FOR THE p-LAPLACIAN

(1)

FOR THE p-LAPLACIAN

SIEGFRIED CARL AND KANISHKA PERERA Received 25 June 2002

We obtain a positive solution, a negative solution, and a sign-changing solution for a class ofp-Laplacian problems with jumping nonlinearities using variational and super-subsolution methods.

1. Introduction

LetΩ⊂R^N, N≥1, be a bounded domain with a smooth boundary∂Ω. In this paper, we consider the quasilinear elliptic boundary value problem

u∈W₀^1,p(Ω) :−∆pu=f(x, u) inW⁻^1,p(Ω), (1.1) where∆pu=div(|∇u|^p⁻²∇u) is thep-Laplacian, 1< p <∞. ByW^1,p(Ω) we denote the usual Sobolev space with dual space (W^1,p(Ω))^∗, andW₀^1,p(Ω) denotes its subspace whose elements have generalized homogeneous boundary values and whose dual space is given byW⁻^1,p(Ω). We assume the following growth and asymptotic behaviour of the nonlinear right-hand side f of (1.1):

(H1) f :Ω×R→Ris a Carath´eodory function satisfying f(x, t)≤C|t|^p⁻¹+ 1,

f(x, t)=at⁺^p⁻¹−bt⁻^p⁻¹+g(x, t), (1.2) where

limt→0

g(x, t)

|t_|^p⁻¹ ⁼0 uniformly inx. (1.3) HereCdenotes some generic positive constants.

Copyright©2002 Hindawi Publishing Corporation Abstract and Applied Analysis 7:12 (2002) 613–625 2000 Mathematics Subject Classification: 35J65, 35J20, 35B05 URL:http://dx.doi.org/10.1155/S1085337502207010

(2)

The setΣpof those points (a, b)∈R²for which the asymptotic problem u∈W0^1,p(Ω) :−∆pu=au⁺^p⁻¹−bu⁻^p⁻¹ inW⁻^1,p(Ω) (1.4) has a nontrivial solution, is called the Fuˇc´ık spectrum of the p-Laplacian on Ω, whereu⁺=max{u,0}andu⁻=max{−u,0}. The Fuˇc´ık spectrum was intro- duced in the semilinear casep=2 by Dancer [8] and Fuˇc´ık [14] who recognized its significance for the solvability of problems with jumping nonlinearities. In the semilinear ODE case p=2,N=1, Fuˇc´ık [14] showed that Σ2 consists of a sequence of hyperbolic-like curves passing through the points (λl, λl), where {λl}l∈Nare the eigenvalues of−d²/dx², with one or two curves going through each point. Dr´abek [12] has recently shown thatΣphas this same general shape for allp >1 in the ODE case.

In the PDE case N≥2, much of the work to date on Σp has been done for the semilinear case p=2. It is now known that Σ2 consists, at least locally, of curves emanating from the points (λ_l, λ_l) (see, e.g., [2, 7,8,10, 14, 25]). Schechter [28] has shown that Σ2 contains two continuous and strictly decreasing curves through (λ_l, λ_l), which may coincide, such that the points in the square (λ_l₋1, λ_l+1)²that are either below the lower curve or above the upper curve are not inΣ2, while the points between them may or may not belong toΣ2

when they do not coincide.

In the quasilinear PDE case p =2,N≥2, it is known that the first eigen- valueλ1of−∆pis positive, simple, and admits a positive eigenfunctionϕ1(see Lindqvist [20]), soΣpclearly contains the two linesλ1×RandR×λ1. In addi- tion,σ(−∆p) has an unbounded sequence of variational eigenvalues{λ_l}satisfying a standard min-max characterization, andΣpcontains the corresponding sequence of points{(λl, λl)}. A first nontrivial curveᏯin Σp through (λ2, λ2) asymptotic toλ1×RandR×λ1 at infinity was recently constructed and variationally characterized by a mountain-pass procedure by Cuesta et al. [6] (see Figure 1.1). More recently, unbounded sequences of curves (analogous to the lower and upper curves of Schechter) have been constructed and variationally characterized by min-max procedures by Micheletti and Pistoia [26] for p≥2 and by the second author [27] for allp >1.

