# Existence of multiple positive solutions of quasilinear elliptic problems in [R.sup.N].

Abstract

This paper concerns quasilinear elliptic equations of the form

-div ([[absolute value of [nabla]u].sup.p-2] [nabla]u) = [lambda]a (x)u(x) [[absolute value of u].sup.p-2] (1 - [[absolute value of u].sup.[gamma]]

in [R.sup.N] with p > 1 and a(x) changes sign. We discuss the question of existence and multiplicity of solutions when a(x) has some specific properties.

AMS subject classification: 35J70, 35B32, 35B40.

Keywords: The p-Laplacian, variational method, multiple solutions.

1. Introduction

In this paper we study the problem of existence of solutions for quasilinear Elliptic equations in [R.sup.N] of the type

-[[DELTA].sub.p]u = [lambda]a(x)u[[absolute value of u].sup.p-2](1 - [[absolute value of u].sup.[gamma]]), (1.1)

where p > 1, [lambda] > 0, [gamma] < [p.sup.2]/N - p, [[DELTA].sub.p]u = div([[absolute value of [nabla]u].sup.p-2][nabla]u) is the p-Laplacian operator and a(x) is a smooth weight function which changes sign in [R.sup.N]. Here we say a function a(x) changes sign if the measures of the sets {x [member of] [R.sup.N] : a(x) > 0} and {x [member of] [R.sup.N] : a(x) < 0} are both positive.

It is known  that when a(x) satisfies proper conditions, the eigenvalue Problem

-[[DELTA].sub.p]u = [lambda]a(x)u [[absolute value of u].sup.p-2]

in [R.sup.N] allows positive eigenvalue [[lambda].sup.+.sub.1] with positive eigenfunction [u.sup.+.sub.1]. Thus we can study the bifurcation problem when [lambda] is near [[lambda].sup.+.sub.1].

The bifurcation problem of this type on bounded domains has received extensive attention recently, and we refer to [4, 6] and [7, 8] for details. The variational method was used there to prove the main results. On the other hand, the study of the existence of global positive solutions of the p-Laplacian also sees great increase in number of papers published. We mention [9, 12, 13], to name a few. For the case p = 2, [11, 15] studied the bifurcation from the first eigenvalue in [R.sup.N] and obtained the existence of bifurcating branches, where a(x) was assumed positive. We note that topological degree arguments and fixed point theory are employed in  and  respectively. However, their principal operator is defined via a Green's function which is not available to the p-Laplacian.

Also we can mention the work of Alama and Tarantello , Berestycki et al.  and Ouyang . Bifurcation results are also obtained in [1, 14]. For p [not equal to] 2, Le and Schmitt  study this equation on bounded domain as an example in their more General framework and obtain the existence of nontrivial solutions.

In this work we investigate the situation where a(x) decays as [absolute value of x] [right arrow] [infinity] and Satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using variational arguments we prove that [[lambda].sup.+.sub.1] is a bifurcation point of (1.1) and there exists [[lambda].sup.*] > [[lambda].sup.+.sub.1], such that (1.1) has at least two positive solutions for [lambda] [member of] ([[lambda].sup.+.sub.1]; [[lambda].sup.*]). Moreover, under proper conditions, we give information about the bifurcating branches.

This paper is organized as follows: In Section [member of] we introduce some assumptions and notations which we use in this paper. In Section 3 we prove the existence of multiple solutions in a certain range of [lambda]. We then verify the case [lambda] = [[lambda].sup.+.sub.1] in this section.

