# The first nonprincipal eigenvalue for a Steklov problem.

Introduction

In a previous work [1], we investigated the following asymmetric Steklov problem with weights:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where v denotes the unit exterior normal, 1 < p < [infinity] and [[DELTA].sub.p]u = div([[absolute value of [nabla]u].sup.p-2] [nabla]u) indicate the p-Laplacian. [OMEGA] [subset] [R.sup.N] be a bounded domain with a Lipschitz continuous boundary, where N [greater than or equal to] 2, m, n [member of] [L.sup.q] ([partial derivative][OMEGA]) with [N - 1/p - 1] < q < [infinity] if p < N and q [greater than or equal to] 1 if p [greater than or equal to] N. We proved the existence of a first nonprincipal positive eigenvalue for (1.1) in case where the weights m and n have meanvalues nonzero. As an application we gave another variational characterization for the second eigenvalue of the problem (1.1) with m = n.

The construction of this distinguished eigenvalue was obtained by applying a version of the mountain pass theorem to the functional [phi](u) := [1/p] [[integral].sub.[OMEGA]] [[absolute value of [nabla]u].sup.p] dx restricted to the manifold [M.sub.m,n] := {u [member of] [W.sup.1,p]([OMEGA]); [1/p] [[integral].sub.[partial derivative][OMEGA]] [m[([u.sup.+]).sup.p] + n[([u.sup.-]).sup.p]]d[sigma] = 1}. In this process the (PS) condition was show to hold at all levels and the geometry of the mountain pass was derived from the observation that [phi](m) and -[phi](n) were strict local minima (where [phi](m) denotes the normalized positive first eigenvalue of the problem (1.1) with m = n). In [2], we are interested in the singular case (in case where one of the weights has meanvalue zero). In this case the Palais Smale condition is not satisfied any more at level 0 and at least one of the two naturals candidates for local minimum fails to belong to the manifold [M.sub.m,n]. To by pass this difficulty we applied a version of the mountain pass theorem for a local [C.sup.1] functional restricted to a [C.sup.1] manifold and which satisfies the Palais Smale condition of Cerami at certain levels.

Our purpose in the present paper is to study the following asymmetric Steklov problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

When trying to adapt the approach followed in [1] to the present situation, this relevant functional is A(u) = [1/p] [[parallel]u[parallel].sup.p.sub.1,p] restricted to the manifold [M.sub.m,n] where [[parallel]u[parallel].sub.1,p] = [([[integral].sub.[OMEGA]] [[absolute value of [nabla]u].sup.p] dx + [[integral].sub.[OMEGA]] [[absolute value of u].sup.p] dx).sup.1/p] is the [W.sup.1,p] ([OMEGA])-norm. We show the existence of a first nonprincipal positive eigenvalue, as an application we give another variational characterization for the second eigenvalue for (1.2) with m = n.

Preliminaries

throughout this paper [OMEGA] will be bounded domain in [R.sup.N]. We assume that m, n [member of] [L.sup.q]([partial derivative][OMEGA]), [m.sup.+] = max(m,0) [not equal to] 0, [n.sup.+] = max(n, 0) [not equal to] 0, where q > [N - 1/p -1] if 1 < p [less than or equal to] N and q [greater than or equal to] 1 if p > N. We are interested in weak solution of (1.2) i.e., functions u [member of] [W.sup.1,p]([OMEGA]) satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

