数理解析研究所講究録
第 1779 巻 2012 年 1-10

1

Existence and non-existence results of
the Fucik type spectrum for thegeneralized -Laplace operators$p$

東京理科大学理学部二部数学科

田中

視英子 (Mieko Tanaka)

Department of Mathematics, Tokyo University of Science

1

Introduction

In this paper, we consider the existence offor which the following
quasilinear elliptic equation has a non-trivial solution:$(\alpha, \beta)\in \mathbb{R}^{2}$$(F)_{(\alpha,\beta)}$

where$\{\begin{array}{ll}-divA(x, \nabla u)=\alpha u_{+}^{p-1}-\beta u_{-}^{p-1} in \Omega,\frac{\partial u}{\partial\nu}=0 on \partial\Omega,\end{array}$denotes the outward unit normal vector on$1<p<\infty,$. Here,boundaryisa map which is strictly monotone in the second variable and satisfies certain
regularity conditions (see the following assumption $(A)$ ). The equation
contains the corresponding p-Laplacian problem as a special case, and in thiscase,admitting a non-trivial solution tois said to belong to
the Fu\v{c}lk $spectr^{v}um$ of the p-Laplacian. Although the p-Laplace operator is$(p-1)$ -homogeneous, the operator
is not supposed generally to be $(p-1)-$
homogeneous in the second variable.
Here, we say that $u\in W^{1,p}(\Omega)$ is a (weak) solution ofif$\mathbb{R}^{N}$$\nu$$\Omega\subset$$\partial\Omega,$

is a bounded domain with$C^{2}$$A:\overline{\Omega}\cross \mathbb{R}^{N}arrow \mathbb{R}^{N}$$\partial\Omega$$(F)_{(\alpha,\beta)}$$(\alpha, \beta)$$(F)_{(\alpha,\beta)}$$A$$(F)_{(\alpha,\beta)}$$\int_{\Omega}A(x, \nabla u)\nabla\varphi dx=\int_{\Omega}\alpha u_{+}^{p-1}\varphi dx-\int_{\Omega}\beta u_{-}^{p-1}\varphi dx$

for all.
Throughout this paper, we assume that the opemtor
assumption $(A)$ :$\varphi\in W^{1,p}(\Omega)$$A$satisfies the following$(A)A(x, y)=a(x, |y|)y$ , where $a(x, t)>0$ for all $(x, t)\in\overline{\Omega}\cross(0, +\infty)$ and(i)$A\in C^{0}(\overline{\Omega}\cross \mathbb{R}^{N}, \mathbb{R}^{N})\cap C^{1}(\overline{\Omega}\cross(\mathbb{R}^{N}\backslash \{0\}), \mathbb{R}^{N})$;(ii) there exists a $C_{1}>0$ such that$|D_{y}A(x, y)|\leq C_{1}|y|^{p-2}$

for every$x\in\overline{\Omega}$, and$y\in \mathbb{R}^{N}\backslash \{0\}$;(iii) there exists a $C_{0}>0$ such that
$D_{y}A(x, y)\xi\cdot\xi\geq C_{0}|y|^{p-2}|\xi|^{2}$for every$x\in\overline{\Omega},$$y\in \mathbb{R}^{N}\backslash \{0\}$and$\xi\in \mathbb{R}^{N}$.

2(iv) there exists a $C_{2}>0$ such thatfor every$|D_{x}A(x, y)|\leq C_{2}(1+|y|^{p-1})$$x\in\overline{\Omega},$$y\in \mathbb{R}^{N}\backslash \{0\}$.

Throughout this paper, we assume $C_{0}\leq p-1\leq C_{1}$ because we can take suchandanew if necessary.
desiredThe hypothesis $(A)$ has been considered in the study of the quasilinear elliptic problems (cf. [6], [12], [13]). For example, we can treat the operators likethe p-Laplacian with the positive weight and$C_{1}$$C_{0}$$div((|\nabla u|^{p-2}+|\nabla u|^{q-2})(1+|\nabla u|^{q})^{R=_{q}Z}\nabla u)$

for $1<p\leq q<\infty$ .

