Abstract and Applied Analysis

Volume 2012 (2012), Article ID 532430, 19 pages

doi:10.1155/2012/532430

Research Article

Multiple Solutions for Degenerate Elliptic Systems Near Resonance at Higher Eigenvalues

Yu-Cheng An1,2 and Hong-Min Suo¹

¹School of Sciences, Guizhou Minzu University, Guiyang 550025, China

²School of Mathematics and Computer Science, Bijie University, Bijie 551700, China

Received 24 February 2012; Accepted 29 April 2012

Academic Editor: Shaoyong Lai

Copyright © 2012 Yu-Cheng An and Hong-Min Suo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the degenerate semilinear elliptic systems of the form , where is an open bounded domain with smooth boundary , the measurable, nonnegative diffusion coefficients , are allowed to vanish in (as well as at the boundary ) and/or to blow up in . Some multiplicity results of solutions are obtained for the degenerate elliptic systems which are near resonance at higher eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory.

1. Introduction

In this paper, we study a class of degenerate elliptic systems:

where is an open bounded domain with smooth boundary , satisfies the following sublinear growth condition:

uniformly in , where denotes the gradient of with respect to . The degeneracy of this system is considered in the sense that the measurable, nonnegative diffusion coefficients , are allowed to vanish in (as well as at the boundary ) and/or to blow up in . The consideration of suitable assumptions on the diffusion coefficients will be based on the work [1], where the degenerate scalar equation was studied. We introduce the function space , which consists of functions , such that , , and , for some .

Then for the weight functions , we assume the following hypothesis. there exist functions satisfying condition , for some , and satisfying condition , for some , such that

a.e. in , for some constants and .

The mathematical modeling of various physical processes, ranging from physics to biology, where spatial heterogeneity plays a primary role, is reduced to nonlinear evolution equations with variable diffusion or dispersion. Also note that problem (1.1) is closely related (see [1]) to the following system:

Problems of such a type have been successfully applied to the heat propagation in heterogeneous materials, to the study of transport of electron temperature in a confined plasma, to the propagation of varying amplitude waves in a nonlinear medium, to the study of electromagnetic phenomena in nonhomogeneous superconductors and the dynamics of Josephson junctions, to electrochemistry, to nuclear reaction kinetics, to image segmentation, to the spread of microorganisms, to the growth and control of brain tumors, and to population dynamics (see [2–4] and the references therein).

An example of the physical motivation of the assumptions , may be found in [3]. These assumptions are related to the modeling of reaction diffusion processes in composite materials occupying a bounded domain , which at some point they behave as perfect insulators. When at some point the medium is perfectly insulated, it is natural to assume that and/or vanish in . For more information we refer the reader to [4] and the references therein.

For the perturbed problem, Mawhin and Schmitt [5] first considered the following two point boundary value problem:

Under the assumption that is bounded and satisfies a sign condition, if the parameter is sufficiently close to from left, problem (1.5) has at least three solutions, if , problem (1.5) has at least one solution, where are the first and the second eigenvalues of the corresponding linear problem. Ma et al. [6] considered the boundary value problem defined on a bounded open set , no matter whether the boundary conditions are Dirichlet or Neumann condition, as the parameter approaches from left, there exist three solutions. Moreover, existence of three solutions was obtained for the quasilinear problem in bounded domains as the parameter approaches from left. In [7, 8], these results were extended to the perturbed -Laplacian equation in . In [9], Ou and Tang extended above some results to some elliptic systems with the Dirichlet boundary conditions. Especially, de Paiva and Massa in [10] studied the semilinear elliptic boundary value problem in any spatial dimension and by using variational techniques, they showed that a suitable perturbation will turn the almost resonant situation ( near to , i.e., near resonance with a nonprincipal eigenvalue) in a situation where the solutions are at least two. In [11], those results were extended to the cooperative elliptic systems in the bounded domain. Motivated by the idea above, we have the goal in this paper of extending these results in [10, 11] to some degenerate elliptic systems with the Dirichlet boundary conditions.

2. Preliminaries and Main Results

Let be a nonnegative weight function in which satisfies condition . We consider the weighted Sobolev space to be defined as the closure of with respect to the following norm:

and the following scalar product:

for all . The space is a Hilbert space. For a discussion about the space setting we refer to [1] and the references therein. Let

Lemma 2.1. Assume that is a bounded domain in and the weight satisfies . Then the following embeddings hold:

(i) continuously,(ii) compactly for any .

In the sequel one denotes by and the quantities and , respectively, where and are induced by condition , recall that , satisfy . The assumptions concerning the coefficient functions of systems (1.1) are as follows.

The functions and there exists , such that , . The function .

The space setting for our problem is the product space equipped with the following norm:

and the following scalar product:

for all . Observe that inequalities (1.3) in condition imply that the functional spaces and are equivalent. Especially, by Lemma 2.1 we know that for any , the embedding is continuous and there is a positive constant such that for all . Moreover, if , the embedding above is also compact. Let

Assume that hypothesis is satisfied and the coefficient functions , and satisfy conditions and , respectively.

We consider the eigenvalue problem with weight ,

A simple calculation shows that is an eigenvalue of (2.7) if and only if

where is the symmetric bounded linear operator defined by

Since the coefficient of are continuous functions and the embedding is compact, we can check that the operator is also compact. Thus, we may invoke the spectral theory for compact operators to conclude that possesses a Hilbertian basis formed by eigenfunctions of (2.7).