The main goal of this paper is to identify the set of points (a, b) relative to the Fuˇc´ık spectrum which ensure the existence of sign-changing solutions of (1.1). More precisely, assuming the existence of a positive supersolutionuand a negative subsolutionuof (1.1) and (a, b) located above the curveᏯ, we prove the existence of at least three nontrivial solutions within the order interval [u, u];

a positive solution, a negative solution, and a sign-changing solution.

There are many existence and multiplicity results for (1.1) in the literature (see, e.g., [5,6,9,13,23,27]). However, to the best of our knowledge, the first results on sign-changing solutions were obtained only recently by Li and Zhang [17]. In their paper the authors assume that p > Nand f is independent ofx and locally Lipschitz int. All these assumptions can be relaxed by our approach

(3)

a b

(a, b)

(λ1, λ1)

Ꮿ (λ2, λ2)

Figure 1.1

which is very diﬀerent from that of Li and Zhang. Our main result, which will be proved inSection 3(Theorem 3.1), improves upon their results.

2. Preliminaries

We denote the norm inW^1,p(Ω) andL^p(Ω) by · and · p, respectively, and recall the notion of sub- and supersolutions.

Definition 2.1. A functionu∈W^1,p(Ω) is asupersolutionof (1.1) if the following holds:

(i)u≥0 on∂Ω,

(ii)_Ω|∇u|^p⁻²∇u· ∇ϕ dx≥

Ωf(x, u)ϕ dxfor allϕ∈W₀^1,p(Ω)∩L+^p(Ω).

Similarly,uis asubsolutionof (1.1) if the reversed inequalities ofDefinition 2.1hold withureplaced byu. HereL+^p(Ω) stands for the positive cone ofL^p(Ω).

Consider the boundary value problem

u∈W₀^1,p(Ω) :−∆pu=h inW⁻^1,p(Ω). (2.1) Besides the hypothesis (H1) we will assume the following hypotheses to hold throughout the rest of the paper.

(H2) There exist a positive supersolutionuand a negative subsolutionuof (1.1), and the point (a, b)∈R²is above the curveᏯof the Fuˇc´ık spectrum.

(H3) Any solutionuof (2.1) withh∈L^∞(Ω) belongs toC¹(Ω).

Remark 2.2. (i) Assuming the existence of super- and subsolutions as in hypothesis (H2) is a weaker assumption than the usual condition on the jumping nonlinearity at infinity.

(4)

(ii) By (H3) we imposeC¹(Ω)-regularity of the solution of (2.1). As for regularity results up to the boundary (C^1,α-regularity) we refer, for example, to Gi- aquinta and Giusti [16], Giaquinta [15], Liu and Barrett [21,22], Lieberman [18,19], or Manfredi [24].

Lemma2.3. Ifu≥(resp.,≤) 0is a solution of (1.1), then eitheru >(resp.,<) 0or u≡0. Moreover, ifu >0, then there is anε >0such thatu≥εϕ1, whereϕ1is the eigenfunction of the first eigenvalue of−∆p.