2. Some Notations and Preliminaries

In this section we introduce some basic assumptions and notations which we will need in this paper. We assume first that 1 < p < N and [gamma] < [p.sup.2]/N - p. Write a(x) = [a.sub.1](x) - [a.sub.2](x) with [a.sub.1]; [a.sub.2] [greater than or equal to] 0, [a.sub.1] [member of] [L.sup.[infinity]]([R.sup.N])[intersection] [L.sup.N/p] ([R.sup.N]) and [a.sub.2] [member of] [L.sup.[infinity]]([R.sup.N]). Let

[omega](x) = [(1 + [absolute value of x]).sup.-p]; x [member of] [R.sup.N];

W(x) = max{[a.sub.2](x); [omega](x)} > 0; x [member of] [R.sup.N]:

The weight function [omega](x) satisfies the inequality

[integral] [(1 + [absolute value of x]).sup.-p] [[absolute value of u].sup.p] [less than or equal to] [(p/ N - p).sup.p] [integral] [[absolute value of [nabla][u.sub.n]].sup.p],

where here and henceforth the integrals are taken over [R.sup.N] unless otherwise specified. We define as in , the norm

[parallel]u[parallel] = [([integral][[absolute value of [nabla][u.sub.n].sup.p] + [integral] W (x)[[absolute value of u].sup.p]).sup.1/p]

and introduce the uniformly convex Banach space V by the completion of [C.sup.[infinity].sub.0] ([R.sup.N]) with respect to the norm [parallel]x[parallel]. We assume that a(x) satisfies

([a.sub.1]) [absolute value of a (x)] [less than or equal to] cW(x) for some c > 0,

([a.sub.2]) a(x) [member of] [L.sup.[gamma]1]([R.sup.N]) where [[gamma].sub.1] = [p.sup.*]/[p.sup.*] - p - [gamma].

To introduce the last condition we first give the following result.

Proposition 2.1. Assume that above conditions are satisfied. The eigenvalue Problem

-[[DELTA].sub.p]u = [lambda]a (x)u [[absolute value of u].supl.p-2] (2.1)

has a pair of principal eigenvalue and eigenfunction ([[lambda].sup.+.sub.1]; [u.sup.+.sub.1]) with [[lambda].sup.+.sub.1] > 0 and 0 < [u.sup.+.sub.1] [member of] V. Moreover, such [[lambda].sup.+.sub.1] is simple, unique. If [a.sup.2] [??] 0 and [a.sub.i] [member of] [L.sup.[infinity]]([R.sup.N]) [intersection] [L.sup.N=p]([R.sup.N]), i = 1; 2; then by symmetry there is also principal eigenpair ([[lambda].sup.-.sub.1], [u.sup.-.sub.1]) with [[lambda].sup.-.sub.1] < 0 and 0 < [u.sup.-.sub.1] [member of] V with analogous properties. Moreover the principal eigenvalue [[lambda].sup.+.sub.1] is isolated ([2, 9]).

Also we have (from [9, Lemma 2.3, Theorem 4.1, 4.4 and 4.5]) the following result.

Proposition 2.2. There is a continuum C of positive decaying solutions of (1.1) such that ([[lambda].sup.+.sub.1], 0) [member of] C, and C is either unbounded in E = R x V , where E is equipped with the norm

[[parallel]([lambda], u)[parallel].sub.E] = [([absolute value of [lambda]].sup.2] + [[parallel]u[parallel].sup.2]).sup.1/2]; ([lambda], u) [member of] E,

or there is another eigenvalue [??] [not equal to] [[lambda].sup.+.sub.1] such that ([??], 0) [member of] [bar.C]. If for some [delta] > 0 the problem (1.1) in [[lambda].sup.+.sub.1] has no nonzero solution u [member of] V for 0 < [parallel]u[parallel] < [delta], then C is unbounded in E. Moreover, for any solution u [member of] V, u [member of] [L.sup.Q]([R.sup.N]), where [p.sup.*] [less than or equal to] Q [less than or equal to] [infinity] and u [member of] [C.suip.1,[alpha].sub.loc] ([R.sup.N]).

Finally we assume that

([a.sub.3]) [integral] a(x)[([u.sup.+.sub.1]).sup.p+[gamma]] < 0.