Let us recall the known results in the special case of $A(x, y)=|y|^{p-2}y$ thatis, -Laplace problem and $C_{0}=C_{1}=p-1$ . The set of all points
for which the equation$(\alpha, \beta)\in \mathbb{R}^{2}$

$p$

in

$-\triangle_{P}u=\alpha u_{+}^{p-1}-\beta u_{-}^{p-1}$,$\Omega$$\frac{\partial u}{\partial\nu}=0$

on

(1)

$\partial\Omega$

has a non-trivial solution is called the Fu\v{c}lk spectrum of the p-Laplacian under
the Neumann boundary condition. In this paper, we denote the Fu\v{c}\’ik spectrum
isof p-Laplacian by . It is well known that the first eigenvalue
simple and every eigenfunction corresponding to $\mu_{1}=0$ is a constant function.
(we call these lines as
andTherefore,
contains the lines“the trivial lines“). Furthermore, by the same argument as in [5], it can bewhich isproved that there exists a Lipschitz continuous curve contained in(see Section 2). In the p-Laplacian case,called “the first nontrivial curve”
many authors have treated the Fu\v{c}\’ik spectrum (see [5], [7], [8], [10] under the
Dirichlet boundary condition and [2], [3] for Neumann boundary condition).
Let us return to the general case. In [14], D. Motreanu and the present
author treated the equation$\mu_{1}=0of-\triangle_{p}$$\Theta_{p}$$\mathbb{R}\cross\{0\}$$\{0\}\cross \mathbb{R}$$\Theta_{p}$$\Theta_{p}$$\mathscr{C}$$\partial u$in$-divA(x, \nabla u)=f(x, u)$$\Omega$,$–=0$ on(2)$\partial\Omega$$\partial\nu$with the following nonlinearity:$f(x, u)=\{\begin{array}{ll}\alpha_{0}u_{+}^{p-1}-\beta_{0}u_{-}^{p-1}+o(|u|^{p-1}) at 0,\alpha u_{+}^{p-1}-\beta u_{-}^{p-1}+o(|u|^{p-1}) at \infty\end{array}$. Roughly speaking, by constructing two curves andforrelated to the map (see section 3), it was shown that the equation (2) has ais below the curve andsign-changing solution in the case whereis above the curve . In the p-Laplacian case, we see that two curves andcoincide with the first nontrivial curve . Moreover, if the first nontrivial curve, then $equation-\triangle_{p}u=f(x, u)$ in (under theandlies between
Dirichlet boundary condition) has a non-trivial solution. Therefore, even for thegeneral case of , it seems reasonable to expect the existence of uncountably
many Fu\v{c}\’ik type spectrum between and .
Mainly, this paper consists of results in [14] and [15]. In the final section,
we see further results and several questions concerning our problem.
$\tilde{\mathscr{C}}$$(\alpha_{0}, \beta_{0}),$$(\alpha, \beta)\in \mathbb{R}^{2}$$A$$\underline{\mathscr{C}}$$(\alpha, \beta)$$\underline{\mathscr{C}}$$(\alpha_{0}, \beta_{0})$$\tilde{\mathscr{C}}$$\tilde{\mathscr{C}}$$\underline{\mathscr{C}}$$\mathscr{C}$$(\alpha_{0}, \beta_{0})$$\Omega$$(\alpha, \beta)$$A$$\tilde{\mathscr{C}}$$\underline{\mathscr{C}}$3
2The first nontrivial curve contained in$\Theta_{p}$Here, we recall the result for the special case of $A(x, y)=|y|^{p-2}y$ , that is, pLaplacian problems (note that we can take $C_{0}=C_{1}=p-1$ in $(A)$ ). The
construction of the curve
contained in the Fu\v{c}\’ik spectrum is carried out by
the same argument as in [5]: For $s\geq 0$ , we define$\mathscr{C}$$J_{s}(u);= \int_{\Omega}|\nabla u|^{p}$$S$$\Sigma$dx–s$\int_{\Omega}u_{+}^{p}dx$

for $u\in W^{1,p}(\Omega)$ ,$:= \{u\in W^{1,p}(\Omega);\int_{\Omega}|u|^{p}dx=1\}$$\tilde{J}_{s}:=J_{s}|s$,$:=\{\gamma\in C([0,1], S);\gamma(0)=\psi_{1}, \gamma(1)=-\psi_{1}\}$

,

where
. Here, the set $C([0,1], S)$ denotes the set ofcontinuous functions $hom[0,1]$ to with the topology induced by thenorm. Finally, we set.
(3)$\psi_{1}=1/|\Omega|^{1/P}$$(so \Vert\psi_{1}\Vert_{p}=1)$$S$$W^{1,p}(\Omega)$$c(s):= \inf_{\gamma\in}\max_{t\in[0,1]}\tilde{J}_{s}(\gamma(t))$

Then, it can be proved that $c(s)$ is a positive critical value ofwith $c(O)=$, whereis the second eigenvalue of the p-Laplacian under the Neumannboundary condition. Moreover, we can see that $c(s)$ is continuous, strictly
decreasing in $s\geq 0$ and $c(s)+s$ is strictly increasing in $s\geq 0$ (refer to [1,Lemma2.2] and [5, Proposition 4.1] . Then,is defined as follows:$J_{s}$$\mu_{2}$$\mu_{2}$$)$$\mathscr{C}$$\mathscr{C}:=\{(c(s)+s, c(s));s\geq 0\}\cup\{(c(s), c(s)+s);s\geq 0\}$.