Where d[sigma] is the N - 1 dimensional Hausdorff measure. Let us formulate variationally the problem (1.2). For that purpose we introduce the [C.sup.1] functionals A and [B.sub.m,n]: [W.sup.1,p]([OMEGA]) [right arrow] R, defined by A(u) = [1/p] [[parallel]u[parallel].sup.p.sub.1,p] and [B.sub.m,n](u) = [1/p] [[integral].sub.[partial derivative][OMEGA]] [m[([u.sup.+]).sup.p] + n[([u.sup.- ]).sup.p]]d[sigma]. At this point let us introduce the set [M.sub.m,n] := {u [member of] [W.sup.1,p]([OMEGA]); [B.sub.m,n](u) = 1}. The condition [m.sup.+] [not equal to] 0 implies that [M.sub.m,n] [not equal to] [empty set]. Moreover the set [M.sub.m,n] is a [C.sup.1] manifold in [W.sup.1,p] ([OMEGA]); for any u [member of] [M.sub.m,n] the tangent space of [M.sub.m,n] at u, [T.sub.u][M.sub.m,n] is the set [T.sub.u][M.sub.m,n] := {w [member of] [W.sup.1,p]([OMEGA): <[B'.sub.m,n](u), w) = 0}. Let us denote by [??] the restriction of A to [M.sub.m,n]. We recall that a value c is a critical value of [??] if A'(u)/[T.sub.u][M.sub.m,n] = 0 and [??](u) = c for some u [member of] [M.sub.m,n]. Let us briefly recall some properties relative to the following Steklov problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

Each eigenvalue of (2.2) is called a Steklov eigenvalue. In [3] Bonder and Rossi proved by using a general result from the infinitely dimensional Ljuternik- Schnirelman theory (see [5]) that for any k [member of] [N.sup.*], [[lambda].sub.k](m) defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a nondecreasing and unbounded sequence of the positive eigenvalue of the problem (2.2), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] compact, symmetric and [gamma](C) [greater than or equal to] k} with [gamma](C) denotes the Krasnoselski's genus on [W.sub.1,p]([OMEGA]). He showed the following proposition concerning the first Steklov eigenvalue.

Proposition 2.1. The first eigenvalue [[lambda].sub.1](m) of (2.2) is simple and isolated. Moreover any associated eigenfunction does not change sign in [OMEGA].

We can easily show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where

[[GAMMA].sub.k] = {C [subset] [M.sub.m,n]: C compact, symmetric and [gamma](C) [greater than or equal to] k}.

Let E be a Banach space and let M = {u [member of] E; g(u) = 1}, where g [member of] [C.sup.1](E, R) and 1 is a regular value of g. Let f [member of] [C.sup.1](E, R) and consider the restriction [??] to f. The differential of [??] at u [member of] M has a norm which will be denoted by [[parallel][??][parallel].sub.*] and which is given by the norm of the restriction [??] to the tangent space [T.sub.u](M) := {v[member of] E;<g'(u), v> = 0}; where <,> denotes the duality pairing between [E.sup.*] and E. We recall that [??] is said to satisfy the Palais-Smale condition on M if, for any sequences [u.sub.k] [member of] M such that [??]([u.sub.k]) is bounded and [[parallel][??]'([u.sub.k])[parallel].sub.*] [right arrow] 0, one has that [u.sup.k] admits a converging subsequences. The following Lemma will be used in the proof of our theorem. It guarantees the existence of a critical point in any component of any sublevel

set.

Lemma 2.2 ([4]): Let E, g, M, f and [??] be as considered previously. Assume [??] is bounded from below on M and satisfy the Palais-Smale condition on M. Let d [member of] R and consider O = {u [member of] M; [??](u) < d}. Then any component (i.e. a nonempty maximal open connected subset) O1 of O contains a critical point of [??].

We now recall a version of the classical mountain pass theorem on a [C.sup.1] manifold.

Proposition 2.3 ([4]): Let u, v [member of] M with u [not equal to] v and suppose that

H = {h [member of] C([0,1],M); h(0) = u and h(1) = v}

is nonempty. Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and that [??] satisfies the Palais-Smale condition on M. Then c is a critical point of [??].

We will apply Proposition (2.3) with E = [W.sup.1,p]([OMEGA), M = [M.sub.m,n] f = A and g = [B.sub.m,n]

A first nonprincipal eigenvalue

Now consider the family of paths in [M.sub.m,n]:

[GAMMA] = {[gamma] [member of] C([0,1], [M.sub.m,n]): [gamma](0) [less than or equal to] 0 and [gamma](1) [greater than or equal to] 0},

and define the minimax value

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

Since [GAMMA] [not equal to] [empty set] (see [1]), then c(m, n) is finite.