Now we define the functionals [I.sub.1]; [I.sub.2]; [I.sub.3] : V [right arrow] R as follows: for u [member of] V

[I.sub.1] (u) = [integral] [[absolute value of [nabla]u].sup.p],

[I.sub.2](u) = [integral] a(x)[[absolute value of u].sup.p],

[I.sup.3](u) = [integral] a(x)[[absolute value of u].sup.p+[gamma]],

Sometimes we split [I.sup.2] as [I.sup.2] = [I.sup.+.sub.2] - [I.sup.-.sub.2], where

[I.sup.+.sub.2] (u) = [integral] [a.sub.1](x)[[absolute value of u].sup.p], [I.sup.-.sub.2] (u) = [integral] [a.sub.2] (x)[[absolute value of u].sup.p].

The situation for [I.sub.3] is similar by symmetry. We use some properties of these operators in the next section.

By a (weak) solution of problem (1.1), we mean a function u [member of] V such that for every v [member of] [C.sup.[infinity].sub.0] ([R.sup.N]), we have

[integral] [[absolute value of [nabla]u].sup.p-2] [nabla]u[nabla]u - [lambda] [integral] a (x) [[absolute value of u].sup.p-2] uv + [lambda] [integral] a (x) [[absolute value of u].sup.p+[gamma]-2] uv = 0. (2.2)

Since the seminal work of Drabek and Huang , problems like (1.1) have captured great interest.

Now let us define the variational functional corresponding to problem (1.1). We set [J.sub.[lambda]] : V [right arrow] R as

[J.sub.[lambda]](u) = 1/p ([I.sub.1](u) - [lambda][I.sub.2](u)) - [lambda]/p + [gamma] [I.sub.3](u): (2.3)

It is easy to see that J [member of] [C.sup.1](V,R), and for all v [member of] V we have

([J'.sub.[lambda]] (u); v) = [integral] [[absolute value of u].sup.p-2] [nabla]u[nabla]u - [lambda] [integral] a(x) [[absolute value of u].sup.p-2]uv + [lambda] [integral] a(x)[[absolute value of u].sup.p+[gamma]-2] uv = 0: (2.4)

Since [C.sup.[infinity].sub.0] ([R.sup.N]) [subset] V, we know that critical points of [J.sub.[lambda]](u) are weak solutions of (1.1).

When [J.sub.[lambda]] is bounded below on V, [J.sub.[lambda]] has a minimizer on V which is a critical point of [J.sub.[lambda]]. In many problems such as (1.1), [J.sub.[lambda]] is not bounded below on V . In order to obtain an existence result in this case, motivated by Brown and Zhang , we introduce the Nehari manifold

S([lambda]) = {u [member of] V : ([J'.sub.[lambda]] (u), u) = 0}.

It is clear that u [member of] S([lambda]) if and only if

[integral] [[absolute value of [nabla]u].sup.p] - [lambda] [integral] a(x)[[absolute value of u].sup.p] = [lambda] [integral] a(x)[[absolute value of u]p+[gamma]],

and so

[J.sub.[lambda]](u) = (1/p - 1/p + [gamma]) ([I.sub.1](u) - [lambda][I.sub.2](u))

= (1/p - 1/p + [gamma]) ([lambda][I.sub.3](u)).

It is useful to study S([lambda]) it terms of the stationary points of the functions of the form [[phi].sub.u] : t [right arrow] [J.sub.[lambda]](tu) (t > 0). Such maps are known as fibrering maps and were introduced by Drabek and Pohozaev in , and also mentioned in Brown and Zhang . In this case we have

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Hence if we define

[S.sup.+]([lambda]) = {u [member of] S : (p - 1)([I.sub.1](u) - [lambda][I.sub.2](u)) > [lambda](p + [gamma])[I.sub.3](u)};

[S.sup.-]([lambda]) = {u [member of] S : (p - 1)([I.sub.1](u) - [lambda][I.sub.2](u)) < [lambda](p + [gamma])[I.sub.3](u)},

and

[S.sup.0]([lambda]) = {u [member of] S : (p - 1)([I.sub.1](u) - [lambda][I.sub.2](u)) = [lambda](p + [gamma])[I.sub.3](u)},

then for u [member of] S([lambda]) we have

(i) [[phi]'.sub.u](1) = 0.