Finally, we remark that in the case of $N\geq p$ , it is shown in [3] that $c(s)arrow 0$ as$sarrow\infty$ , whence the asymptotic lines of the first
nontrivial curve are the trivial
lines
and
. However, if $N<p$ , thenas $sarrow\infty$ , whereis a positive constant defined by$\mathbb{R}\cross\{0\}$$c(s)arrow\overline{\lambda}$$\{0\}\cross \mathbb{R}$$\overline{\lambda}$$\overline{\lambda}=\inf_{B}\int_{\Omega}|\nabla u|^{p}dx$,

where $B:=\{u\in S;u(x_{0})=0$ for some $x_{0}\in 7\}$ .

This yields that the trivial lines are not the asymptotic lines of the first nontrivial
curve.

3

Existence and non-existence results

To state the results for$(F)_{(\alpha,\beta)}$, we define curves$\underline{\mathscr{C}}$and$\tilde{\mathscr{C}}$by,$\underline{\mathscr{C}}:=\frac{C_{0}}{p-1}\mathscr{C}:=\{(aC_{0}/(p-1), bC_{0}/(p-1));(a, b)\in \mathscr{C}\}$$\tilde{\mathscr{C}}:=\frac{C_{1}}{p-1}\mathscr{C}=\{(aC_{1}/(p-1), bC_{1}/(p-1));(a, b)\in \mathscr{C}\}$

,

where
and
are positive constants satisfying $(A)$ . First, we state theelementary results for the equation
which is shown in [14].
$C_{0}$$C_{1}$$(F)_{(\alpha,\beta)}$

Proposition 1 ([14, Proposition 2]) The following assertions hold:4(i) iftions;$\alpha\beta<0$

or $\max\{\alpha, \beta\}<0$ holds, then(ii) $ifu$ is a non-trivial solution ofsign;$(F)_{(\alpha,\beta)}$

is a non-trivial solution offunction;(iii) if$u$$(F)_{(\alpha,\beta)}$

has no non-trivial solu-

with $\min\{\alpha, \beta\}>0$ , then$(F)_{(\alpha,\beta)}$(iv) if $0<\alpha<\alpha’$ and $0<\beta<\beta’$ for somewith$\alpha\beta=0$$(\alpha’, \beta^{l})\in\underline{\mathscr{C}}$, then, then$u$$u$changes

is a constant$(F)_{(\alpha,\beta)}$

has nonon-trivial solutions.

Define$\beta_{0}(s)$and$\beta_{1}(s)$for $s\geq 0$ by$\beta_{0}(s):=\frac{C_{0}}{p-1}c(\frac{p-1}{C_{0}}s)$

where$c(\cdot)$,$\beta_{1}(s):=\frac{C_{1}}{p-1}c(\frac{p-1}{C_{1}}s)$,

is a function defined by (3) (see the following figure):.

$\cdot$Now, we state existence results.

Theorem 2 ([15]) For every $s\geq 0$ and $R>0$ , there exists a
have at least one sign-changing solution
andsuch that.
with$\beta\in[\beta_{0}(s), \beta_{1}(s)]$$(F)_{(\beta+s,\beta)}$$u\in C^{1}(\overline{\Omega})$$(F)_{(\beta,\beta+s)}$$\int_{\Omega}|u|^{P}dx\leq R^{p}$

Theorem 3 ([15]) Let $s\geq 0,$$R_{2}> \max\{\frac{\beta_{1}(s)+s+\epsilon}{\min\{\beta_{0}(s),\epsilon\}},$$\epsilon>0$

and $R_{2}>R_{1}>0$ be constants satisfying$\frac{C_{1}(\beta_{1}(s)+s+\epsilon)^{2}}{C_{0}(\beta_{1}(s)+\epsilon)^{2}},$vists ahave at least one sign-changing solution
Then, there$e$$\beta\in[\beta_{0}(s), \beta_{1}(s)+\epsilon]$$\frac{s(C_{1}-C_{0})}{C_{0}(\beta_{1}(s)+\epsilon)}\}^{1/p}R_{1}$such thatwith$u\in C^{1}(\overline{\Omega})$$(F)_{(\beta+s,\beta)}$and.

$(F)_{(\beta,\beta+s)}$$R_{1}^{p} \leq\int_{\Omega}|u|^{p}dx\leq R_{2}^{p}$

.

