EXISTENCE RESULT FOR A p-LAPLACIAN EQUATION WITH INDEFINITE WEIGHTS
PETRE SORIN ILIAS¸
We consider the Dirichlet problem−∆pu=λV(x)|u|p−2u+f(x, u),u∈W01,p(Ω) wherep >1,λ∈R, andV is a given function inLs(Ω). We prove the existence of weak solutions for this problem whenλlies between the first two eigenvalues of thep-Laplacian.
AMS 2000 Subject Classification: 47H05, 35B38.
Key words: p-Laplacian, critical point.
1. INTRODUCTION
In this paper we shall investigate the existence of weak solutions for the Dirichlet problem
(P)
−∆pu=λV(x)|u|p−2u+f(x, u) in Ω,
u= 0 on ∂Ω,
wherep >1, Ω⊆RN is a bounded domain,λ∈(λ1, λ2),f is a Carath´eodory function which satisfies the growth condition
(1.1) |f(x, t)| ≤h(x) forx∈Ω, t∈R,
whith h ∈ Lp(Ω) and 1p + p1 = 1. The weight function V belongs to Ls(Ω) withs > N if p∈(1, N] or V belongs toL1(Ω) if p > N. At the same time, we assume that
(1.2) V(x)>0 a.e. x∈Ω.
Some spectral properties of thep-Laplacian with indefinite weights were established, but much work remains to be done. Cuesta [1] proved that the least positive eigenvalue λ1 is simple, isolated in the spectrum, and is the unique eigenvalue associated with a nonnegative function. Furthermore, this author established the properties of the next smallest eigenvalue λ2 > λ1
giving a variational characterization through a minimax formula.
The aim of this article is to prove the following result.
REV. ROUMAINE MATH. PURES APPL.,52(2007),4, 389–395
Theorem 1.1. If hypotheses (1.1) and (1.2) are satisfied and the real numberλ∈(λ1, λ2), then problem(P)has at least a weak solution inW01,p(Ω).
2. VARIATIONAL FORMULATION OF THE PROBLEM We can formulate problem (P) variationally. For that purpose we intro- duce theC1 functionals Ψ, J,Φ and H:W01,p(Ω)→Rdefined by
Ψ(u) = 1
pup1,p, J(u) =
ΩV(x)|u|pdx, Φ(u) =
ΩF(x, u)dx, H(u) = Ψ(u)−λ
pJ(u)−Φ(u), whereF(x, t) =
Ωf(x, s)ds.
Definition 2.1. A functionu ∈W01,p(Ω) is said to be a weak solution of problem (P) if and only if
Ω|∇u|p−2∇u· ∇vdx=λ
ΩV(x)|u|p−2uvdx+
Ωf(x, u)vdx for allv∈W01,p(Ω).
It is well-known that H(u), v
=
Ω|∇u|p−2∇u· ∇vdx−λ
ΩV(x)|u|p−2uvdx−
Ωf(x, u)vdx for allu, v ∈W01,p(Ω).It follows that the weak solutions of problem (P) corre- spond to critical points ofH. In order to apply the standard methods of varia- tional theory, an important first step is to prove that, under the assumptions in Theorem 1.1,Hsatisfies the Palais-Smale condition, i.e., if (un)n∈N⊆W01,p(Ω) such that (H(un))n∈N ⊆ R is bounded and H(un) → 0 in W−1,p(Ω), then (un)n∈N⊆W01,p(Ω) has a convergent subsequence.
Theorem 2.1. If the hypotheses of Theorem 1.1 are satisfied, then H satisfies the Palais-Smale condition.
Proof. Let a sequence (un)n∈N ⊆W01,p(Ω) for which (H(un))n∈N ⊆Ris bounded and H(un) → 0 in W−1,p(Ω). Then there exists c > 0 such that
|H(un)| ≤cfor all n∈N.
In order to simplify the notation, consider the operatorsA, Nf :W01,p(Ω)
→W−1,p(Ω) defined by A(u), v=
ΩV(x)|u|p−2uvdx, Nf(u), v=
Ωf(x, u)vdx.
Notice that A is well-defined, odd, (p−1)-homogeneous and strongly continuous. Also, the growth condition (1.1) ensures that the operatorNf is strongly continuous and
Nf(u) ≤chp
for allu∈W01,p(Ω), where c >0 is the constant from Poincare’s inequality.
