ1, M1 , respectively. We can infer that Ju is bounded from below on −M2, M1 , so we get a mountain pass point u1 ∈
−M2, M1 \−M2,−εϕ1 ∪εϕ1, M1 according to mountain pass theorem in order interval.
From the definition of mountain pass point, we have thatC1J, u 1is nontrivial.
From assumptiong2, we know that the left and the right derivatives offat 0 are different, we consider the problem
−Δpufx, u −α|u|p−2u, inΩ,
|∇u|p−2∂u
∂ν bx|u|p−2u0, on∂Ω, 3.25
wheref∈CΩ×R, and asu → 0 we have fx, u −α|u|p−2u
fx,0−α
up−1−
f−x,0−α u−p−1
◦
|u|p−1
. 3.26
We takeafx,0−α,bf−x,0−α; then also from assumptiong2and the definition of Σp, we know thata, b∈/Σp.
Then we consider the following cases.
1Ifa, bdoes not belong toΣp, but lies aboveC2, then Cq
J, 0
0 forq0,1 3.27
byLemma 2.6iv. In this case,C1J, 0 0, soCqJ,0 CqJ, u 1, we haveu1/0.
2Denote
Ja−bu
Ω|∇u|p−a−bup
∂Ωbx|u|pds, u∈X, 3.28 andJa−bis the restriction ofJa−bto theC1manifold
S
u∈X:
Ω|u|p1
, 3.29
whereafx,0−α,bf−x,0−αas shown above.
Fromg2, we know thata, bdoes not belong toΣp, ifJa−bu> b, a.e.Ja−bb ∅, then Cq
J, 0
δq0Z 3.30
byLemma 2.7. In this case,C1J, 0 0, soCqJ,0 CqJ, u 1, and we haveu1/0.
Now, we discuss the solutions in −M2, M1 more deeply. We already have four solutions 0,u1,u2,u3, whereu1is the mountain pass point andu2,u3are the local minimum points ofJ2. For the minimum pointsu2,u3, we have
Cq
J, u 2
∼δq0G, Cq
J, u 3
∼δq0G. 3.31
Sinceu1is a mountain pass point, for the Leray-Schauder degree ofid−K, we have the computing formular
deg
id−K, Bu 1, r,0
−1, 3.32 wherer > 0 is small enough,K −Δp mαgp·−1f∗|X : X → X is of classC0 and
strongly order preserving,f∗x, u fx, u mgpu seeRemark 2.5b. Then according to Poincar´e-Hopf formula forC1case and the computation ofCqJ, 0, we have
index J,0
−1dl−1. 3.33
From the additivity of Leray-Schauder degree and Theorem 1.1 in10 , we can get
1deg
id−K, −M, M ,0 deg
id−K,
−M,−εϕ1
,0 deg
id−K,
εϕ1, M ,0 deg
id−K, B0, r, 0 deg
id−K, Bu 1, r,0 11 −1dl−1 −1.
3.34
It is impossible. From the above discussion, we conclude that there must exist another critical pointu∗∈−M2, M1 , which satisfiesu∗/u1and is nontrivial.
This completes the proof ofTheorem 1.3.
Proof ofTheorem 1.5. Consider the truncated function
fix, t
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
0, t <0, fx, t, 0≤t≤ai, fx, ai, t > ai.
3.35
Corresponding functional is
Jiu 1 p
Ω|∇u|pdxα p
Ω|u|pdx1 p
∂Ωbx|u|pds−
ΩFix, udx, 3.36
whereFix, u u
0fix, sds,i1,2, . . ..
It is known that the solution ofp2is also a solution of the following equation as in the same discussion inLemma 3.2:
−Δpuα|u|p−2ufix, u, inΩ,
|∇u|p−2∂u
∂ν bx|u|p−2u0, on∂Ω. 3.37
By the standard argument we know thatJisatisfiesJ1–J3and the order intervals consisted by sub-super-solutions are admissible invariant set ofJi. Takingv0 −M2,v1a1 >0, then Jiuhas a minimizeru1∈v0, v1 . By assumptionFthere exists at1>0 such that
J2 t1ϕ1
tp1 p
Ω
∇ϕ1pdxα ptp1
Ωϕ1pdxtp1 p
∂Ωbxϕ1pds−
ΩF x, t1ϕ1
dx
≤ λ1αtp1 p
Ωϕ1pdx−
λ1αε0C
tp1 p
Ω ϕ1pdx < J2u1.
3.38
If we takev2an1 > t1ϕ1, wheren1< i, then Ji
t1ϕ1
J2
t1ϕ1
< Jiu1 3.39
which implies thatJiuhas a minimizeru2 ∈v0, v2 \v0, v1 such thatJiu2< Jiu1. By Lemma 2.3we get a mountain pass point u3. Moreover,v0 < ui < v2,i 1,2,3, anduiare positive.
Next, we take v1 an1, v0 εϕ1. Then Jiu has a minimizer u2 ∈ v0, v1 . By assumptionFthere is at2>0 such that
J2
t2ϕ1
< J2u2. 3.40
If we takev2an2 > t2ϕ1, wheren2< i, then Ji
t2ϕ1 J2
t2ϕ1
< Jiu2 3.41
which implies thatJiuhas a minimizeru4 ∈v0, v2 \v0, v1 such thatJiu4< Jiu2. By Lemma 2.3we get a mountain pass pointu5. Moreover,v0< ui< v2,i1,2,3,4,5, anduiare all positive. Continue making the procedure we obtain the result.
The proof is complete.
Corollary 3.5. Moreover, p1 has infinitely many nonconstant negative energy solutions {uk}, which are mountain pass types, if the conditions as in Theorem 1.1 hold and J1a2k → −∞ or J1b2k → −∞ask → ∞.
Proof. Assume thatJ1a2k → −∞ask → ∞. Letc infγ∈ΓmaxγI∩SJ1ut, whereΓ {γ∈CI, W|γ0 a2k−1, γ1 a2k1}, andI 0,1 ,SW\W1∪W2,W u2k−1, u2k1 , W1 u2k−1, u2k−1 ,W2 u2k1, u2k1 , c∗ Ja2k,k 1,2, . . .. We discuss the problem inW which have two minimum pointsa2k−1 anda2k1. We have thata2k−1 anda2k1 are in the same radial directionA {ke1 |k ∈ R},e1 is the first eigenvalue function of−Δpα with Neumann boundary. In fact,e1is a constant. We conclude thatc∗ ≥csee Corollary 3.4 of C. Li and S. Li21 . Furthermore, if f3,f4hold, thenc∗ > c. In fact, if c∗ c, then c∗ maxu∈γ∗I∩SJu infγ∈ΓmaxγI∩SJut Ja2k, whereγ∗ is a special path between a2k−1 anda2k1, which is a path of radial directionA {ke1 | k ∈ R}. Soa2k is a mountain pass point. But according to assumptionsf3andf4, we know thatC1J1, a2k 0,l /2, that is,a2kis not a mountain pass type. This is a contradiction. We draw the conclusion.