5
3.1

Variational setting and notations

In what follows, we define the norm of $W$ $:=W^{1,p}(\Omega)$ by
,
where
$u\in
L^{q}(\Omega)(1\leq
q\leq\infty)$
denotes the norm offor. Define$G(x, y):= \int_{0}^{|y|}a(x, t)tdt$ , thenwe can easily see that$\Vert u\Vert^{p}$$\Vert u\Vert_{q}$and $G(x, 0)=0$$\nabla_{y}G(x, y)=A(x, y)$for every$:=\Vert\nabla u\Vert_{p}^{p}+\Vert u\Vert_{p}^{p}$$L^{q}(\Omega)$$x\in\overline{\Omega}$.

Remark 4 The following assertions hold:(i) for all$x\in\overline{\Omega},$$A(x, y)$is maximal monotone and strictly monotone in ;$y$(ii) $|A(x, y)| \leq\frac{C_{1}}{p-1}|y|^{p-1}$ for every(iii)$A(x, y)y\geq\overline{p}-\overline{1}C_{\Delta}|y|^{p}$for every(iv) $G(x, y)$ is convex in$y$for all$(x, y)\in\overline{\Omega}\cross \mathbb{R}^{N}$$(x, y)\in\overline{\Omega}\cross \mathbb{R}^{N}$

$x$

where

$C_{0}$

and$(x, y)\in\overline{\Omega}\cross \mathbb{R}^{N}$$C_{1}$;and satisfies the following inequalities:$A(x, y)y \geq G(x, y)\geq\frac{C_{0}}{p(p-1)}|y|^{p}$for every

;

and

(4)

$G(x, y) \leq\frac{C_{1}}{p(p-1)}|y|^{p}$,

are the positive constants described in $(A)$ .

For parameters $s\geq 0$ andonby$\beta\in \mathbb{R}$, we define the$C^{1}$

functionals

and$I_{\beta,s}$$I_{\beta,s}^{+}$$W^{1,p}(\Omega)$$I_{\beta,s}(u)$$:= \int_{\Omega}G(x, \nabla u)dx-\frac{\beta+s}{p}\int_{\Omega}u_{+}^{p}dx-\frac{\beta}{p}\int_{\Omega}u_{-}^{p}dx$

with$\langle I_{\beta,s}^{l}(u),$$I_{\beta,s}^{+}(u)$for,$v \}=\int_{\Omega}A(x, \nabla u)\nabla vdx-(\beta+s)\int_{\Omega}u_{+}^{p-1}vdx+\beta\int_{\Omega}u_{-}^{p-1}vdx$$:= \int_{\Omega}G(x, \nabla u)dx-\frac{\beta+s}{p}\int_{\Omega}u_{+}^{p}dx$$u,$ $v\in W^{1,p}(\Omega)$. In this paper, we use the following notations:$B(r):=\{u\in W;\Vert u\Vert\leq r\}$,$D(r, r’):=\{u\in W;r\leq\Vert u\Vert\leq r^{l}\}$$rS:=\{u\in W;\Vert u\Vert_{p}=r\}$,$B_{p}(r):=\{u\in W;\Vert u\Vert_{p}\leq r\}$,,$D_{p}(r, r’)$ $:=\{u\in W;r\leq\Vert u\Vert_{p}\leq r’\}$$rS_{+}:=\{u\in W;\Vert u_{+}\Vert_{p}=r\}$for $r’\geq r>0$ . Here, we note that the topology of all subsets above are inducedby thenorm. We set$W^{1,p}(\Omega)$$K(I_{\beta,s}):=\{u\in W;I_{\beta,s}’(u)=0\}$

for$c\in \mathbb{R}$.

and$I_{\beta,s}^{c}:=\{u\in W;I_{\beta,s}(u)\leq c\}$

6Remark 5 Let $u\in W^{1,p}(\Omega)$ be a critical point ofequality$I_{\beta,s}$, namely,$u$satisfies the$\int_{\Omega}A(x, \nabla u)\nabla\varphi dx=(\beta+s)\int_{\Omega}u_{+}^{p-1}\varphi dx-\beta\int_{\Omega}u_{-}^{p-2}\varphi dx$

for everysee$\varphi\in W^{1,p}(\Omega)$. Then, because of(see Appendix in [14]), $we$
[1 $1J)$ .
(cf.
regularity
resultby thein the distribution sense and thesatisfies$u\in L^{\infty}(\Omega)$$u\in C^{1,\gamma}(\overline{\Omega})(0<\gamma<1)$