First, we prove that the sequence (un)n∈N⊆W01,p(Ω) is bounded. Sup- posing for a contradiction that (un)n∈N⊆W01,p(Ω) is unbounded, we can con- sider without any loss of generality that lim
n→∞un1,p= +∞ and un1,p>0 for all n ∈ N. Consider the sequence (vn)n∈N ⊆ W01,p(Ω) defined by vn =
un
un1,p for any n∈N. It is obvious that vn1,p = 1 for all n∈ Nand, using Smuljian’s theorem, we can extract a subsequence (vnk)k∈Nweakly convergent to somev∈W01,p(Ω). We have
(1) H(unk)
unkp−11,p = (−∆p)(vnk)−λA(vnk)− Nf(unk) unkp−11,p for allk∈N. As H(unk)→0 and unk1,p→ ∞, we obtain
(2) lim
k→∞
H(unk) unkp−11,p = 0.
From the weak convergence vnk v in W01,p(Ω), using the properties of the operatorsA and Nf, we obtain that
k→∞lim A(vnk) =A(v) and
(3) lim
k→∞
Nf(unk) unkp−11,p = 0. Using (1), (2) and (3), we conclude that
(4) lim
k→∞(−∆p)(vnk) =λA(v).
We know that−∆pis a homeomorphism ofW01,p(Ω) intoW−1,p(Ω). It is now clear, using the weak convergencevnk v inW01,p(Ω), that (−∆p)(v) = λA(v) and lim
k→∞vnk =vinW01,p(Ω). As the real numberλ∈(λ1, λ2) is not an eigenvalue of the p-Laplacian with indefinite weights, we deduce that v = 0.
But lim
k→∞vnk = v in W01,p(Ω) and vnk1,p = 1 for all k ∈ N and, because v= 0, we get a contradiction.
Second, we prove that the sequence (un)n∈N ⊆ W01,p(Ω) has a conver- gent subsequence. Since the sequence (un)n∈N ⊆ W01,p(Ω) is bounded, we
can extract a subsequence (unk)k∈N weakly convergent to some u ∈W01,p(Ω).
Using again the properties of the operators A and Nf, we conclude that
k→∞lim A(unk) =A(u) and lim
k→∞Nf(unk) =Nf(u). On the other hand, because H(unk)→0 as k→ ∞, we have
(5) lim
k→∞(−∆p)(unk) =λA(u) +Nf(u).
Finally, by (5) and taking into account that−∆p is a homeomorphism, we get
k→∞lim unk = (−∆p)−1(λA(u) +Nf(u)), and the proof is complete.
3. THE MAIN RESULT
In this section we prove Theorem 1.1. The proof will establish the ex- istence of a critical value of the functional H, characterized by a minimax formula. For that purpose we need to define the first two eigenvalues λ1 and λ2. Let us introduce the set M =
u ∈W01,p(Ω)|J(u) = 1
. It is simple to verify thatM =∅. Moreover, the setM is a manifold in W01,p(Ω) of classC1. Fork∈ {1,2}consider the set
Γk={A⊆M |A is compact, symmetric andγ(A)≥k}, whereγ(A) denotes Krasnoselski’s genus onW01,p(Ω).
The first two eigenvalues of the p-Laplacian are given by the formulae λ1= inf
A∈Γ1
supu∈A
Ω|∇u|pdx, λ2 = inf
A∈Γ2
supu∈A
Ω|∇u|pdx.
Cuesta [1] gave new characterizations ofλ1 andλ2, namely, λ1= inf
Ω|∇u|pdx|u∈W01,p(Ω) and J(u) = 1 , λ2= inf{λ∈R|λis an eigenvalue andλ > λ1}. Now, let
S =
u∈W01,p(Ω)|
Ω|∇u|pdx≥λ2J(u) and notice that there existsu1 ∈M such that
Ω|∇u1|pdx∈(λ1, λ). Consider the setsA ={−u1, u1} and T A ={tu1,−tu1 |t≥T}, whereT ≥0, and the real numberm=
Ω|∇u1|pdx.
Remark 3.1. There exist α = inf
u∈SH(u) and a real number T0 > 0 such thatα < γ= sup
u∈T0AH(u).
For the real numberT0 >0 appearing in Remark 3.1, consider the family of mappings Γ = {h : [−1,1] → W01,p(Ω) | h is continuous and there exists t≥T0 such that h(−1) =−tu1,h(1) =tu1 or vice-versa}. The properties of Γ are established in the two lemmas below.
Lemma 3.1. Γ is nonempty.
Proof. Let the function h : [−1,1]→ W01,p(Ω) defined by h(t) =tT0u1. It is obvious thath is continuous, h(−1) =−T0u1, and h(1) =T0u1. Clearly, the functionh belongs to Γ.
Lemma 3.2. For all h∈Γ we have h([−1,1])∩S=∅.
Proof. If 0∈h([−1,1]), then we are done. Otherwise, consider the closed setY =h([−1,1])⊆W01,p(Ω). It is obvious that 0∈/ Y.