Theorem 3.1. c(m, n) is an eigenvalue of (1.2) which satisfies

max{[[lambda].sub.1](m), [[lambda].sub.1](n)} < c(m, n)..

Moreover there is no eigenvalue of (1.2) between max{[[lambda].sub.1](m), [[lambda].sub.1](n)} and c(m, n).

As the demonstration is comparatively long, we organize it in several propositions and lemmas.

Proposition 3.2. [??] satisfies the Palais-Smale condition (PS) on [M.sub.m,n].

Proof: Let ([u.sub.k]) [member of] [M.sub.m,n] be a sequence such that [??]([u.sub.k]) [right arrow] c, where c is a positive constant and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since the sequence ([u.sub.k]) is bounded then there exists a subsequence still denoted by ([u.sub.k]) such that [u.sub.k] [right arrow] [u.sub.0] weakly in [W.sup.1,p]([OMEGA]) and [u.sub.k] [right arrow] [u.sub.0] strongly in [L.sup.p]([OMEGA]). Let us write for w[member of] [W.sup.1,p] ([OMEGA])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking [phi] = [a.sub.k] (w) in (3.2), one deduces

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

for some constant C. Indeed: [parallel][a.sub.k](w)[parallel] = [parallel]w - [c.sub.k][u.sub.k][parallel], where [c.sub.k] = [[integral].sub.[partial derivative][OMEGA]] (m([u.sup.+.sub.k]) - n[([u.sup.-.sub.k]).sup.p-1])wd[sigma], so [parallel][a.sub.k](w)[parallel] [less than or equal to] [parallel]w[parallel] + [absolute value of [c.sub.k]]. [parallel][u.sub.k][parallel] and [absolute value of [c.sub.k]] [less than or equal to] [[integral].sub.[partial derivative][OMEGA]] m[([u.sup.+]).sup.p- 1] wd[sigma] [absolute value of +] [[integral].sub.[partial derivative][OMEGA]] n[([u.sup.-.sub.k]).sup.p-1] wd[sigma]]. Using the compacity of the trace mapping [W.sub.1,p]([OMEGA]) [right arrow] [L.sup.q-1] ([partial derivative][OMEGA]), we obtain [absolute value of [[integral].sub.[partial derivative][OMEGA]] m[([u.sup.+.sub.k]).sup.p-1] wd[sigma]] [less than or equal to] [c.sub.1](m). [parallel]w[parallel]. [[parallel][u.sup.+.sub.k][parallel].sup.p-1], where [c.sub.1](m) is a constant depending of m. Similarly we have [absolute value of [[integral].sub.[partial derivative][OMEGA]] m[([u.sup.- .sub.k]).sup.p-1] wd[sigma]] [less than or equal to] [c.sub.1](m). [parallel]w[parallel]. [[parallel][u.sup.+.sub.k][parallel].sup.p-1],

Consequently [absolute value of [c.sub.k]] [less than or equal to] C [parallel]w[parallel]. [[parallel][u.sub.k][parallel].sup.p-1], where C [greater than or equal to] max{[c.sub.1](m), [c.sub.1](n)}. Thus

[parallel][a.sub.k](w)[parallel] [less than or equal to] [parallel]w[parallel](1 + C [[parallel][u.sub.k][parallel].sup.p]).

Finally, we have (3.3). Put w = [u.sub.k] = [u.sub.0] in inequality (3.3), one obtains

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It then follows from the ([S.sup.+]) property that [u.sub.k] [right arrow] [u.sub.0] strongly in [W.sup.1,p]([OMEGA]).