By Theorem 3 in $l41,$
boundary condition$0= \frac{\partial u}{\partial\nu_{A}}$$u$$(F)_{(\beta+s,\beta)}$$:=A( \cdot, \nabla u)\nu=a(\cdot, |\nabla u|)\frac{\partial u}{\partial\nu}$in$W^{-1/q,q}(\partial\Omega)$). Sincefor every $1<q<\infty$ (see for the definition of$y\neq0,$
Neumanntheevery
and $a(x, y)>0$ forboundaw
satisfies.
condition, that is,for every$u\in$$W^{-1/q,q}(\partial\Omega)$

$l41$

$u$

$C^{1,\gamma}(\overline{\Omega})$$x\in\partial\Omega$$\frac{\partial u}{\partial\nu}(x)=0$

By Proposition 1 and the remark above (note also that $A(x,$ is odd in ), it
is sufficient to prove the following theorems for the proofs of Theorem 2 and 3.
$y)$Theorem 6 ([15]) For every $s\geq 0$ and $R>0$ , there exists a.
such that$y$$\beta\in[\beta_{0}(s), \beta_{1}(s)]$$K(I_{\beta,s})\cap B_{p}(R)\backslash \{0\}\neq\emptyset$

and $R_{2}>R_{1}>0$ be constants satisfying
Theorem 7 ([15]) Let $s\geq 0,$
such that(3) as in Theorem 3. Then, there exists a$\epsilon>0$$\beta\in[\beta_{0}(s), \beta_{1}(s)+\epsilon]$$K(I_{\beta,s})\cap D_{p}(R_{1}, R_{2})\neq\emptyset$.

Roughly speaking, to show the existence of a non-trivial critical point nearwhen a, we see the variation of the critical groups at for
zero of. Moreover, it is necessary to constructtoparameter changes froma flow for which $B_{p}(R)$ $(or D_{p}(R_{1}, R_{2}))$ is invariant. IFUrthermore, we shallproduce suitable paths to see that 0-th reduced homology group is trivial. For
this purpose, we need to consider the constrained variational problems. The
key point of our proof is to introduce a Finsler manifold
Finally, we state the result characterizing $c(s)$ by Morse theory.
$0$$I_{\beta,s}$$\beta_{0}(s)$$\beta$$I_{\beta,s}$$\beta_{1}(s)$$rS+\cdot$

Corollary 8 ([15]) Let $C_{0}=C_{1}=p-1$ (that is, the case of p-Laplace operator). Then, for every $s\geq 0$$c(s)= \min\{\beta>0;\tilde{H}_{0}(I_{\beta,s}^{0}\backslash \{0\})=0\}$

holds, where $c(s)$ is a function defined by (3) and$\tilde{H}_{*}$

denotes the reduced ho-mology groups.

This corollary means that the mountain pass value $c(s)$ is attained by somefor each $s\geq 0$ .
continuous path$\gamma_{s}\in\Sigma$

7

4

The constrained variational problems

Throughout this section, we fix any $s\geq 0$ . Thus, setforto simplih the notation. First, we definefunctionals andon $W$ by
$u\in
W$
and
for
. Because
is a regularvalue of andfor each $r>0$ , it is well known that the norm of the derivative
at $u\in(rS)$ or $u\in(rS_{+})$ of the restriction ofor
to $rS$ or $rS+$ is defined
as follows:$I_{\beta,s}(\cdot)=I_{\beta}(\cdot)$$C^{1}$$\Phi(u)$$\Phi_{+}(u)$$:= \frac{1}{p}\Vert u\Vert_{p}^{p}$$\Phi$$\Phi$$\beta\in \mathbb{R}$$\Phi+$$r^{p}/p$$:= \frac{1}{p}\Vert u_{+}\Vert_{p}^{p}$$\Phi+$$I_{\beta}$$I_{\beta}^{+}$$\Vert I_{\beta}^{\sim/}(u)\Vert_{*};=\min\{\Vert I_{\beta}’(u)-t\Phi’(u)\Vert_{W^{*}};t\in \mathbb{R}\}$$= \sup\{\langle I_{\beta}’(u), v\};v\in T_{u}(rS),$$\Vert v\Vert=1\}$,$\Vert(\tilde{I}_{\beta}^{+})’(u)\Vert_{*};=\min\{\Vert(I_{\beta}^{+})’(u)-t\Phi_{+}’(u)\Vert_{W*};t\in \mathbb{R}\}$(5),

where $T_{u}(rS)$ denotes the tangent space of $rS$ at , that is, $T_{u}(rS)=\{v\in$$W; \int_{\Omega}|u|^{p-2}uvdx=0\}$ (cf. section 5.3 in [17] for (5)). It is known that $rS$
and $rS+$ are
Finsler manifolds (cf. section 27.4 and 27.5 in [9]). Hence, $rS$
and $rS+$ are locally path connected. Concerning $rS+$ , the following result is$u$$C^{1}$

proved.