Proof. First, we note that by the results of Anane [1] and DiBenedetto [11] any solutionuof (1.1) belongs toL^∞(Ω)∩C¹(Ω), and thus the right-hand side of (1.1) yields a functionh∈L^∞(Ω), which by (H2) implies that u∈C¹(Ω). If u≥0 is a solution of (1.1) which is not identically zero, then by means of the Harnack inequality (Trudinger [29, Theorem 1.1])umust be positive inΩ. For >0, letΩ= {x∈Ω: dist(x, ∂Ω)≤}. Then forsuﬃciently small, we have f(x, u(x))≥0 for allx∈Ωby (H1) and (H2). This allows us to apply the strong maximum principle due to V´azquez [30] to get the strict inequality∂u/∂ν(x)>

0 for allx∈∂Ω, where ν is the interior normal atx. The eigenfunctionϕ1 of the first eigenvalue of−∆pis positive, is of classC^1,α(Ω) forα∈(0,1), and also satisfies∂ϕ1/∂ν(x)>0 (see [1,20]). Therefore, forεsuﬃciently small, we obtain

u≥εϕ1inΩ.

Lemma2.4. Given a bounded sequence{un} ⊂W₀^1,p(Ω)and a sequence of positive reals{εn}withεn→0asn→ ∞, for a subsequence,

1 εn^p⁻¹

Ω

gx, εnun(x)dx−→0 asn−→ ∞. (2.2)

Further, ifGis the primitive ofg, that is,G(x, t)=_t

0g(x, s)ds, then 1

εn^p

Ω

Gx, εnun(x)dx−→0 asn−→ ∞ (2.3)

for a subsequence.

Proof. Passing to a subsequence (again denoted by (un)), we may assume that u_n→ua.e. and inL^p(Ω). By Egoroﬀ’s theorem, for anyµ >0 there is a measur- able subsetΩµ ofΩsuch that|Ω\Ωµ| ≤µandun→uuniformly onΩµ. Thus εnun→0 a.e. inΩµ. We have

1 εn^p⁻¹

Ω

gx, εnun(x)dx

=

Ωµ

gx, εnun(x) εn^p⁻¹un(x)^p⁻¹

un(x)^p⁻¹dx

+

Ω\Ωµ

gx, ε_nu_n(x) εn^p⁻¹u_n(x)^p⁻¹

u_n(x)^p⁻¹dx.

(2.4)

(5)

By (1.2) and (1.3),

g(x, t)

|t|^p⁻¹ ≤C. (2.5)

The first integral on the right-hand side of (2.4) tends to zero by the asymptotic behavior (1.3) ofg, (2.5), and Lebesgue’s dominated convergence theorem (ob- serve that the integrand is majorized byC(|u(x)|+δ)^p⁻¹for anyδ >0 due to the uniform convergence inΩµ). The second integral is bounded by

CΩ\Ωµ^{1/ p}u_n^(p⁻^{1)/ p}≤Cµ^{1/ p}−→0 asµ−→0, (2.6) which proves (2.2). Observing that the elementary inequality

Gx, ε_nu_n(x)≤ε_nu_n(x)gx, τ_n(x)ε_nu_n(x) (2.7) holds, where 0≤τn(x)≤1, which yields

1 εn^p

Ω

Gx, εnun(x)dx≤

Ω

gx, τ_n(x)ε_nu_n(x) τn(x)^p⁻¹εn^p⁻¹un(x)^p⁻¹

un(x)^pdx, (2.8)

we see that (2.3) follows similarly.

Lemma2.5. Problem (1.1) has a positive solutionu >0within the order interval [0, u]and a negative solutionu <0within the order interval[u,0].

Proof. In the proof we focus on the existence of a positive solution only since the existence of a negative solution can be shown in a similar way.

As is well known, solutions of (1.1) are the critical points of Φ(u)=

Ω

|∇u|^p−p F(x, u)dx, u∈W₀^1,p(Ω), (2.9)

whereF(x, t)=_t

0f(x, s)ds. Let f be the following truncated nonlinearity:

f(x, t)=











0, t≤0,

f(x, t), 0< t < u(x), fx, u(x), t≥u(x),

(2.10)

andFits associated primitive given by F(x, t)=

_t

0 f(x, s)ds. (2.11)

Consider the functional Φ(u)=

Ω

|∇u|^p−pF(x, u)dx (2.12)