(ii) u [member of] [S.sup.+]([lambda]), [S.sup.-]([lambda]), [S.sup.0]([lambda]) if [[phi]'.sub.u](1) > 0, [[phi]'.sub.u](1) < 0, [[phi]'.sub.u] (1) = 0 respectively.

(iii) [S.sup.+]([lambda]) ([S.sup.-]([lambda]), [S.sup.0]([lambda]), resp.) = {u [member of] S([lambda]) : [I.sub.3](u) < (>, = , resp.) 0} so that [S.sup.+]([lambda]), [S.sup.-]([lambda]); [S.sup.0]([lambda]) correspond to minima, maxima and points of inflection of fibrering map, respectively.

(iv) The condition ([a.sub.3]) on a(x) implies that [u.sup.+.sub.1] [not member of] [S.sup.-]([lambda]).

Remark 2.3. If u [member of] S([lambda]) is a minimizer of [J.sub.[lambda]] on S([lambda]), then [absolute value of u] [member of] S([lambda]) is also a minimizer of [J.sub.[lambda]] on S([lambda]).

3. Properties of the Bifurcation Diagram

In this section we will consider the problem (1.1) in viewpoint of the bifurcation theory.

By using Proposition 2.2 we consider ([u.sub.n], [[lambda].sub.n]) on the bifurcation diagram with [[lambda].sub.n] [right arrow] 0 and [[lambda].sub.n] [less than or equal to] [[lambda].sup.+.sub.1], [[lambda].sub.n] [right arrow] [[lambda].sup.+.sub.1]. By the means of the structure of the Nehari manifold S([lambda]), we have

[integral] [[absolute value [nabla][u.sub.n]].sup.p] - [[lambda].sub.n] [integral] a(x)[[absolute value of [u.sub.n]].sup.p+[gamma]] = [lambda] [integral] a(x)[[absolute value of [u.sub.n]].sup.p+[gamma]]. (3.1)

Let [v.sub.n] = [u.sub.n]/[parallel][u.sub.n][parallel]. Observe that, by the uniform convexity of V, we may assume that [v.sub.n] [right arrow] v for some v [member of] V. By dividing (2.3) by [[parallel][u.sub.n][parallel].sup.p] we have

[integral] [[absolute value of [nabla][v.sub.n]].sup.p] - [[lambda].sub.n] [integral] a(x)[[absolute value of [v.sub.n].sup.p] = [[lambda].sub.n] [[parallel][u.sub.n][parallel].sup.[gamma]] [integral] a(x)[[absolute value of [v.sub.n]].sup.p+[gamma]].

Using the compactness argument mentioned in , we have [I.sup.+.sub.2] ([v.sub.n) [right arrow] [I.sup.+.sub.2] (v) and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that we use the variational characteristic of [[lambda].sup.+.sub.1] in the first inequality. It then follows that v = 0 or v = t(v)[u.sup.+.sub.1] for some positive constant t(v). We show that the first is impossible. Suppose otherwise, then [I.sup.+.sub.2] ([v.sub.n]) [right arrow] 0 and 0 [less than or equal to] [integral] [[absolute value of [nabla]v].sup.p] - [[lambda].sup.+.sub.1] [integral] a(x)[[absolute value of v].sup.p] [right arrow] 0. So we conclude [[lambda].sub.n] [integral] [a.sub.2](x)[[absolute value of [v.sub.n].sup.p] [right arrow] 0, hence [integral] [a.sub.2](x)[[absolute value of [v.sub.n].sup.p] [right arrow] 0. To obtain [parallel][v.sub.n][parallel] [right arrow] 0, that is our contradiction, it suffices to show that [integral] W(x)[[absolute value of [v.sub.n].sup.p] [right arrow] 0. This follows From

0 [less than or equal to] [integral] W(x)[[absolute value of [v.sub.n].sup.p] [less than or equal to] [integral] ([[bar.a].sub.2](x) + [omega](x))[[absolute value of [v.sub.n].sup.p],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and Hardy's inequality. Hence [parallel][v.sub.n][parallel] [right arrow] 0, contradicting the fact [parallel][v.sub.n][parallel] = 1.

