# Principal Eigenvalues of a Second-Order Difference Operator with Sign-Changing Weight and Its Applications.

1. Introduction

In 1997, Constantin  studied the following linear periodic eigenvalue problem:

u" (t)-q(t)u(t) = [lambda]m(t)u(t), t [member of] (0, w), (1)

u (0) = u (w), u'(0) = u'(w), (2)

where m, q [member of] C[0, w], q(t) [greater than or equal to] 0, and q(t) [not equivalent to] 0. He obtained that if m changes its sign, then (1) and (2) have infinite real eigenvalues, [[lambda].sup.[+ or -].sub.k], such that

[mathematical expression not reproducible]. (3)

Equation (1) with q [equivalent to] 1/4 plays a crucial role in the study of the water shallow equation; see [2-5].

Let T >2 be an integer and T = {1,2,..., T}. In 2005, Wang and Shi  discussed the eigenvalues of a discrete periodic boundary value problem

[DELTA] [p (j - 1) [DELTA]u (j -1)] + q (j) u (j) + [lambda]m (j) u (j) = 0, j [member of] T, (4)

u(0) = u(T), u(1) = u(T + 1), (5)

where p(j), q(j), and m(j) are real functions with p(j) > 0 for j [member of] {0,1,..., T}, m(j) > 0 for j [member of] T, and p(0) = p(T) = 1, and [lambda] is the spectral parameter. They showed the existence of eigenvalues of (4) and (5) and calculated the numbers of eigenvalues.

Ji and Yang  considered a class of boundary value problems of the second-order difference equation (4) with the more general boundary conditions

u (0) = [alpha]u (T), u(T + 1) = [alpha]u (1). (6)

The class of problems considered include those with antiperiodic, Dirichlet, and periodic boundary conditions. They focused on the structure of eigenvalues and comparisons of all eigenvalues of (4) and (6), as the coefficients p(t), q(t), and m(t) change their signs. They got a very interesting result: the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function.

Gao and Ma  studied the eigenvalues of periodic and antiperiodic eigenvalue problems of discrete linear secondorder difference equation (4) with sign-changing weight. They find that these two problems have T real eigenvalues (including the multiplicity), respectively. Furthermore, the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function. Furthermore, these eigenvalues, including the eigenvalues of Neumann problem, satisfy the certain order relation.

However, all of above papers provide no information about the sign of the eigenfunctions of (4) and (5). In particular, they give no information about the sign of the eigenfunctions corresponding to two simple eigenvalues [[lambda].sup.+.sub.1] and [[lambda].sup.-.sub.1].

It is the purpose of this paper to show the existence of the principal eigenvalues and determine the sign of the corresponding eigenfunctions for linear periodic eigenvalue problem

[mathematical expression not reproducible], (7)

where q(j) [greater than or equal to] 0 and q(j) [not equivalent to] 0 in T, and the weight function g changes its sign in T. Our approach is motivated by Brown and Lin  and Smoller , where the infimum of Rayleigh quotient is used to characterize the principal eigenvalues for diverse linear eigenvalue problems of elliptic equations with infinite weight.

For the other recent results on the spectrum structure of discrete linear eigenvalue problems with one-sign weight, see Sun and Shi , Shi and Chen , Jirari , Bohner , and Agarwal et al.  and the references therein.

The rest of the paper is organized as follows. In Section 2, we show the existence of the principal eigenvalues of (7) and determine the sign of the corresponding eigenfunctions. In Section 3, we apply our spectrum theory and the wellknown Rabinowitz bifurcation theorem to show the existence of positive solutions for nonlinear discrete periodic boundary value problem

-[[DELTA].sup.2] u (j - 1 ) + q (j) u (j) = f (j, u (j)), j [member of] T, (8)

u (0) = u(T), u(1) = u(T + 1), (9)

under [mathematical expression not reproducible] is a continuous function, and g : T [right arrow] R changes its sign.

2. Existence of Principal Eigenvalues

In this section, we consider the linear eigenvalue problem (7) under the following assumptions:

(H0) q: T [right arrow] [0, [infinity]), and q([j.sub.0]) > 0 for some [j.sub.0] [member of] T.