(6)

whose critical points are the solutions of the auxiliary boundary value problem u∈W₀^1,p(Ω) :−∆pu=f(x, u) inW⁻^1,p(Ω). (2.13) Obviously,Φ:W₀^1,p(Ω)→Ris bounded from below, weakly lower semicontin- uous, and coercive. Thus, there is a global minimizer, that is, a critical pointuof Φ

0=

Φ(u), ϕ=

Ω

|∇u|^p⁻²∇u· ∇ϕ−f(x, u)ϕdx. (2.14)

We will show that this global minimizer is in fact a positive solution of (1.1) within [0, u]. Taking in (2.14) the special test functionϕ=u⁻:=max(−u,0) we get in view of the definition of f the equation

0=

Ω

|∇u|^p⁻²∇u· ∇u⁻−f(x, u)u⁻dx=u⁻^p, (2.15)

which shows thatu⁻=0, and thusu≥0. Sinceuis a supersolution,ϕ=(u− u)⁺∈W0^1,p(Ω)∩L+^p(Ω), so byDefinition 2.1and (2.14) we obtain

0≥

Ω

|∇u|^p⁻²∇u− |∇u|^p⁻²∇u· ∇(u−u)⁺

−

f(x, u)−f(x, u)(u−u)⁺dx

=

{u>u}

|∇u|^p⁻²∇u− |∇u|^p⁻²∇u·(∇u− ∇u)dx≥0,

(2.16)

which implies that∇(u−u)⁺=0, and thusu≤u. This shows that the global minimizeruof the functionalΦsatisfiesu∈[0, u], and thusuis a solution of (1.1) due to the definition of f. Sincea > λ1, we get by hypothesis (H1) that

Φεϕ1

<0, ε >0 small. (2.17) Asuis a global minimizer ofΦ, it followsΦ(u)≤Φ(εϕ)<0, and thusumust be

a positive solution of (1.1).

Definition 2.6. A solutionu+ is called theleast positive solutionof (1.1) if any other positive solutionuof problem (1.1) satisfiesu≥u+. Similarly,u₋is the greatest negative solutionof (1.1) if any other negative solutionusatisfiesu≤u₋. Lemma2.7. Problem (1.1) has a least positive solutionu+and a greatest negative solutionu₋.

Proof. We are going to prove the existence of the least positive solution only, since the proof of the existence of the greatest negative solution is analogous.

In view ofLemma 2.5, there exists a positive solutionu∈[0, u], and applying

(7)

Lemma 2.3there is aε >0 small enough such thatεϕ1≤u, whereϕ1is the eigenfunction that belongs to the first eigenvalueλ1of−∆p. Sincea > λ1, one readily verifies thatεϕ1 is a subsolution of problem (1.1) for suﬃciently smallε >0.

Thus there is anε0>0 such thatε0ϕ1anduforms an ordered pair of sub- and supersolutions. Applying [3, Corollary 5.1.2] on the existence of extremal solutions for general quasilinear elliptic problems, we obtain the existence of a least and greatest solution of (1.1) with respect to the order interval [ε0ϕ1, u]. We denote the least solution within this interval byu0. Now let (εn)^∞_n₌0be a decreasing sequence withεn→0 asn→ ∞, and denote byunthe corresponding least solution of (1.1) with respect to the order interval [ε_nϕ1, u]. Then obviously (u_n) is a decreasing sequence of least positive solutions of (1.1) which converges to its nonnegative pointwise limitu_∗inL^p(Ω). We will show thatu_∗is in fact the least positive solution, that is,u_∗=u+. First we verify thatu_∗is a solution of (1.1).