Lemma 3.3. Let [v.sub.k] [member of] [W.sup.1,p]([OMEGA]) with [v.sub.k] [greater than or equal to] 0, [v.sub.k] [not equal to] 0 and [absolute value of [v.sub.k] > 0] [right arrow] 0. Let [n.sub.k] be a bounded in [L.sup.q]([partial derivative][OMEGA]). Then [[integral].sub.[partial derivative][OMEGA]] [n.sub.k][v.sup.p.sub.k]d[sigma]/[[parallel][v.sub.k][parallel].sup.p.sub.1,p] [right arrow] 0..

Proof: Without loss of generality, one can assume that [[parallel][v.sub.k][parallel].sub.1,p] = 1. So for a subsequence, [v.sub.k] [right arrow] v weakly in [W.sup.1,p]([OMEGA]) strongly in [L.sup.p]([OMEGA]). In addition [v.sub.k] [right arrow] v strongly in [[L.sup.pq/q-1]] ([partial derivative][OMEGA]) and [v.sub.k](x) [right arrow] v(x) a.e. in [OMEGA]. The assumption [absolute value of [v.sub.k] > 0] [right arrow] 0 implies v(x) = 0 a.e. x [member of] [OMEGA] since [v.sub.k](x) [right arrow] 0 at measure. Consequently [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (since [v.sub.k] [right arrow] 0 strongly in [L.sup.pq/q- 1]([partial derivative][OMEGA])).

We now turn to the geometry of [??]. Let [[phi].sub.m] be the normalized positive eigenfunction associated to the first positive eigenvalue [[lambda].sub.1](m).

Proposition 3.4. If m [member of] [L.sup.q]([partial derivative][OMEGA]) and [m.sup.+] [not equal to] 0, then [[phi].sub.m] [member of] [M.sub.m,n] is a strict local minimum of [??], with in addition for some [[epsilon].sub.0] > 0 and all 0 < [epsilon] < [[epsilon].sub.0],

[??]([[phi].sub.m]) = [[lambda].sub.1](m) < inf{[??](u); u [member of] [M.sub.m,n] [intersection] [partial derivative]B([[phi].sub.m], [epsilon])} (3.4)

where B([[phi].sub.m], [epsilon]) denotes the ball in [W.sup.1,p]([OMEGA]) of center [[phi].sub.m] and radius [epsilon]. Similar conclusion for -[[phi].sub.n] if [n.sup.+] [not equal to] 0.

Proof: One first shows for some [[epsilon].sub.0] > 0,,

[??]([[phi].sub.m]) < [??](u) [for all]u [member of] [M.sub.m,n] [intersection] B([[phi].sub.m], [[epsilon].sub.0]}, u [not equal to] [[phi].sub.m]. (3.5)

Assume by contradiction the existence of a sequence [u.sub.k] [member of] [M.sub.m,n] with [u.sub.k] [not equal to] [[phi].sub.m], [u.sub.k] [right arrow] [[phi].sub.m] strongly in [W.sup.1,p]([OMEGA]) and [??]([u.sub.k]) [less than or equal to] [[lambda].sub.1](m). We first observe that [u.sub.k] changes sign for k sufficiently large. Indeed, since [u.sub.k] [right arrow] [[phi].sub.m], [u.sub.k] must be > 0 somewhere. If [u.sub.k] [greater than or equal to] 0 in [OMEGA], then

[??]([u.sub.k]) = [1/p] [[parallel][u.sub.k][parallel].sup.p.sub.1,p] > [[lambda].sub.1](m)/p] [[integral].sub.[partial derivative][OMEGA]] m [[absolute value of [u.sub.k]].sup.p] d[sigma] = [[lambda].sub.1](m)

since [u.sub.k] [not equal to] [[phi].sub.m], but this conradicts [??]([u.sub.k]) [less than or equal to] [[lambda].sub.1](m). So [u.sub.k] changes sign for k sufficiently large. Now we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

Thus (3.6) implies

[[integral].sub.[partial derivative][OMEGA]] n[([u.sup.- .sub.k]).sup.p]d[sigma]/[[parallel][u.sup.- .sub.k][parallel].sup.p.sub.1,p] [greater than or equal to] 1/[[lambda].sub.1](m)