We now turn our attention to

0 [less than or equal to] 1/[[parallel][u.sub.n][parallel].sup.[gamma]] [integral] [[absolute value of [nabla][v.sub.n].sup.p] - [[lambda].sub.n] [integral] a(x)[absolute value of [v.sub.n].sup.p] = [[lambda].sup.n] [integral] a(x)[[absolute value of [v.sub.n].sup.p+[gamma],

and conclude that [integral] a(x)[([u.sup.+.sub.1).sup.p+[gamma]] [greater than or equal to] [lambda] 0, contradicting ([a.sub.3).

Note that as [u.sub.n] [right arrow] 0, [v.sub.n] [right arrow] t(v)[u.sup.+.sub.1] and

1/[[parallel][u.sub.n][parallel].sup.p+[gamma]] [integral] a(x)[[absolute value of [u.sub.n].sup.p+[gamma]] [right arrow] [t(v).sup.p+[gamma]] [integral] a(x)[([u.sup.+.sub.1]).sup.p+[gamma]] [greater than or equal to] 0.

Thus [u.sub.n] [member of] S+([lambda]). So we have proved the following result. Theorem 3.1. The solution branch C bends to the right of [[lambda].sup.+.sub.1] at ([[lambda].sup.+.sub.1], 0) and for ([lambda], u) close enough to ([[lambda].sup.+.sub.1], 0), we have u [member of] [S.sup.+]([lambda]).

Now we turn our attention to [S.sup.-]([lambda]) and investigate the behavior of [J.sub.[lambda]] on [S.sup.-]([lambda]). For u [member of] [S.sup.-]([lambda]) we have

[J.sub.[lambda]](u) = (1/p - 1/p + [gamma])([I.sub.1](u) - [lambda][I.sub.2](u)) = (1/p - 1/p + [gamma])([lambda][I.sub.3](u)) > 0:

So [J.sub.[lambda]] is bounded below by 0 on [S.sup.-]([lambda]). We now show that there exists a minimizer on [S.sup.-]([lambda]) which is a critical point of [J.sub.[lambda]] and so another nontrivial solution of (1.1).

Theorem 3.2. Suppose ([a.sub.1])-([a.sub.3]) hold. There exists [delta] > 0 such that the problem (1.1) has two positive solutions whenever [[lambda].sup.+.sub.1] < [lambda] < [[lambda].sup.+.sub.1] + [delta].

Proof. Step 1. First we claim that there exists [delta] > 0 such that [S.sup.-]([lambda]) is closed in V and open in S([lambda]) whenever [[lambda].sup.+.sub.1] < [lambda] < [[lambda].sup.+.sub.1] + [delta].

Suppose otherwise. Then there exist [[lambda].sub.n] and [u.sub.n] [member of] [S.sup.-]([lambda]) such that [[lambda].sub.n] [right arrow] [[lambda].sup.+.sub.1] and [u.sub.n] [right arrow] [u.sub.0] [member of] [S.sup.-]([lambda]), i.e.,

0 < [[lambda].sup.+.sub.1] [integral] a(x)[[absolute value of [u.sub.n].sup.p+[gamma]] = [integral] ([[absolute value of [nabla][u.sub.n]].sup.p] - [[lambda].sub.n]a(x)[[absolute value of [u.sub.n].sup.p]) [right arrow] 0.