Since theunare solutions of (1.1) we get from the equation un^p=

Ω

∇un^p⁻²∇un· ∇undx=

Ωfx, un

undx, (2.18) which by the growth condition (H1) and the boundedness inL^p(Ω) of the sequence (un) implies its boundedness inW0^1,p(Ω), that is,un ≤c. Thus there exists a subsequence weakly convergent inW₀^1,p(Ω), and due to the strong convergence of (un) in L^p(Ω) even the entire sequence is weakly convergent in W₀^1,p(Ω) with weak limitu_∗. From (1.1) with the test functionu_n−u_∗, we obtain

−∆pu_n, u_n−u_∗=

Ω

∇u_n^p⁻²∇u_n· ∇

u_n−u_∗dx

=

Ωfx, u_nu_n−u_∗dx,

(2.19)

which implies

lim sup

n

−∆pu_n, u_n−u_∗≤0. (2.20)

The weak convergence of (un) and (2.20) along with the S+-property of the operator−∆p(see, e.g., [3, Chapter D]) yield its strong convergence inW₀^1,p(Ω). This allows the passage to the limit in (1.1) withureplaced byun, and henceu_∗is a solution of problem (1.1). To show thatu_∗>0, our argument is by contradiction. Supposeu_∗=0, that is,u_n→0 inW₀^1,p(Ω). Sinceu_n>0 we may consider

˜

un:=un/unwhich satisfies

Ω

∇u˜_n^p⁻²∇u˜_n· ∇ϕ dx=

Ω

a˜u^pn⁻¹+gx, u_n u_n^p⁻¹

ϕ dx. (2.21)

(8)

By definitionu˜n =1, so there is a subsequence (˜un) that converges weakly in W0^1,p(Ω) and strongly inL^p(Ω) to ˜udue to the compact embedding ofW0^1,p(Ω)

⊂L^p(Ω). Taking in (2.21) as special test functionϕ=u˜_n−u, we get for the right-˜ hand side of (2.21)

Ω

a˜un^p⁻¹+ gx, u_n u_n(x)^p⁻¹

u˜_n(x)^p⁻¹

˜

u_n−u˜dx−→0, (2.22) asn→ ∞, because the terms in parentheses areL^q(Ω)-bounded. Hence (2.21) implies

lim sup

n

−∆pu˜_n,u˜_n−u˜≤0, (2.23)

which due to the S+-property of−∆p implies the strong convergence of ˜un→u˜ inW0^1,p(Ω). Moreover, the third integral term on the right-hand side of (2.21) converges to zero byLemma 2.4, so we may pass to the limit to get

Ω|∇u˜|^p⁻²∇u˜· ∇ϕ dx=

Ωa˜u^p⁻¹ϕ dx ∀ϕ∈C^∞₀(Ω), (2.24) that is, ˜usatisfies the boundary value problem

˜

u∈W₀^1,p(Ω) :−∆pu˜=a˜u^p⁻¹ inW⁻^1,p(Ω). (2.25) Since u˜n =1 and ˜un>0,by Lemma 2.3 we have the same properties for ˜u, which, however, contradicts the fact that a nontrivial solution of (2.25) changes sign. So far we have shown that the limitu_∗of the least solutionsu_n∈[ε_nϕ1, u] is a positive solution of (1.1). Finally, to prove thatu_∗is the least positive solution, letwbe any positive solution of (1.1). Then byLemma 2.3there is aεn>0 forn suﬃciently large such thatε_nϕ1≤wwhich by definition of the sequence of least solutions (un) yieldsu_∗≤un≤w, which proves thatu_∗=u+is in fact the least

positive one.

3. Main result

Theorem3.1. Let hypotheses (H1), (H2), and (H3) be satisfied. Then the bound- ary value problem (1.1) has at least three nontrivial solutions: a positive solution, a negative solution, and a sign-changing solution.