Corollary 9 ([15]) $rS+is$ path connected for each $r>0$ .

To state our results for constrained variational problems, we set the following
open subsets of $rS$ or $rS+$ as follows:$\mathcal{O}(I_{\beta}, r, b):=\{u\in rS;I_{\beta}(u)<b\}$for $r>0$ and$\beta,$$b\in \mathbb{R}$,$\mathcal{O}^{+}(I_{\beta}^{+}, r, b):=\{u\in rS_{+};I_{\beta}^{+}(u)<b\}$. Then, we have the following existence result.

Lemma 10 ([15]) Let$r>0$ andopen connected subset oforpoint of, respectively.
or$\beta\in \mathbb{R},$$\mathcal{O}(I_{\beta}, r, b)$$I_{\beta}|_{rS}$$b\in \mathbb{R}$. Then, any nonempty maximal
contains at least one critical$\mathcal{O}^{+}(I_{\beta}^{+}, r, b)$$I_{\beta}^{+}|_{rS_{+}}$

The above lemma plays an important role for the proof of constructing a suitable
path. It is the developed result $hom$ one as in [5] for the manifold .
$S$

5Further results and remaining questions

Finally, the present author would like to take up two questions. First one is “Isclosed¿‘ where
the setdenotes the set of all
such thathas
a non-trivial solution. Of course, in the case where is $(p-1)$ -homogeneous in
the second variable, we know that the above question is true. Second is “When”dosecontain a similar curve to the first nontrivial curve
We state thefollowing result related to the first question.
$\Theta_{A}$$\Theta_{A}$$(\alpha, \beta)$$(F)_{(\alpha,\beta)}$$A$$\Theta_{A}$$\mathscr{C}?$

Proposition 11 For $R_{2}\geq R_{1}>0$ , we set$\Theta_{A}(R_{1}, R_{2});=$$\Theta_{A}(R_{1}, R_{2})_{p}:=$

Then,$\Theta_{A}(R_{1}, R_{2})${{$(\alpha,$$\beta)\in \mathbb{R}^{2};(F)_{(\alpha,\beta)}$

has a solution in $D(R_{1},$$(\alpha,$$\beta)\in \mathbb{R}^{2};(F)_{(\alpha,\beta)}$

has a solution in

and$\Theta_{A}(R_{1}, R_{2})_{p}$$R_{2})$},$D_{p}(R_{1},$ $R_{2})$are closed for any $R_{2}\geq R_{1}>0$ .

}.

8) be a sequence satisfy(resp.
(resp.
. Because of
ing
as
and
$D(R_{1},$$u_{n}\inD_{p}(R_{1},R_{2})$(resp.
)
being
a solution, there exists aon, that is, $-divA(x, \nabla u_{n})=\alpha_{n}u_{n+}^{p-1}-\beta_{n}u_{n-}^{p-1}$ inof. Indeed, by taking as
is bounded in. Then, we can see thattest function, we have

Proof. Let$\{(\alpha_{n},\beta_{n})\}\subset\Theta_{A}(R_{1}, R_{2})_{p}$$\beta_{n}arrow\beta_{0}$$\alpha_{n}arrow\alpha_{0}$$\Theta_{A}(R_{1},$ $R_{2})$$(\alpha_{n}, \beta_{n})\in\Theta_{A}(R_{1}, R_{2})_{p}$$narrow\infty$$R_{2})$$\Theta_{A}(R_{1}, R_{2}))$$\Omega,$$(F)_{(\alpha_{n},\beta_{n})}$$\{u_{n}\}$$\partial\Omega$$\partial u_{n}/\partial\nu=0$$L^{\infty}(\Omega)$$u_{n}$$\frac{C_{0}}{p-1}\Vert\nabla u_{n}\Vert_{p}^{p}\leq\int_{\Omega}A(x, \nabla u_{n})\nabla u_{n}d_{X}\leq\max\{|\alpha_{n}|, |\beta_{n}|\}\Vert u_{n}\Vert_{p}^{p}\leq\max\{|\alpha_{n}|, |\beta_{n}|\}R_{2}^{p}$. Moreover, it is known
by Remark 4 (iii). This implies the boundedness of 1
that there exists a positive constant independ of such that 1
andis a solution of
because$u_{n}\Vert$$C$$u_{n}\Vert_{\infty}\leq C\Vert u_{n}\Vert$$n$$(F)_{(\alpha_{n},\beta_{n})}$$u_{n}$(6)$| \alpha_{n}t_{+}^{p-1}-\beta_{n}t_{-}^{p-1}|\leq\max\{|\alpha_{0}|+1, |\beta_{0}|+1\}|t|^{p-1}$

for everyand sufficiently large (see Appendix in [14]). Thus, our claim
is shown.
and (6), the regularity result in [11]Because of the boundedness of$\gamma\in(0,1)$
and $M>0$ independ of such thatguarantees that there exist(St). Since the inclusion oftoand(St) by choosing
converges someinis compact, we may assume that$u_{0}\inD_{p}(R_{1}, R_{2})$result,is a solution ofanda subsequence. As a$t\in \mathbb{R}$$n$$\Vert u_{n}\Vert_{\infty}$$n$$u_{n}\in C^{1,\gamma}(\overline{\Omega})$$\Vert u_{n}\Vert_{C^{1,\gamma}(\overline{\Omega})}\leq M$$u_{n}$(resp.