(H1) g : T [right arrow] R, and there exists a proper subset, [T.sup.+], of T, such that g(j) > 0 for j [member of] [T.sup.+] and g(j) <0 for j [member of] T \ [T.sup.+]. Let n be the number of elements in [T.sup.+]. Then T - n is the number of elements in T [T.sup.+].

The difference operator is [L.sub.0], defined by

[L.sub.0] U(j) = -[[DELTA].sup.2]u(j - 1) + q(j) u (j) for u [member of] D, j [member of] T, (10)

where

D = {(u(0),u(1),..., u(T), u(T + 1)) : u (0) = u(T), u(1) = u(T + ]1)}. (11)

Define a linear operator [T.sub.0] : D [right arrow] D by

[mathematical expression not reproducible]. (12)

Then, it is easy to see that [T.sub.0] : D [right arrow] D is isomorphism. Moreover, [T.sub.0] : D [right arrow] D is a self-adjoint operator whose spectrum consists only of the real eigenvalues.

Let us denote the norm and the inner product of D by

[mathematical expression not reproducible], (13)

respectively. For v [member of] D, we have from [16, Lemma 2.1] that

{-[[DELTA].sup.2] v (j - 1), v (j)) = {[DELTA]v (j), [DELTA]v (j)). (14)

Thus, we may define a functional

[Q.sub.[lambda]] (v) = {[DELTA]V (j), [DELTA]V (j)) + {q (j) V (j) , V (j)) -[lambda] [T.summation over (j=1)] g(j) [v.sup.2] (j). (15)

To study the principal eigenvalues of (7), we need the following preliminary lemmas.

Lemma 1. Let (H0) and (H1) hold. Assume there is a nonnegative eigenfunction corresponding to an eigenvalue [lambda] of (7). Then, [Q.sub.]lambda]](v) [greater than or equal to] 0for all v [member of] D.

Proof. Suppose that u is a nonnegative eigenfunction corresponding to the eigenvalue [lambda]. Then, u is a nonnegative eigenfunction corresponding to the eigenvalue r = k of

-[[DELTA].sup.2] u(j - 1) + q (j) u (j) - [lambda]g (j) u (j) + ku (j) = ru (j), j [member of] T, (16)

u(0) = u(T), u(1) = u(T + 1). (17)

Let

[T.sub.0] u(j) = -[[DELTA].sup.2]u (j - 1) + q (j) u (j) - [lambda]g (j) u (j), U [member of] D, (18)

and [T.sub.k]u(j) := ([T.sub.0]u) (j) + ku(j). Then, [T.sup.-1.sub.k] : D [right arrow] D is strongly positive if k is large enough. Clearly, r and [psi] are the eigenvalue and eigenfunction of [T.sub.k] if and only if [mu] = r - k and [psi] are the corresponding eigenvalue and eigenfunction of the problem

[mathematical expression not reproducible], (19)

u(0) = u(T), u(1) = u(T + 1). (20)

Since [T.sub.k] : D [right arrow] D is a self-adjoint operator, its spectrum contains only real eigenvalues:

[[eta].sub.1] [less than or equal to] [[eta].sub.2] [less than or equal to] ... [less than or equal to] [[eta].sub.T]. (21)

Moreover, by the well-known Krein-Rutman theorem [17, Theorem 19.2], [[eta].sub.1] > 0 is simple, and its corresponding eigenfunction [[psi].sub.1] is of one sign.

So, the eigenvalue [[mu].sub.1](= [[eta].sub.1] - k) is simple and the corresponding eigenfunction [[psi].sub.1] does not change sign in T. Notice k and u are eigenvalue and the corresponding eigenfunction of (16) and (17), respectively, and u is not orthogonal to [[psi].sub.1]. This together with the fact that eigenfunctions corresponding to distinct eigenvalues are orthogonal implies that u must be an eigenfunction of [T.sub.k] corresponding to [[eta].sub.1](= [[mu].sub.1] + k). Hence, [[mu].sub.1] = 0. By the spectral theorem

([T.sub.0] v, V) [greater than or equal to] [[mu].sub.1] (v, v) = 0 [for all]v [member of] D; (22)

that is, [Q.sub.[lambda]] (v) [greater than or equal to] 0 for all v [member of] D.