Proof. Let

f+(x, t)=











0, t≤0,

f(x, t), 0< t < u+(x),F+(x, t)=_t

0 f+(x, s)ds, fx, u+(x), t≥u+(x),

(3.1)

(9)

and consider

Φ+(u)=

Ω

|∇u|^p−pF+(x, u)dx. (3.2)

Arguments similar to those in the proof ofLemma 2.5show that critical points ofΦ+are solutions of (1.1) in the order interval [0, u+], so 0 andu+are the only critical points ofΦ+by Lemmas2.3and2.7. Now,Φ+is bounded from below and coercive, and

Φ+

εϕ1

<0, ε >0 small (3.3)

sincea > λ1, soΦ+has a global minimizer at a negative critical level. It follows thatu+is the (strict) global minimizer ofΦ+andΦ+(u+)<0.

Now let

f(x, t)=











fx, u₋(x), t≤u₋(x),

f(x, t), u₋(x)< t < u+(x),F(x, t) =_t

0f(x, s)ds, fx, u+(x), t≥u+(x),

Φ(u)=

Ω

|∇u|^p−pF(x, u) dx.

(3.4)

As before, critical points ofΦare solutions of (1.1) in the order interval [u₋, u+], so it follows from Lemmas2.3 and2.7that any nontrivial critical point other thanu_±is a sign-changing solution.

Lemma3.2. The solutionsu_±are strict local minimizers ofΦ, and Φ(u ±)<0.

Proof. We only consideru+as the argument foru₋is similar. Suppose that there is a sequenceu_j→u+inW₀^1,p(Ω), u_j =u+withΦ(u_j)≤Φ(u+). By (1.2) and (1.3) we have

F(x, t)≤C|t|^p, (3.5)

so Φuj

=

Ω

∇u⁺_j^p−pFx, u⁺_jdx+

Ω

∇u⁻_j^p−pFx, u⁻_jdx

≥Φ+

u⁺_j+u⁻_j^p−Cu⁻_j^p_p.

(3.6)

Ifu⁻_j =0, thenu⁺_j =u+and Φ+

u⁺_j≤Φuj

≤Φu+

=Φ+ u+

, (3.7)

(10)

contradicting the fact thatu+is the unique global minimizer ofΦ+, sou⁻_j =0.

We will show that

u⁻_j^p> Cu⁻_j^p_p, jlarge. (3.8)

Assuming this for the moment, we have the contradictionΦ+(u⁺_j)<Φ+(u+).

To see that (3.8) holds, we first note that the measure of the setΩj= {x∈Ω: u_j(x)<0}goes to zero. To see this, givenε >0, take a compact subsetΩ^ε ofΩ such that|Ω\Ω^ε|< εand letΩ^ε_j=Ω^ε∩Ωj. Then

u_j−u+^p

p≥

Ω^εj

u_j−u+^pdx≥

Ω^εj

u+^pdx≥c^pΩ^εj, (3.9)

wherec₌minΩ^εu+>0, so|Ω^εj| →0. SinceΩj⊂Ω^εj∪(Ω\Ω^ε) andε >0 is arbi- trary, the claim follows.

If (3.8) does not hold, settinguj=u⁻_j/u⁻_jp,ujis bounded for a subsequence, sou_j→uinL^p(Ω) and a.e. for a further subsequence, whereup=1 andu≥0. But thenΩµ= {x∈Ω:u(x) ≥µ}has positive measure for all suﬃ- ciently smallµ >0 and

uj−u^p_p≥

Ωµ\Ωj

uj−u^pdx=

Ωµ\Ωj

u^pdx≥µ^pΩµ−Ωj, (3.10)

a contradiction.

Now a standard deformation argument gives a mountain-pass pointu1at the critical value

c:=inf

γ∈Γ max

u∈γ([−1,1])Φ(u) >Φu_±, (3.11) where Γ= {γ∈C([−1,1], W0^1,p(Ω)) :γ(±1)=u_±} is the class of paths joining u_±. To show that u1 =0, we will construct a path that lies in Φ⁰:= {u∈ W₀^1,p(Ω) :Φ(u)<0}.