Let [v.sub.n] = [u.sub.n]/[parallel][u.sub.n][parallel]. Then we can assume [v.sub.n] [??] [v.sub.0] in V for some [v.sub.0] [member of] V. By dividing the last relation by [[parallel][u.sub.n][parallel].sup.p] we get

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From the weak convergence of [v.sub.n] to [v.sub.0] in V and [integral] a(x)[[absolute value of [v.sub.n]].sup.p+[gamma]] [right arrow] [integral] a(x)[[absolute value of [v.sub.0]].sup.p+[gamma]], we conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [v.sub.0] = 0, we then derive that [integral] a(x)[[absolute value of [v.sub.n]].sup.p] [right arrow] 0 and [integral] ([[absolute value of [nabla][v.sub.n]].sup.p] - [[lambda].sub.n][a.sub.1](x)[[absolute value of [v.sub.n]].sup.p]) [right arrow] 0, the latter contradicting the fact that [parallel][v.sub.n][parallel] = 1. It then follows from the uniqueness of [u.sup.+.sub.1] that [v.sub.0] = t([v.sub.0])[u.sup.+.sub.1] for some positive constant t([v.sub.0]). We now have by the compactness argument

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which is impossible due to ([a.sub.2]).

Step 2. Now we claim that there exist M > 0 and [[delta].sub.1] > 0 such that for all u [member of] [S.sup.-]([lambda]) and [[lambda].sup.+.sub.1] < [lambda] < [[lambda].sup.+.sub.1] + [[delta].sub.1]

[integral] ([[absolute value of [nabla]u].sup.p] - [[lambda].sub.a](x)[[absolute value of u].sup.p]) [greater than or equal to] M[[parallel]u[parallel].sup.p]. (3.2)

We prove the claim by contradiction. Assume there exist [[lambda].sub.n] [right arrow][[lambda].sup.+.sub.1] and [u.sub.n] [member of] [S.sup.-]([[lambda].sub.n]) such that

[integral] ([[absolute value of [nabla]u].sup.p] - [[lambda].sub.n]a(x)[[absolute value of [u.sub.n]].sup.p]) < 1/n [[parallel][u.sub.n[parallel].sup.p].

Let [v.sub.n] = [u.sub.n]/[parallel][u.sub.n][parallel]. We may assume [v.sup.n] [??] [v.sub.0] in V for some [v.sub.0] [member of] V. On the other hand 0 < [integral] ([[absolute value of [nabla][v.sub.n]].sup.p] - [[lambda].sub.n] a(x)[[absolute value of [v.sub.n]].sup.p]) < 1/n [right arrow] 0. Note that the first inequality follows from the variational characteristic of [[lambda].sup.+.sub.1]. So

[integral] ([[absolute value of [nabla][v.sub.n]].sup.p] - [[lambda].sub.n] a(x)[[absolute value of [v.sub.n]].sup.p]) [right arrow] 0. (3.3)

Using the compactness argument we have [I.sup.+.sub.2] ([v.sub.n]) [right arrow] [I.sup.+.sub.2] ([v.sub.0]). It then follows from [v.sub.n] [??] [v.sub.0] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So [integral] [[absolute value of [nabla][v.sub.0]].sup.p] = [[lambda].sup.+.sub.1] [integral] a(x)[[absolute value of [v.sub.0]].sup.p]. If [v.sub.0] = 0, then we arrive at a contradiction like in Step 1, and so the possibility of [v.sub.0] = 0 is excluded. Hence there exists a positive constant t([v.sub.0]) [member of] (0, 1] such that [v.sub.0] = t([v.sub.0])[u.sup.+.sub.1]. Again by the compactness argument, we obtain [I.sub.3]([v.sub.n]) [right arrow] [I.sub.3]([v.sub.0]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From [u.sub.n] [member of] [S.sup.-]([[lambda].sub.n]), we get