$D(R_{1},$). Thus,$C^{1}$$u_{0}$$(F)_{(\alpha_{O},\beta_{0})}$$u_{0}$$R_{2})$$C^{1}$$C^{1,\gamma}(\overline{\Omega})$$(\alpha_{0}, \beta_{0})\in\Theta_{A}(R_{1}, R_{2})_{p}$(resp.

$\Theta_{A}(R_{1},$ $R_{2})$) holds,1

whence our conclusion is shown.

For any $s\geq 0$ and $R_{2}\geq R_{1}>0$ such that $K(I_{\beta,s})\cap D_{p}(R_{1}, R_{2})\neq 0$ for some$\beta>0$ , we can defineby$c_{A}(s, R_{1}, R_{2})$$c_{A}(s, R_{1}, R_{2}):= \inf\{\beta\geq\beta_{0}(s);K(I_{\beta,s})\cap D_{p}(R_{1}, R_{2})\neq\emptyset\}$.

It follows from Proposition 11 that the above infimum is attained, that is,$c_{A}(s, R_{1}, R_{2})= \min\{\beta\geq\beta_{0}(s);K(I_{\beta,s})\cap D_{p}(R_{1}, R_{2})\neq\emptyset\}$.

Then, the present author would like to consider the problem “What properties
does $c_{A}(s, R_{1}, R_{2})$ have¿‘ to answer to the second question.

5.1

Asymptotically $(p-1)$ homogeneous caseIn this subsection, we deal with the special case where the map $A(x, y)$ is asymptotically $(p-1)$ homogeneous in the following sense:$(AH)$ there exist a positive function(Ki, ) and a functiononStsuch that$a_{\infty}\in C^{1}$$\tilde{a}(x, t)$$\mathbb{R}$$\cross \mathbb{R}$$A(x, y)=a_{\infty}(x)|y|^{p-2}y+\tilde{a}(x, |y|)y$and$\lim_{tarrow+\infty}\frac{\tilde{a}(x,t)}{t^{p-2}}=0$

uniformly infor every$x\in\overline{\Omega}$.

$x\in\Omega,$$y\in \mathbb{R}^{N}$,

9
For this weight, we can define the following mountain pass value
by the same argument as in $c(s)$ , namely$a_{\infty}$$c_{a_{\infty}}(s):= \inf_{\gamma\in\Sigma}\max_{t\in[0,1]}\tilde{J}_{a_{\infty},s}(\gamma(t))$$J_{a_{\infty},s}(u)$$:= \int_{\Omega}a_{\infty}(x)|\nabla u|^{p}$

$dx$

–

,

(7)

$s$ $\int_{\Omega}u_{+}^{p}dx$,$\tilde{J}_{a_{\infty},s}$$:=J_{a_{\infty},s}|_{S}$It can be proved that the interval
has no critical values ofUnder the hypothesis $(AH)$ , we have the following result.
$(0, c_{a_{\infty}}(s))$

Proposition 12 Assume $(AH)$ . Let $s\geq 0,$ $\beta>0$ anda solution for. If
oo as $narrow\infty$ , thenwhereis the constant defined by (7).
$\Vert u_{\eta}\Vert_{p}arrow$.

$\tilde{J}_{a_{\infty},s}$.

be a sequence ofholds,$\{u_{n}\}$$(F)_{(s+\beta,\beta)}$$c_{a_{\infty}}(s)$$\beta\geq c_{a_{\infty}}(s)$$c_{a_{\infty}}(s)$. Then, by
Proof. Here, we give the sketch of the proof. Set
the same argument as in [16, Proposition 36], we can prove that
has asubsequence strongly convergent to a solution of$v_{n}$$:=u_{n}/\Vert u_{n}\Vert_{p}$$\{v_{n}\}$$v$$\partial u$$-div(a_{\infty}(x)|\nabla u|^{p-2}\nabla u)=(s+\beta)u_{+}^{p-1}-\beta u_{-}^{p-1}$