First we show that, for all suﬃciently smallε >0,±εϕ1can be joined by a path γεinΦ⁰. We have

Φ(u) =I(a,b)(u)−

ΩG(x, u)dx, (3.12)

(11)

where

I(a,b)(u)=

Ω

|∇u|^p−au⁺^p−bu⁻^pdx (3.13)

is the functional associated with (1.4) and

G(x, t) ₌F(x, t) ₋at⁺^p₋bt⁻^p₌o_|t_|^p ast_−→0. (3.14) Since (a, b) is aboveᏯ, there is a pathγ0in{u∈W0^1,p(Ω) :I(a,b)(u)<0,up= 1}joining±ϕ1by the construction ofᏯ[6]. Foru∈γ0([−1,1]),

Φ(εu)≤ε^p

maxI(a,b) γ0

[−1,1]+

Ω

G(x, εu) ε^p dx

, (3.15)

and the last integral goes to 0 uniformly on the compact setγ0([−1,1]) asε→0 byLemma 2.4, so we can takeγε=εγ0.

We complete the proof by showing that±εϕ1andu_±can be joined by paths in Φ⁰. Again we only considerεϕ1 and u+. Settingα=infΦ+=Φ+(u+) and β=Φ+(εϕ1)=Φ(εϕ1)<0, by the second deformation lemma (see, e.g., Chang [4]), the sublevel setΦ^α+:= {u∈W0^1,p(Ω) :Φ+(u)≤α} = {u+}is a strong deformation retract ofΦ^β+, that is, there is anη∈C([0,1]×Φ^β+,Φ^β+) such that

(i)η(0, u)=ufor allu∈Φ^β+, (ii)η(t, u+)=u+for allt∈[0,1], (iii)η(1, u)=u+for allu∈Φ^β+.

In particular,γ=η(·, εϕ1) is a path inΦ^β+joiningεϕ1andu+. Now the pathγ+

defined byγ+(t)=γ(t)⁺also joinsεϕ1andu+, and Φγ+(t)=Φ+(γ(t))−

Ω

∇γ(t)⁻^p≤β. (3.16)

References

[1] A. Anane,Simplicit´e et isolation de la premi`ere valeur propre dup-laplacien avec poids [Simplicity and isolation of the first eigenvalue of thep-Laplacian with weight], C.

R. Acad. Sci. Paris S´er. I Math.305(1987), no. 16, 725–728 (French).

[2] N. C´ac,On nontrivial solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue, J. Diﬀerential Equations80(1989), no. 2, 379–404.

[3] S. Carl and S. Heikkil¨a,Nonlinear Diﬀerential Equations in Ordered Spaces, Mono- graphs and Surveys in Pure and Applied Mathematics, vol. 111, Chapman &

Hall/CRC, Florida, 2000.

[4] K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Diﬀerential Equations and Their Applications, vol. 6, Birkh¨auser Boston, Massachusetts, 1993.

[5] D. G. Costa and O. H. Miyagaki,Nontrivial solutions for perturbations of the p- Laplacian on unbounded domains, J. Math. Anal. Appl.193(1995), no. 3, 737–

755.

(12)

[6] M. Cuesta, D. de Figueiredo, and J.-P. Gossez,The beginning of the Fuˇc´ık spectrum for thep-Laplacian, J. Diﬀerential Equations159(1999), no. 1, 212–238.

[7] M. Cuesta and J.-P. Gossez,A variational approach to nonresonance with respect to the Fuˇc´ık spectrum, Nonlinear Anal.19(1992), no. 5, 487–500.

[8] E. N. Dancer,On the Dirichlet problem for weakly non-linear elliptic partial diﬀeren- tial equations, Proc. Roy. Soc. Edinburgh Sect. A76(1976/77), no. 4, 283–300, Erratum: Proc. Roy. Soc. Edinburgh Sect. A89(1981), no. 1-2, 15.

[9] N. Dancer and K. Perera,Some remarks on the Fuˇc´ık spectrum of thep-Laplacian and critical groups, J. Math. Anal. Appl.254(2001), no. 1, 164–177.