Since [u.sub.k] [right arrow] [[phi].sub.m], [absolute value of [u.sup.-.sub.k] > 0] [right arrow] 0. By lemma 3.3, we have [[integral].sub.[partial derivative][OMEGA]] n[([u.sup.- .sub.k]).sup.p]d[sigma]/[[parallel][u.sup.- .sub.k][parallel].sup.p.sub.1,p] [right arrow] 0. It follows that 0 [greater than or equal to] 1/[lambda](m), this is a contradiction. Thus the proposition (3.2) is proved. Now we show (3.4), one assumes by contradiction the existence of 0 < [epsilon] < [[epsilon].sub.0] and of a sequence [u.sub.k] [member of] [M.sub.m,n] such that [[parallel][u.sub.k] - [[phi].sub.m][parallel].sub.1,p] = [epsilon] and [??]([u.sub.k]) [less than or equal to] [??]([[phi].sub.m]) + 1/2[k.sup.2]. Consider

C := {u [member of] [M.sub.m,n]; [epsilon] - [delta] [less than or equal to] [parallel][u.sub.k] - [[phi].sub.m][parallel] [less than or equal to] [epsilon] + [delta]}

where 0 < [delta] < [epsilon] and [delta] + [epsilon] < [[epsilon].sub.0]. Clearly inf{[??](u); u [member of] C} = [??]([[phi].sub.m]). We apply for each k Ekland's variational principle to the functional [??] on C to get the existence of a sequence [v.sub.k] [member of] C such that

[??]{[v.sub.k]) [less than or equal to] [??]([u.sub.k]) [less than or equal to] [??]([[phi].sub.m]) + 1/2[k.sup.2] (3.7)

[[parallel][v.sub.k] - [u.sub.k])).sub.1,p] [less than or equal to] 1/k (3.8)

[??]{[v.sub.k]) [less than or equal to] [??]([u.sub.k]) + [1/k] [[parallel]u - [v.sub.k][parallel].sub.1,p] [for all]u [member of] C. (3.9)

Our purpose is to show that [v.sub.k] is a Palais-Smale sequence for [??], i.e. that [??]([v.sub.k]) is bounded (which is clearly by (3.7)) and that [[parallel][??]([v.sub.k])[parallel].sub.*] [right arrow] 0. We fix k with [1/k] < [delta], take [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and consider a [C.sup.1] path y:]-[eta],[eta][[right arrow] [M.sub.m,n] such that [gamma](0) = [v.sub.k] and y'(0) = [omega]. For [absolute value of t] sufficiently small, [gamma](t) [member of] C. Indeed

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.10)

and it is easily seen using (3.8), 0 < [1/k] < [delta] and [absolute value of [u.sub.k] - [[phi].sub.m][[parallel].sub.1,p] = [epsilon] that the right-hand side of (3.10) is > [epsilon] - [delta] and < [epsilon] + [delta]. So we can take u = [gamma](t) in (3.9). This gives, for t > 0,

[[??]([v.sub.k]) - [??]([gamma](t))/t] [less than or equal to] [1/k] [parallel][gamma](t) - [v.sub.k]/t[parallel],

and so, going to the limit as t [right arrow] 0, we get

-<[??]([v.sub.k]), [omega]> [less than or equal to] [1/k] [parallel][omega][parallel].

Consequently [[parallel][??]'([v.sub.k])[parallel].sub.*] [less than or equal to] 1/k. Thus [[parallel][??]'([v.sub.k])[parallel].sub.*] [right arrow] 0 and [v.sub.k] is Palais-Smale sequence for [??]. It follows that, for a subsequence, [v.sub.k] [right arrow] v strongly in [W.sup.1,p]([OMEGA]). Clearly v [member of] C and satisfies [[parallel]v - [[phi].sub.m][parallel].sub.1,p] = [epsilon] and [??](v) = [??]([[phi].sub.m]) which contradicts (3.5), similar argument when [n.sup.+] [not equal to] 0.

Lemma 3.5. c(m, n) > max{[[lambda].sub.1](m), [[lambda].sub.1](n)}.