Lemma 2. Let (H0) and (H1) hold. Let

[mathematical expression not reproducible]. (23)

Then, [[lambda].sup.+.sub.1] [member of] (0, [infinity]).

Proof. By the spectral theorem, ([L.sub.0]v, v) [greater than or equal to] [[gamma].sub.1] (v, v) for all v [member of] D, where [[gamma].sub.1] is the first eigenvalue of [L.sub.0]. Note that (H0) implies that

[[gamma].sub.1] > 0. (24)

Hence, if v [member of] D with [[summation].sup.T.sub.j=1] g(j)[v.sup.2](j) > 0, then

[mathematical expression not reproducible]. (25)

Hence

[[lambda].sup.+.sub.1] [greater than or equal to] [[gamma].sub.1]/[sup.sub.j[member of]T] | g(j)] > 0. (26)

Lemma 3. If [lambda] > [[lambda].sup.+.sub.1], then [lambda] is not an eigenvalue of (7) possessing a nonnegative eigenfunction.

Proof. If [lambda] > [[lambda].sup.+.sub.1], there exists v [member of] D such that [[summation].sup.T.sub.j=1] g(j) [v.sup.2] (j) > 0 and K(v) < [lambda]; that is,

[mathematical expression not reproducible] (27)

and so [Q.sub.[lambda]](v) < 0. The required result is now an immediate consequence of Lemma 1.

Lemma 4. Assume (H1) and 0 < [lambda] < [[lambda].sup.+.sub.1]. Then, there exists a > 0 (a depends on [lambda]) such that [Q.sub.[lambda]](v) [greater than or equal to] a [[parallel]v[parallel].sup.2] for all v [member of] D.

Proof. Let [lambda] = (1 - s) [[lambda].sup.+.sub.1], where 0 < s < 1.

We claim that

[Q.sub.[lambda]] (v) [greater than or equal to] s[[gamma].sub.1] [[parallel]v[parallel].sup.2]. (28)

In fact, for v [member of] D, we have from the fact that

[mathematical expression not reproducible] (29)

that

[mathematical expression not reproducible]. (30)

Lemma 5. Let (H0) and (H1) hold. Let

[mathematical expression not reproducible]. (31)

Then, [[lambda].sup.-.sub.1] [member of] (-[infinity], 0).

Proof. If v [member of] D with [[summation].sup.T.sub.j=1] g(j) [v.sup.2] (j) < 0, then it follows from the fact that

{[L.sub.0]v, v) [greater than or equal to] [[gamma].sub.1] (v, v) v [member of] D (32)

that

[mathematical expression not reproducible]. (33)

Hence,

[[lambda].sup.-.sub.1] [less than or equal to] - [[gamma].sub.1]/[sup.sub.j[member of]T] [absolute value of g(j)] < 0. (34)

Lemma 6. If [lambda] < [[lambda].sup.-.sub.1], then [lambda] is not an eigenvalue of (7) possessing a nonnegative eigenfunction.

Proof. If [lambda] < [[lambda].sup.-.sub.1], then there exists v [member of] D such that [[summation].sup.T.sub.j=1] g(i)[v.sup.2](i) < 0 and K(v) > [lambda]; that is,

[mathematical expression not reproducible] (35)

and so [Q.sub.[lambda]](v) < 0. The required result is now an immediate consequence of Lemma 1.

Lemma 7. Assume (H1) and [[lambda].sup.-.sub.1] < [lambda] < 0. Then, there exists a > 0 (a depends on [lambda]) such that [Q.sub.[lambda]](v) [greater than or equal to] a[[parallel]v[parallel].sup.12] for all v [member of] D.

Proof. Let [lambda] = (1 - s)[[lambda].sup.-.sub.1], where 0 < s < 1.