[10] D. de Figueiredo and J.-P. Gossez,On the first curve of the Fuˇc´ık spectrum of an elliptic operator, Diﬀerential Integral Equations7(1994), no. 5-6, 1285–1302.

[11] E. DiBenedetto,C^1+αlocal regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal.7(1983), no. 8, 827–850.

[12] P. Dr´abek,Solvability and Bifurcations of Nonlinear Equations, Pitman Research Notes in Mathematics Series, vol. 264, Longman Scientific & Technical, Harlow, 1992.

[13] P. Dr´abek and S. B. Robinson,Resonance problems for thep-Laplacian, J. Funct. Anal.

169(1999), no. 1, 189–200.

[14] S. Fuˇc´ık,Boundary value problems with jumping nonlinearities, ˇCasopis Pˇest. Mat.101 (1976), no. 1, 69–87.

[15] M. Giaquinta,The problem of the regularity of minimizers, Proceedings of the Inter- national Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), American Mathematical Society, Rhode Island, 1987, pp. 1072–1083.

[16] M. Giaquinta and E. Giusti,GlobalC^1,α-regularity for second order quasilinear elliptic equations in divergence form, J. reine angew. Math.351(1984), 55–65.

[17] S. J. Li and Z. Zhang, On sign-changing and multiple solutions of p-Laplacian, preprint, 2001.

[18] G. M. Lieberman,Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal.12(1988), no. 11, 1203–1219.

[19] ,Regularity of solutions to some degenerate double obstacle problems, Indiana Univ. Math. J.40(1991), no. 3, 1009–1028.

[20] P. Lindqvist,On the equationdiv(|∇u|^p−2∇u) +λ|u|^p−2u=0, Proc. Amer. Math. Soc.

109(1990), no. 1, 157–164, Addendum: Proc. Amer. Math. Soc.116(1992), no.

2, 583–584.

[21] W. B. Liu and J. W. Barrett,Higher-order regularity for the solutions of some degenerate quasilinear elliptic equations in the plane, SIAM J. Math. Anal.24(1993), no. 6, 1522–1536.

[22] ,A remark on the regularity of the solutions of thep-Laplacian and its applica- tion to their finite element approximation, J. Math. Anal. Appl.178(1993), no. 2, 470–487.

[23] Z. Liu and J. Su,Nontrivial solutions for perturbedp-Laplacian onR^N, preprint, 2001.

[24] J. J. Manfredi,Regularity for minima of functionals with p-growth, J. Diﬀerential Equations76(1988), no. 2, 203–212.

[25] C. A. Margulies and W. Margulies,An example of the Fuˇc´ık spectrum, Nonlinear Anal.

29(1997), no. 12, 1373–1378.

[26] A. M. Micheletti and A. Pistoia,On the Fuˇc´ık spectrum for thep-Laplacian, Diﬀeren- tial Integral Equations14(2001), no. 7, 867–882.

[27] K. Perera,On the Fuˇc´ık spectrum of thep-Laplacian, to appear in NoDEA Nonlinear Diﬀerential Equations Appl.

(13)

[28] M. Schechter,The Fuˇc´ık spectrum, Indiana Univ. Math. J.43(1994), no. 4, 1139–

1157.

[29] N. S. Trudinger,On Harnack type inequalities and their application to quasilinear el- liptic equations, Comm. Pure Appl. Math.20(1967), 721–747.

[30] J. L. V´azquez,A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim.12(1984), no. 3, 191–202.

Siegfried Carl: Fachbereich Mathematik und Informatik, Institut f ¨ur Analy- sis, Martin-Luther-Universit¨at Halle-Wittenberg, D-06099Halle, Germany

E-mail address:[email protected]

Kanishka Perera: Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL32901, USA

E-mail address:[email protected] URL:http://winnie.fit.edu/∼kperera/