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Step 3. [J.sub.[lambda]](u) satisfies the Palais-Smale condition on [S.sup.-]([lambda]) for [[lambda].sup.+.sub.1] < [lambda] < [[lambda].sup.+.sub.1] + [[delta].sub.1]. Suppose there is a sequence [u.sub.n] [member of] [S.sup.-]([lambda]) such that [J.sub.[lambda]]([u.sub.n]) [less than or equal to] c and [J'.sub.[lambda]] ([u.sub.n]) [right arrow] 0. Note that Step 2 implies that such sequence {[u.sub.n]} is bounded by c/M (1/p - 1/p + [gamma]), and so e may assume [u.sub.n] [??] [u.sub.0] in V for some [u.sub.0] [member of] V. Using the compactness argument we then derive that [I.sup.+.sub.2] ([u.sub.n]) [right arrow] [I.sup.+.sub.2] ([u.sub.0]) and [I.sub.3]([u.sub.n]) [right arrow] [I.sub.3]([u.sub.0]). Now we can estimate ([J'.sub.[lambda]] ([u.sub.n]) - [J'.sub.[lambda]] ([u.sub.m]), [u.sub.n] - [u.sub.m]) as in the proof of [9, Lemmas 2.3 and 3.3] and derive that [I.sub.1]([u.sub.n]) [right arrow] [I.sub.1]([u.sub.0]) and [I.sup.-.sub.2] ([u.sub.n]) [right arrow] [I.su.- .sub.2] ([u.sub.0]). We thus obtain by Hardy's inequality that [parallel][u.sub.n][parallel] [right arrow] [parallel][u.sub.0[parallel] and hence a subsequence of [u.sub.n] converges to [u.sub.0] strongly in V.

Step 4. Existence of a positive solution on [S.sup.-]([lambda]). From the above steps we obtain that [J.sub.[lambda]] has a nonnegative minimizer [u.sup.*] [member of] [S.sup.-]([lambda]). Hence by the Lagrange multiplier theorem there exists [mu] [member of] R such that

([J'.sub.[lambda]] ([u.sup.*]), [phi]) = [mu]([I'.sub.1] ([u.sup.*]) - [lambda][I'.sub.2] ([u.sup.*]) - [lambda][I'.sup.3] ([u.sup.*), [phi])

for all [phi] [member of] V . Taking [phi] = [u.sup.*] and using he fact that [u.sup.*] [member of] [S.sup.-]([lambda]), we get

-[gamma][mu][lambda][I.sub.3]([u.sup.*]) = 0,

which implies [mu] = 0 and hence [u.sup.*] is a solution of (1.1) on [S.sup.-]([lambda]). [16, Theorem 1.2] implies that [u.sup.*] > 0 in [R.sup.N]. This concludes the proof.

Now we study the existence of positive solutions at the point [[lambda].sup.+.sub.1].

Lemma 3.3. S([[lambda].sup.+.sub.1])\{0} is a closed nonempty set.

Proof. First we show that S([[lambda].sup.+.sub.1])\{0} is nonempty. Note that [a.sub.1](x) [??] 0. So there exists a set B [subset] [R.sup.N] with a(x) > 0 in B. Take u(x) [??] 0 such that [??] [??] suppu [subset] B and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consider the auxiliary function

h(t) = [absolute value of t].sup.p] [integral] ([[absolute value of [nabla]u]].sup.p] - [[lambda].sup.+.sub.1] a(x)[[absolute value of u].sup.p]) - [[absolute value of t]].sup.p+[gamma]] [lambda] [integral] a(x)[[absolute value of u]].sup.p+[gamma]],

and observe that if t [right arrow] [+ or -][infinity], we have h(t) [right arrow] -[infinity]. Using the facts that h(0) = 0 and h'(0) = 0 and considering the sign of h'(t) when t [right arrow] [0.sup.+], we obtain that h([t.sub.0]) = 0 for some [t.sub.0] > 0 and hence 0 [not equal to] [t.sub.0]u [member of] S([[lambda].sup.+.sub.1]) and the claim is proved.