By hypothesis (1.2), we have J(u) ≥0 for all u ∈ W01,p(Ω). Moreover, J(u) = 0 if and only ifu= 0. Let π:Y →M be the function defined by
π(u) = u J(u)1p.
Note that π is a continuous odd map on Y. Next, consider the function H : S1→M defined by
H(x1, x2) =
π(h(x1)) ifx2 ≥0
−π(h(−x1)) ifx2 <0, whereS1 is the unit sphere in RN.
Remark thatH is a continuous odd map and the setA0 =H(S1)⊆M is compact. From the equivalent characterization of λ2 gived by Drabek [4], we conclude that
u∈Asup0
Ω|∇u|pdx≥λ2.
Moreover, from the compactness ofA0 we deduce the existence of an element u0 ∈A0 for which
(6)
Ω|∇u0|pdx≥λ2.
On the other hand, there exist (x1, x2)∈S1 such thatu0 =H(x1, x2). With- out any loss of generality, we can suppose thatx2 ≥0, so that
u0 = h(x1) [J(h(x1))]1p
.
From (6) we obtain that (7)
Ω|∇(h(x1))|pdx≥λ2J(h(x1)).
It is clear from (7) thath(x1)∈S. Buth(x1)∈h([−1,1]). From the last two assertions we conclude that theh([−1,1])∩S =∅.
Before proving Theorem 1.1, we state a deformation theorem (see [5], p. 75) which plays a fundamental role in showing that the C1 functional H has critical points.
Theorem3.1. Let E be aC1 functional which satisfies the Palais-Smale condition on a Banach space X. Let β ∈ R be a regular value of E and let ε >0. Then there exist ε0 ∈(0, ε) and a continuous one-parameter family of homeomorphisms σ:X×[0,1]→X with the properties below:
a) σ(x, t) =x if t= 0 or |E(x)−β| ≥ε;
b)E(σ(x, t)) is non-increasing int for anyx ∈X; c)if E(x)≤β+ε0, then E(σ(x,1))≤β−ε0. Now, we can prove our main result.
Proof of Theorem 1.1. It suffices to show that the C1 functional H : W01,p(Ω)→R defined by
H(u) =
Ω|∇u|pdx−λ p
ΩV(x)|u|pdx−
ΩF(x, u)dx has at least a critical point. Consider the real number
c= inf
h∈Γ sup
t∈[−1,1]H(h(t)).
We shall prove by contradiction thatcis a critical value of H.
Suppose that cis a regular value of H. Lemma 3.2 ensures that for any h∈Γ there existt0 ∈[−1,1] such thath(t0)∈S. We have
H(h(t0))≥ inf
u∈SH(u) =α which gives the inequality
sup
t∈[−1,1]H(h(t))≥α for allh∈Γ, hencec≥α.
As theC1functionalHsatisfies the Palais-Smale condition, we can apply Theorem 3.1 for β = c and 0 < ε < c−γ to get a deformation σ and a correspondingε0. Notice that if u∈T0A, then H(u)≤γ≤c−ε, so
(8) σ(u, t) =u for all u∈T0A andt∈[0,1].
By the definition ofc, there ish0∈Γ such that sup
t∈[−1,1]H(h0(t))< c+ε0. By Theorem 3.1 we have
(9) H(σ(h0(t),1))< c−ε0 for all t∈[−1,1]. Consider the functionh1 : [−1,1]→W01,p(Ω) defined by
h1(t) =σ(h0(t),1).
It is obvious that h1 is continuous on [−1,1]. Moreover, taking into account thath0(−1)∈T0A andh0(1)∈T0A, using (8) we obtain
h1(1) =h0(1) and h1(−1) =h0(−1) and, clearly,
(10) h1∈Γ.
We can rewrite (9) asH(h1(t))< c−ε0 for allt∈[−1,1], hence
(11) sup
t∈[−1,1]H(h1(t))≤c−ε0.
Combining (10) and (11), we conclude thatc≤c−ε0 and the contradiction is obvious. Consequently, c is a critical value ofH and the existence of critical points forH is proved.
REFERENCES
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[3] G. Dinc˘a and P. Jebelean, Some existence results for a class of nonlinear equations involving a duality mapping. Nonlinear Anal. Ser. A: Theory Methods 46 (2001), 3, 347–363.
[4] P. Drabek,Resonance problems for the p-Laplacian.J. Funct. Anal.169(1999), 189–200.
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Springer-Verlag, New York, 1990.
Received 5 October 2006 University of Bucharest
Faculty of Mathematics and Computer Science Str. Academiei 14
010014 Bucharest, Romania ilias@fmi.unibuc.ro