in$\Omega$,$–=0$ on$\partial\Omega$,$\partial\nu$

whereis the positive function as in $(AH)$ . This means thatpoint ofwith
. Because $\beta>0$ andcritical values of, we obtain.
$a_{\infty}$$\beta=\tilde{J}_{a_{\infty},s}(v)$$\tilde{J}_{a_{\infty},s}$$\tilde{J}_{a_{\infty},s}$$v$$(0, c_{a_{\infty}}(s))$

is a criticalcontains noI$\beta\geq c_{a_{\infty}}(s)$Corollary 13 Assume $(AH)$ and $s\geq 0$ . Then, we have$\lim_{Rarrow}\inf_{\infty}c_{A}(s, R, \infty)\geq c_{a_{\infty}}(s)$where $c_{A}(s, R, \infty)$,$:= \inf\{\beta\geq\beta_{0}(s);K(I_{\beta,s})\cap D_{f^{l}}(R, \infty)\neq\emptyset\}$.

Proof. By way of contradiction, we prove our assertion. So, we assume that
there exists $s\geq 0$ such that $(0<\beta_{0}(s)\leq)\beta$ $:= \lim\inf_{Rarrow\infty}c_{A}(s, R, \infty)<$. Then, by choosing a subsequence, we can take a sequenceof asolution forwith 1and. By the same argument
as in [16, Proposition 36], we can show that is a critical value of.
Therefore, we have a contradiction because of $0<\beta<c_{a_{\infty}}(s)$ .
1$c_{a_{\infty}}(s)$$\{u_{n}\}$$(F)_{(\beta_{n}+s,\beta_{n})}$$u_{n}\Vert_{p}arrow\infty$$\beta_{n}arrow\beta$$\tilde{J}_{a_{\infty},s}$$\beta$The present author expect that in Theorem 3, we can choose
under the additional hypothesis $(AH)$ .

$\beta$close to$c_{a_{\infty}}(s)$References[1] M. Alif and P. Omari, On a p-Laplace Neumann problem with asymptotically asymmetric perturbations, Nonlinear Analysis TMA 51 (2002), 369-

389.

[2] M. Arias, J. Campos, M. Cuesta and J.-P. Gossez, An asymmetric Neumann problem with weights, Ann. Inst. Henri Poincar\’e 25 (2008), 267-280.

10[3] M. Arias, J. Campos and J.-P. Gossez, On the antimaximum principle andthe Fu\v{c}ik spectrum for the Neumann p-Laplacian, Differential Int. Equations 13(2000), 217-226.
[4] E. Casas and L. A. Fernandez, A Green’s formula for quasilinear elliptic
operators, J. Math. Anal. Appl. 142 (1989), 62-73.

[5] M. Cuesta, D. de Figueiredo, and J.-P. Gossez, The beginning of the Fu\v{c}ik
spectrum for the p-Laplacian, J. Differential Equations 159 (1999), 212-

238.

[6] L. Damascelli, Comparison theorems for some quasilinear degenemte elliptic opemtors and applications to symmetry and monotonicity results, Ann.
Inst. Henri Poincar\’e 15 (1998), 493-516.

[7] E. Dancer, On the Dirichlet problem for weak nonlinear elliptic partial differential equations, Proc. Royal Soc. Edinburgh, $76A(1977),$ 283-300.
[8] N. Dancer and K. Perera, Some Remarks on the Fu\v{c}lk Spectreum of the pLaplacian and Critical Groups, J. Math. Anal. Appl. 254 (2001), 164-177[9] K. Deimling, “Nonlinear Functional Analysis”, Springer-Verlag, New York,1985.
[10] S. Fu\v{c}\’ik, Boundary value problems with jumping nonlinearities, Casopis
Pest. Mat. 101 (1976), 69-87.

[11] G. M. Lieberman, Boundary regularity for solutions-of degenemte elliptic
equations, Nonlinear Anal. 12 (1988), 1203-1219.
[12] M. Montenegro, Strong maximum principles for supersolutions of quasilinear elliptic equations, Nonlinear Anal. 37 (1999), 431-448.
[13] D. Motreanu and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator, Proc.
Amer. Math. Soc., to appear.
[14] D. Motreanu and M. Tanaka, Existence of solutions for quasilinear elliptic
equations with jumping nonlinearities under the Neumann boundary condition, to appear in Calc. Var. Partial Differential Equations.

[15] M. Tanaka, Existence of the Fu\v{c}lk type spectrums for the genemlized pLaplace opemtors, submitted.
[16] M. Tanaka, The antimaximum principle and the existence of a solution for
the genemlized p-Laplace equations with indefinite weight, submitted.
[17] M. Willem, “Minimax Theorem”, Birkh\"auser, Boston, 1996.