Proof: One first shows that [[lambda].sub.1](m) [less than or equal to] c(m, n) and [[lambda].sub.1](n) [less than or equal to] c(m, n). For any y [member of] [GAMMA], [gamma](1) [member of] [M.sub.m,n] is [greater than or equal to] 0 and so satisfies the contraint in

[[lambda].sub.1](m) = inf{[1/p] [[parallel]u[parallel].sup.p.sub.1,p]; u [member of] [W.sup.1,p] (Q[OMEGA]) and [1/p] [[integral].sub.[partial derivative][OMEGA]] m [[absolute value of u].sup.p] d[sigma] = 1}.

Consequently [[lambda].sub.1](m) [less than or equal to] c(m, n), and a similar argument applies to [[lambda].sub.1](n). Now one shows the strict inequality [[lambda].sub.1](m) < c(m, n). Assume by contradiction that [[lambda].sub.1](m) = c(m, n), so there exists a sequence [[gamma].sub.k] [member of] [GAMMA] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Put [u.sub.k] = [[gamma].sub.k] (1), since [u.sub.k] [greater than or equal to] 0 one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.11)

and consequently [1/p][[parallel][u.sub.k][parallel].sup.p.sub.1,p] [right arrow] [[lambda].sub.1](m). Thus [parallel][u.sub.k][parallel] remains bounded, for a subsequence and for some [u.sub.0] [member of] [W.sup.1,p] ([OMEGA]), one has [u.sub.k] [right arrow] [u.sub.0] weakly in [W.sup.1,p] ([OMEGA]). Since [u.sub.k] [greater than or equal to] 0 and using the trace mapping, one has [1/p] [[integral].sub.[partial derivative][OMEGA]] m [[absolute value of [u.sub.0]].sup.p] d[sigma] = 1. Thus [u.sub.0] e[member of] [M.sub.m,n] and so [[lambda].sub.1](m) [less than or equal to] [1/p][[parallel][u.sub.0][parallel].sup.p.sub.1,p] [less than or equal to] lim inf [1/p][[parallel][u.sub.k][parallel].sup.p.sub.1,p] = [[lambda].sub.1](m), which implies that [1/p] [[parallel][u.sub.0][parallel].sup.p.sub.1,p] = [[lambda].sub.1](m). Consequently [u.sub.k] [right arrow] [u.sub.0] strongly in [W.sup.1,p]([OMEGA]) and we conclude that [u.sub.0] = [[phi].sub.m]. Let us now choose [epsilon] > 0 such that (3.5) holds and B([[phi].sub.m], [epsilon]) does not contain any function v with v [less than or equal to] 0, which is clearly possible. For k sufficiently large [u.sub.k] = [[gamma].sub.k] (1) [member of] B([[phi].sub.m], [epsilon]), while [[gamma].sub.k] (0) [not member of] B([[phi].sub.m], [epsilon]), since [y.sub.k] (0) [less than or equal to] 0. It follows that the path [[gamma].sub.k] intersects [partial derivative]B([[phi].sub.m], e) and consequently [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This contradicts (3.11).

Lemma 3.6. If [m.sub.k] [member of] [L.sup.q]([partial derivative][OMEGA]) with [m.sup.+.sub.k] [not equal to] 0 and [m.sup.+.sub.k] [right arrow] 0 in [L.sup.q]([partial derivative][OMEGA]), then [[lambda].sub.1]([m.sub.k]) [right arrow] +[infinity].

Proof: By definition of [[lambda].sub.1]([m.sub.k]) and the trace mapping we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 3.7. For any d > 0, the set

O:={u [member of] [M.sub.m,n]; u [greater than or equal to] 0 et [??](u) < d}

is arcwise connected. Similar conclusion if u [greater than or equal to] 0 is replaced by u [less than or equal to] 0.