For v [member of] D, we have from

[mathematical expression not reproducible] (36)

that

[mathematical expression not reproducible]. (37)

Theorem 8. If (H0) and (H1) hold, then (7) has exactly two principal eigenvalues [[lambda].sup.-.sub.1] and [[lambda].sup.+.sub.1], such that

(1) [[lambda].sup.-.sub.1] < 0 < [[lambda].sup.+.sub.1];

(2) the algebraic multiplicity of [[lambda].sup.-.sub.1] and [[lambda].sup.+.sub.1] is 1;

(3) the eigenfunctions [[phi].sup.-] and [[phi].sup.+.sub.1] corresponding to the eigenvalues [[lambda].sup.-.sub.1] and [[lambda].sup.+.sub.1] are of one sign.

Proof. Consider the linear eigenvalue problem

[mathematical expression not reproducible]. (38)

It is easy to see that [[lambda].sup.+.sub.1] is an eigenvalue for (7) with corresponding eigenfunction w if and only if 0 is an eigenvalue of [T.sub.0], and, accordingly, 0 is an eigenvalue of (38) with corresponding eigenfunction w. The least eigenvalue of [T.sub.0] is given by

[mathematical expression not reproducible]. (39)

Since [mathematical expression not reproducible]. Because of how we defined [[lambda].sup.+.sub.1], there exists a sequence {[v.sub.n]} [subset] D such that [[summation].sup.T.sub.j=1] g(j) [v.sup.2.sub.n](j) = 1 and

[mathematical expression not reproducible]. (40)

Therefore,

[mathematical expression not reproducible] (41)

and so [[alpha].sub.1] [less than or equal to] 0. Hence, [[alpha].sub.1] = 0 is the least eigenvalue of (38) and so [[alpha].sub.1] is simple and the corresponding eigenfunction can be chosen to be positive on T.

Using the same method, with obvious changes, we may prove the algebraic multiplicity of [[lambda].sup.-.sub.1] is 1 and the eigenfunction [[phi].sup.-.sub.1] corresponding to the eigenvalue [[lambda].sup.-.sub.1] is of one sign.

3. Existence of Positive Solutions

As an application, we consider the existence of positive solutions of the discrete nonlinear problem (8) and (9).

In this section, we assume that

(H2) [lim.sub.s[right arrow]0](f(j, s)/s) = g(j) for all j [member of] T;

(H3) [lim.sub.s[right arrow][infinity]](f(j,s)/s) = m(j) for all j [member of] T, where m : T [right arrow] [0, [infinity]) and m(j) > 0 for all j [member of] T.

Let [[lambda].sub.1](m) be the principal eigenvalue of

[mathematical expression not reproducible], (42)

and let [phi] be an eigenfunction corresponding to [[lambda].sub.1](m).

Theorem 9. Assume that (H0)-(H3) hold. Then

(1) (8) and (9) have at least one positive solution for [[lambda].sup.+.sub.1] < 1 < [[lambda].sub.1] (m);

(2) (8) and (9) have at least one positive solution for [[lambda].sub.1] (m) < 1 < [[lambda].sup.+.sub.1].

Proof.

Step 1. A bifurcation result.

We extend the function f to a continuous function [??] defined on T x R by setting, for j [member of] T,

[mathematical expression not reproducible]. (43)

Obviously, within the context of positive solutions, problem (8) and (9) is equivalent to the same problem with f replaced by [??]. Furthermore, [??](j, s) is an odd function for j [member of] T. In the sequel of the proof, we shall replace f with [??]. However, for the sake of simplicity, the modified function [??] will still be denoted by f.

Recall that

[L.sub.0] u(j) = -[[DELTA].sup.2]u(j - 1) + q(j) u (j) for u [member of] D, j [member of] T, (44)

where

D = {(u (0), u(1),..., u(T), u(T + 1)) : u (0) = u(T), u(1) = u(T + 1)}. (45)

Let [xi], [zeta] [member of] C(T x R, R) be such that

[mathematical expression not reproducible], (46)

where

[mathematical expression not reproducible]. (47)

Let us consider

[L.sub.0]u - [lambda]g (j) u = [lambda][xi] (j, u) (48)

as a bifurcation problem from the trivial solution u [equivalent to] 0. A solution of (48) is a pair ([lambda], u) [member of] R x D which satisfies (48). It is easy to see that any solution of (48) of the form (1,u) yields a solution u of (8) and (9).