Now suppose there exists a sequence {[u.sub.n} in S([[lambda].sup.+.sub.1])\{0} such that [u.sub.n] [right arrow] 0 in V. Since [u.sub.n] [member of] S([[lambda].sup.+.sub.1]) we have

0 [less than or equal to] [integral] ([[absolute value of [nabla][u.sub.n]].sup.p] - [[lambda].sup.+.sub.1] a(x)[[absolute value of [u.sub.n]].sup.p]) = [[lambda].sup.+.sub.1] [integral] a(x)[[absolute value of [u.sub.n]].sup.p+[gamma]]. (3.4)

The first inequality follows from the variational characteristic of [[lambda].sup.+.sub.1]. Dividing (3.4) by [[parallel][u.sub.n[parallel].sup.p+[gamma]], we arrive at

0 [less than or equal to] [integral] ([[absolute value of [nabla][v.sub.n]].sup.p] - [[lambda].sup.+.sub.1] a(x)[[absolute value of [v.sub.n]].sup.p]) = [[lambda].sup.+.sub.1] [[parallel][u.sub.n][parallel].sup.[gamma]] [integral] a(x)[[absolute value of [v.sub.n]].sup.p+[gamma]], (3.5)

where [v.sub.n] = [u.sub.n]/[parallel][u.sub.n][parallel]. Without loss of generality we can assume [v.sup.n] [??] [v.sub.0] in V for some [v.sub.0] [member of] V. The compactness argument shows that [I.sub.3]([v.sub.n]) [right arrow] [I.sub.3]([v.sub.0]) and so {[I.sub.3]([v.sub.n])} is bounded. Hence the right-hand side of (3.5) tends to 0. From the weak convergence of [v.sub.n] to [v.sub.0] in V and [I.sup.+.sub.2] ([v.sub.n]) [right arrow] [I.sup.+.sub.2] ([v.sub.0]), we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The possibility [v.sub.0] = 0 is excluded, as in Step 2 of Theorem 3.2. So there exists some positive constant t([v.sub.0]) such that [v.sub.0] = t([v.sub.0])[u.sup.+.sub.1]. On the other hand we have

0 [less than or equal to] [[parallel][u.sub.n][parallel][gamma]] [integral] ([[absolute value of [nabla][v.sub.n]].sup.p] - [[lambda].sup.+.sub.1] a(x)[[absolute value of [v.sub.n]].sup.p]) = [[lambda].sup.+.sub.1] [integral] a(x)[[absolute value of [v.sub.n]].sup.p+[gamma].

Letting n [right arrow] [infinity] we conclude that [integral] a(x)[([u.sup.+.sub.1]).sup.p+[gamma]] [greater than or equal to] 0. This contradiction proves the lemma.

Theorem 3.4. Equation (1.1) has a positive solution at [[lambda].sup.+.sub.1].

Proof. We do steps similar to those in the proof of Theorem 3.2. Indeed Step 2 with [lambda] = [[lambda].sup.+.sub.1] implies either [v.sub.0] = 0 or [v.sub.0] = t([v.sub.0])[u.sup.+.sub.1] for some nonzero t([v.sub.0]). In either case we obtain a contradiction. Note that for 0 [not equal to] u [member of] S([[lambda].sup.+.sub.1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (u) is bounded below and we can look for a nontrivial minimizer of this functional on S([[lambda].sup.+.sub.1]). Arguments similar to Steps 3 and 4 yield that the functional [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies the Palais-Smale condition on S([[lambda].sup.+.sub.1])\{0} and [mu] = 0 as in Theorem 3.4. [16, Theorem 1.2] further implies that the solution is positive in [R.sup.N].

Theorem 3.5. Let 0 [less than or equal to] [lambda] < [[lambda].sup.+.sub.1]. Then problem (1.1) has at least one solution.

Proof. To prove that [J.sub.[lambda]] satisfies the Palais-Smale condition on S([lambda]) for 0 [less than or equal to] [lambda] < [[lambda].sup.+.sub.1], we can do it as in Theorem 3.4, step by step. We omit the details.

Remark 3.6. Also in  with the condition (a3), similar to these results by using a different approach in bounded domains are obtained for the case p = 2. In this paper we generalized the results in  for the more general cases p > 1 and whole of [R.sup.N].

Received May 8, 2007; Accepted May 17, 2007

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G.A. Afrouzi and S. Khademloo Department of Mathematics, Faculty of Basic Sciences, Mazandaran University, Babolsar, Iran E-mail: afrouzi@umz.ac.ir