Proof: Since o is empty if d [less than or equal to] [[lambda].sub.1](m), we can assume from now on d > [[lambda].sub.1](m). Using lemma (3.6), one constructs a weight [??] [member of] [L.sup.q]([partial derivative][OMEGA]) such that [[??].sup.+] [not equal to] 0, [[lambda].sub.1]([??]) > d and [??] [less than or equal to] m. It suffices in this construction to take [??] = [epsilon][m.sup.+] - [m.sup.-] with [epsilon] > 0 to be sufficiently small. We then consider the manifold [M.sub.m,[??]] and the sublevel set

[??]:={u [member of] [M.sub.m,n]; A(u) < d}.

By Proposition (2.3), the restriction [??] of A to [M.sub.m,[??]] satisfies the Palais-Smale condition. Lemma (2.2) implies that any (nonempty) component of [??] contains a critical point of [??] But the first two critical levels [[lambda].sub.1](m), [[lambda].sub.1]([??]) of [??] verify [[lambda].sub.1](m) < d < [[lambda].sub.1] ([??]) and consequently [??] admits only one critical point in [??]. We can conclude in this way that [??] is arcwise connected. Let now [u.sub.1], [u.sub.2] [member of] O. Since they are [greater than or equal to] 0, they also belong to [??]. Let [gamma] be a patch in [??] from [u.sub.1], [u.sub.2] and consider the patch

[[gamma].sub.1](t):=[absolute value of [gamma](t)]/[([1/p] [[integral].sub.[partial derivative][OMEGA]] m[[absolute value of [gamma](t)].sup.p]).sup.1/p]

By the choice of n, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.12)

and consequently [[gamma].sub.1] is well defined patch in [M.sub.m,n], which clearly goes from [u.sub.1] to [u.sub.2] and is made of nonnegative functions. Moreover, by (3.12),

A([[gamma].sub.1](t) = A([gamma](t))/[([1/p] [[integral].sub.[partial derivative][OMEGA]] m[[absolute value of [gamma](t)].sup.p]).sup.p] [less than or equal to] A([gamma](t)) < d.

for all t, and we conclude that the patch [[gamma].sub.1] lies in O. This concludes the proof of Lemma (3.7) for O with u [greater than or equal to] 0. similar argument in the case u [less than or equal to] 0

Lemma 3.8. There exist [u.sub.1] [greater than or equal to] 0 and [u.sub.2] [less than or equal to] 0 in [M.sub.m,n] such that [??]([u.sub.1]) < c(m, n) and [??]([u.sub.2]) < c(m, n). Moreover, for any such choice of u1, u2, one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.13)

where

[bar.[GAMMA]]:= {[gamma] [member of] C([0,1], [M.sub.m,n]); [gamma](0) = [u.sub.2] and [gamma](1) = [u.sub.1]}, and c(m, n) is defined in (3.1).

Proof: Since [m.sup.+] [not equal to] 0, one take [u.sub.1] = [[phi].sub.m] and the inequality [??]([u.sub.1]) < c(m, n) follows from Lemma (3.5). Similarly with [u.sub.2] = -[[phi].sub.n]. It remains to prove (3.13). Call [bar.c] the right-hand side of (3.13). One clearly has c(m, n) [less than or equal to] [bar.c]. To prove the converse inequality, let [epsilon] > 0 and take [[gamma].sub.[epsilon]] [member of] [GAMMA] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Lemma (3.7), there exists a patch [e[ta].sub.1] [member of] [M.sub.m,n] joining [[gamma].sub.[epsilon]](1) and [u.sub.1], made of nonnegative functions and such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly there exists a patch [[eta].sub.2] [member of] [M.sub.m,n] joining [[gamma].sub.[epsilon]](0) and [u.sub.2], made of nonpositive functions and such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Gluing together [[eta].sub.2], [[gamma].sub.[epsilon]] and [[eta].sub.1], one gets a path in [M.sub.m,n] joining [u.sub.2] and [u.sub.1] and such that [??] remains < c(m, n) + [epsilon] along this path. This implies [bar.c] [less than or equal to] c(m, n) + [epsilon]. Since [epsilon] > 0 is arbitrary, the conclusion follows.