Equation (48) can be converted into the equivalent equation

u (j) = [lambda][L.sup.-1.sub.0] [g (x) u (x)] (j) + [lambda][L.sup.-1.sub.0] [[xi](x, u (x))] (j). (49)

Further, we note that [lim.sub.u[right arrow]0] ([L.sup.-1.sub.0] [[xi](x,u(x))]/[parallel]u[parallel]) = 0 in D, since

[mathematical expression not reproducible], (50)

where C = [square root of T] [max.sub.j[member of]T] [[summation].sup.T.sub.t=1] G(t, j) and G(t, j) is given by [18, Theorem 2.1]. We denote (49) by

u = lambda]Lu + H ([lambda], u). (51)

Clearly, L and H : R x D [right arrow] D are completely continuous and

[mathematical expression not reproducible] (52)

with respect to [lambda] varying in bounded intervals. Notice that if [lambda] is the eigenvalue of [L.sub.0], then it also is the characteristic value of L.

We say that a solution ([lambda], u) [member of] R x D of (51) is nontrivial if there exists [j.sub.0] [member of] T such that u([j.sub.0]) [not equal to] 0. Denote by S the closure in R x D of the set of all nontrivial solutions (X, u) of (51) with [lambda] > 0.

Theorem 1.3 in  yields the existence of a maximal closed connected set C in S such that ([[lambda].sup.+.sub.1], 0) [member of] C and at least one of the following conditions holds:

(i) C is unbounded in R x D.

(ii) There exists a characteristic value of L, with [mathematical expression not reproducible].

Step 2. In what follows, we prove several properties which will eventually lead to the fact that condition (ii) above does not hold.

Claim 1. Suppose [mathematical expression not reproducible] is a characteristic value of L. Let {([[lambda].sup.[k]], [u.sup.[k]])} be a sequence of nontrivial solutions of (51), converging to ([??], 0) in R x D.

Setting, for all k, [v.sup.[k]] = [u.sup.[k]]/[parallel][u.sup.[k]][parallel], we have

[v.sup.[k]] = [[lambda].sup.[k]] L [v.sup.[k]] + H ([[lambda].sup.[k]], [u.sup.[k]])/[parallel] [u.sup.[k]] [parallel]. (53)

As {[v.sup.[k]]} is bounded in D and L is completely continuous, there exist w [member of] D and a subsequence of {[v.sup.[k]]}, which we denote in the same way, such that

[mathematical expression not reproducible]. (54)

Hence, we conclude by (53) that

[mathematical expression not reproducible]. (55)

Therefore, we have

w = [??]L (w) (56)

with [parallel][??]w[parallel] = 1 and in particular w [not equal to] 0. Accordingly, [??] is a characteristic value of L.

Claim 2. There exists [member of] >0 such that S [subset] [e, +[infinity]) x D.

By contradiction, we can suppose that there exists a sequence {([[lambda].sup.[k]], [u.sup.[k]])} of nontrivial solutions of (51), converging in R x D to some (0, u) [member of] R x D. Arguing as in the proof of Claim 1, we set [v.sup.[k]] = [usup.[k]]/[parallel][u.sup.[k]][parallel], and we have

[mathematical expression not reproducible] (57)

and conclude that, possibly passing to a subsequence, [lim.sub.k[right arrow][infinity]] [v.sup.[k]] = 0 in D, which contradicts [parallel] [v.sup.[k]][parallel] = 1.

Claim 3. ([lambda], u) [member of] C if and only if ([lambda], -u) [member of] C.

This follows from the fact that f, and hence H, is odd with respect to the second variable.

In the sequel, we denote by P the positive cone in D; that is,

P = {u [member of] D : u [greater than or equal to] 0}, (58)

and we denote by int P its interior and by [partial derivative]P its boundary.

Claim 4. There exists a neighborhood U of ([[lambda].sup.+.sub.1], 0) in R x D such that, for all ([lambda], u) [member of] C [intersection] U, either ([lambda], u) = ([[lambda].sup.+.sub.1], 0), or u [member of] int P, or -u [member of] int P.

It is an immediate consequence of the fact that [[phi].sup.+.sub.1](j) > 0 for all j [member of] T and the well-known Crandall-Rabinowitz local bifurcation theorem; see Crandall and Rabinowitz  and Kielhofer .

Claim 5. Assume ([lambda], u) [member of] C and u [member of] [partial derivative]P. Suppose further that ([lambda], u) is the limit of a sequence {([[lambda].sup.[k]], [u.sup.[k]])} in C, with [u.sup.[k]] > 0 for all k. Then, ([lambda], u) = ([[lambda].sup.+.sub.1], 0).

We first show that u = 0.

Suppose, by contradiction, that u > 0. Then, we can take c > 0 such that

[lambda](g(j) + [xi](j, u)/u) + c [greater than or equal to] 1, j [member of] T. (59)

Hence, we get

[mathematical expression not reproducible]. (60)

As m [member of] P satisfies

[mathematical expression not reproducible], (61)

for some [j.sub.0] [member of] T, let G(t, s) be the Green function for the linear boundary value problem

[mathematical expression not reproducible]. (62)

Since the Green function G(j, s) for (62) satisfies G(j, s) > 0 on 0 [less than or equal to] j, s [less than or equal to] T (see [18, Theorems 2.1 and 2.2] for details), this yields u(j) > 0 for j [member of] T, contradicting u [member of] [partial derivative]P. Therefore, we conclude that u = 0. We next show that [lambda] = [[lambda].sup.+.sub.1]. By Claim 1, [lambda] is a characteristic value of L. Setting [v.sup.[k]] = [u.sup.[k]] / [parallel] [u.sup.[k]] [parallel] and arguing as in the proof of Claim 1, we conclude that, possibly passing to a subsequence,

[mathematical expression not reproducible] (63)

in D, where w is an eigenfunction of (7) associated with [lambda]. Since w > 0, we conclude that [lambda] = [[lambda].sup.+.sub.1].

Claim 6. For all ([lambda], u) [member of] C, either u [member of] int P, or -u [member of] int P, or ([lambda], u) = ([[lambda].sup.+.sub.1], 0).

Set

[mathematical expression not reproducible]. (64)

By Claim 4,

E = {([lambda], u) [member of] (C \ U): u [not member of] int P, -u [not member of] int P} (65)

and, subsequently, E is a closed subset of C.

Let us verify that E is open in C. Suppose this is not the case. Then, there exist ([lambda], u) [member of] E and a sequence ([[lambda].sup.[k]], [u.sup.[k]])} in C/E converging to ([lambda],u). We may assume that [u.sup.[k]] [member of] int P for all k; hence, by Claim 5, we obtain ([lambda], u) = ([[lambda].sup.+.sub.1], 0), contradicting the fact that ([lambda], u) [member of] E. As C is connected and ([[lambda].sup.+.sub.1], 0) [member of] C \ E, we conclude that E = 0.

By Claim 6 we have that if [mathematical expression not reproducible] and hence condition (ii) above does not hold. Consequently, condition (i) is valid.

Step 3. Next, let us show that C joins ([[lamba].sup.+.sub.1], 0) to ([[lamba].sub.1] (m), [infinity]). Let {([[mu].sup.[k]], [y.sup.[k]])} [member of] C satisfy

[[mu].sup.[k]] + [parallel] [y.sup.[k]] [parallel] [right arrow] [infinity]. (66)

We note that [[mu].sup.[k]] > 0 for all k [member of] N, since (0,0) is the only solution of (51) for [lambda] = 0 and C [intersection] ({0} x D) = 0.

We first show that there exists a constant M > 0, such that

[[mu].sup.[k]] [member of] (0, M] [for all]k [member of] N. (67)

Suppose (67) does not hold, then choosing a subsequence and relabelling if necessary, it follows that

[mathematical expression not reproducible]. (68)

We divide the equation

[mathematical expression not reproducible] (69)

by [mathematical expression not reproducible], and we have

[mathematical expression not reproducible] (70)

for all j [member of] T. Since {[v.sup.[k]]} is bounded in D, after taking a subsequence and relabelling if necessary, we have that [v.sup.[k]] [right arrow] [bar.v] for all k [member of] N, where [bar.v] [member of] D with [parallel][bar.v][parallel] = 1.