We are now in a position to give the proof of Theorem (3.1).

Proof of Theorem 3.1. By Lemma (3.5), one has c(m, n) > max{[[lambda].sub.1](m), [[lambda].sub.1](n)}. To prove that c(m, n) is an eigenvalue, we pick [u.sub.1], [u.sub.2] as in Lemma (3.8) and we will show that [bar.c], the right-hand side of (3.13), is a critical values of [??]. By Proposition (3.2) Ai satisfies the Palais-Smale condition. Thus the classical mountain pass theorem for a C1 functional manifold (see Proposition (2.1) yields the conclusion.

It remains to show that there is no eigenvalue between max{[[lambda].sub.1](m), [[lambda].sub.1](n)} and c(m, n). Assume by contradiction the existence of such an eigenvalue [lambda] and let u be the corresponding eigenfucntion. We know that u change sign (since [lambda] > max{[[lambda].sub.1](m), [[lambda].sub.1](n)}); moreover

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and we can normalize u so that u [member of] [M.sub.m,n]. The function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

belong to [M.sub.m,n], with [u.sub.1] [greater than or equal to] 0, [u.sub.2] [less than or equal to] 0. We will construct a path [gamma] in [M.sub.m,n] joining [u.sub.1] and [u.sub.2] and such that [??]([gamma](t)) = [lambda] for all t [member of] [0,1]. This will give a contradiction with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. To construct [gamma] we first go from [u.sub.1] to u by the path [[gamma].sub.1] such that u [member of] [gamma] ([0,1])

[[gamma].sub.1](t) := [[u.sup.+] - t[u.sup.-]]/[([B.sub.m,n]([u.sup.+] - t[u.sup.-])).sup.1/p],

and then from u to [u.sub.2] by the path [[gamma].sub.2] such that

[[gamma].sup.2](t) := t[u.sup.+] - [u.sup.-]/[([B.sub.m,n](t[u.sub.+] - [u.sup.- ])).sup.1/p]

It is easily to verified that the path constructed in this way is well defined and satisfies all the required conditions.

We are now in a position to give an application of Theorem (3.1) concerning the second positive eigenvalue [[lambda].sub.2](m) of problem (2.2). In application, for m = n, we obtain another variational characterization for the [[lambda].sub.2](m):

Corollary 3.9.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(for example [gamma](1) = [[phi].sub.m] and [gamma](0) = -[[phi].sub.n]).

References

[1] A. Anane, O. Chakrone, B. Karim, A. Zerouali, An assymmetric problem with weights, Nonlinear analysis 71(2009)614-621, doi:10.1016/j.na.2008.10.123.

[2] A. Anane, O. Chakrone, B. Karim and A. Zerouali, An asymmetric Steklov problem with weights (the singular case), Proceedings of Seminario Internacional Sobre Matematica Aplicada y Repercusion en las Sociedad Actual, Universidad Rey Juan Carlos, 2008, pp. 285-290.

[3] J. Fernandez Bonder and J. D. Rossi, Anonlinear eigenvalue problem with indefinite weights related to the Sobolev trace embedding, Publicacions Matematiques, 46:221-235, 2002.

[4] M. Arias, J. Campos. M.Cuesta. J.P. Gossez, Asymmetric elliptic problems with indefinite weights, ann. Inst. H. Poincar'e. Anal. Nonlinear 19(2002).

[5] Szulkin, "Ljusternik-Schnirelmann theory on C1- manifolds", Annales Inst. H. Poincar'e, Analyse non-lin'eaire, 5, pp. 119-139, 1988.

Aomar Anane, Omar Chakrone, Belhadj Karim and Abdellah Zerouali

Universite Mohamed I, Faculte des Sciences, Departement de Mathematiques et Informatique, Oujda, Maroc

E-mail: ananeomar@yahoo.fr, chakrone@yahoo.fr, karembelf@hotmail.com, abdellahzerouali@yahoo.fr