It follows from (66) and (68) that either

[parallel][y.sup.[k]][parallel] [member of] (0, [infinity]) or [parallel][y.sup.[k]][parallel] [member of] (0, [infinity]). (71)

Combining this fact with (H3) and

[y.sup.[k]] (j) [greater than or equal to] [gamma] [parallel] [y.sup.[k]] [parallel] [for all]j [member of] T, (72)

where [gamma] > 0 with G(j, s) [greater than or equal to] [gamma]G(s, s), we have

[mathematical expression not reproducible], (73)

this is a contradiction, and therefore (67) holds.

According to (67), we have

[parallel][y.sup.[k]][parallel] [right arrow] [infinity] K [right arrow] [infinity]. (74)

Since ([[mu].sup.[k]], [y.sup.[k]]) solve

[mathematical expression not reproducible], (75)

we divide (75) by [mathematical expression not reproducible]. Then, we get

[mathematical expression not reproducible]. (76)

Since {[[bar.y].sup.[k]]} is bounded in D, after taking a subsequence and relabelling if necessary, we have that [mathematical expression not reproducible]. Moreover, by (72) and (74), we can show that

[mathematical expression not reproducible]. (77)

Therefore,

[L.sub.0] [bar.y] = [bar.[mu]]m (j) [bar.y], (78)

where [bar.[mu]] = [lim.sub.k[right arrow][infinity]] [[mu].sup.[k]], again choosing a subsequence and relabelling if necessary.

We claim that

[bar.y] [member of] C. (79)

Suppose on the contrary that [bar.y] [not member of] C. Since [bar.y] [not equal to] 0 is a solution of (78) it follows that [bar.y] [member of] [C.sup.k]. By the openness of D \ C, we know that there exists a neighborhood U([bar.y], [[rho].sub.0]) such that

U([bar.y], [[rho].sub.0]) [subset] D \ C, (80)

which contradicts the facts that [[bar.y].sup.[k]] [right arrow] [bar.y] in D and [[bar.y].sup.[k].sub.y] [member of] C. Therefore, [bar.y] [member of] C. Moreover, by Sturm-Liouville eigenvalue theory, [bar.[mu]] = [[lambda].sub.1](m). Therefore, C joins ([[lambda].sup.+.sub.1], 0) to ([[lambda].sub.1](m), [infinity]).

Therefore, C crosses the hyperplane {1} x D in R x D, and, accordingly, (8) and (9) have at least one positive solution.

Remark 10. Let us consider the nonlinear problem

[mathematical expression not reproducible], (81)

where

[mathematical expression not reproducible]. (82)

Obviously,

[mathematical expression not reproducible] (83)

with

[mathematical expression not reproducible], (84)

and m(j) = 33/4. Thus

[[lambda].sub.1] (m) = 2/33 < 1. (85)

On the other hand, it follows from (26) and the fact [[gamma].sub.1] = 1/2 that

[[lambda].sup.+.sub.1] [greater than or equal to] [[gamma].sub.1]/[sup.sub.j[member of]T] [absolute value of g (j)] = 4/3 > 1. (86)

Therefore, Theorem 9 (2) yields the existence of at least one positive solution of (81).

https://doi.org/10.1155/2018/1949254

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors' Contributions

Ma and Xu completed the main study together and Ma wrote the manuscript; Long was responsible for checking the proofs process and verified the calculation. Moreover, all the authors read and approved the final version of the manuscript.

Acknowledgments

This work was supported by NSFC (no. 11671322) and NSFC (no. 11361054).

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Ruyun Ma (iD), Man Xu, and Yan Long

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Correspondence should be addressed to Ruyun Ma; mary@nwnu.edu.cn

Received 31 October 2017; Revised 14 December 2017; Accepted 24 December 2017; Published 18 January 2018