Let us denote and

Recalling that satisfies and , we can use [2, Theorem  1.1] to conclude that the eigenvalue is positive, simple, and isolated in the spectrum of . Moreover, if we denote by the normalized eigenfunction associated to , we can suppose that the is positive on . By using induction, if we suppose that are the first eigenvalues of and are the associated normalized eigenfunctions, we can define

It is proved in [11, Proposition  1.3], that, if , then it is an eigenvalue of with associated normalized eigenfunction . In view of the condition , we can argue as in the proof of [11, Proposition  1.11(c)], and conclude that . Thus, we obtain a sequence of eigenvalues for (2.7).

such that as . We denote by the eigenfunction space corresponding to . Moreover, if we set , we can decompose . Moreover, the following inequalities hold:

Lemma 2.2 (from Lemma  4.6 of [10]). Let be a Hilbert space with orthonormal direct sum splitting . Moreover, let . For , set

Then links with .

Lemma 2.3 (from Theorem  8.1 of [13]). Let be a Hilbert space where has finite dimension, satisfying the condition and such that, for given ,

where and represent the unit ball and the unit sphere in .

Then there exists a critical point such that .

Next, in order to state our main results, we introduce the following assumptions on the nonlinear term:

uniformly with respect to . uniformly with respect to . uniformly with respect to . uniformly with respect to .

Our main results are given by the following theorems.

Theorem 2.4. Let be an eigenvalue of multiplicity . Suppose that condition and the coefficient functions , and satisfy conditions and , respectively. Assume, in addition, that satisfies (1.2) and for all , and one of the sets of hypotheses or . Then there exists such that for problem (1.1) has at least two solutions.

Theorem 2.5. Let be an eigenvalue of multiplicity . Suppose that conditions and the coefficient functions , and satisfy conditions and , respectively. Assume, in addition, that satisfies (1.2) and for all , and one of the sets of hypotheses or . Then there exists such that for problem (1.1) has at least two solutions.

3. Proof of Theorems

Consider the functional , ,

Since the problems in Theorems 2.4 and 2.5 are not resonant, satisfies the Palais-Smale condition of compactness (see, e.g., in [14]). In addition, is a weak solution of problem (1.1) if and only if is a critical point of .

We set

and we define

and , respectively, their relative boundaries.

Theorems 2.4 and 2.5 will be a consequence of the geometry in Propositions 3.1 and 3.2, whose proofs will be postponed to Sections 4 and 5.

Proposition 3.1. If and hypothesis (1.2) is satisfied, then there exist constants and such that

Moreover, if one of the sets of hypotheses or is satisfied, then there exists such that for there exist , such that, in addition to (3.4) and (3.5),

(The values with index depend on , the others may be fixed uniformly.)

Based on this geometry one gives the following proof.

Proof of Theorem 2.4. Since the functional satisfies the condition, we can apply the saddle point theorem (see, e.g., in [15]) for two times, let

The first solution, that we denote by and may be obtained for any with just hypothesis (1.2), corresponds to a critical point at the level

the criticality of this level is guaranteed by the estimates (3.4) and (3.5), since and link, that is, the image of any map in intersects .

The second solution, that we denote by , corresponds to a critical point at the critical level

actually, this is a critical level because of the estimates (3.6) and (3.7), since and link.

To conclude the proof, we need to show that these two solutions are distinct.

We observe first that by estimate (3.6) we have that , then we observe that we may build a map in such a way that its image is the union between the annulus and the image of a -dimensional ball in whose boundary is . By the estimates (3.7) and (3.8), we deduce that , and as a consequence , proving that the two solutions are distinct, for being at different critical levels.

Proposition 3.2. If and hypothesis (1.2) is satisfied, then there exist constants and such that

Moreover, if one of the sets of hypotheses or is satisfied, then there exists such that for there exist , , such that, in addition to (3.12) and (3.13),

The values with index depend on , the others may be fixed uniformly.

This geometry, along with Lemma 2.2, allows one to give the following.

Proof of Theorem 2.5. Since the functional satisfies the condition, we can apply the saddle point theorem and Lemma 2.3.

The first solution that we denote by and may be obtained for any with just hypothesis (1.2) is again obtained through the saddle point theorem and corresponds to a critical point at the critical level

where now

the criticality is guaranteed by estimates (3.12) and (3.13), since and link.

The second solution that we denote by comes from Lemma 2.3, where we set and , actually we have the following structure:

and then we have a critical point at the level .

Finally, in order to prove that these two solutions are distinct, we need a sharper estimate for than that given by (3.13). For this we use Lemma 2.2 to guarantee that for any map , since , one has that the image of either intersects or has a point with . This implies that , by estimates (3.15) and (3.16), and then proving that the two solutions are distinct, for being at different critical levels.

4. Proof of Estimates

In this section we will prove all the estimates in Propositions 3.1 and 3.2.

From (1.2) and the continuity of the potential , for any , there exists a positive constant such that

for all . By (4.1), Hölder’s inequality, we have

where is the best embedding constant.

4.1. Estimates of the Saddle Geometry

Lemma 4.1. Under hypothesis (1.2), one gets the following:

(i)for , there exists satisfying (3.4) and satisfying (3.6);(ii)for :

(a)there exists satisfying (3.12),(b)for a given , there exists satisfying (3.17).

Proof. Let : using estimates (4.2) and (2.14) we get

For , letting , it follows that is bounded below in , that is, there exists a as in (3.6).

For , then the same estimate holds but the constant cannot be made independent of , giving (3.12).

In the same way, let and set , we get

Letting , it follows that is bounded below in , that is, there exists a such that for all we have (3.4), where again the constant depends on , that is, on .

Finally, (4.4) with implies

then, no matter the value of , is bounded from below in any bounded subset of , giving (3.17) for a suitable value of .

Lemma 4.2. Under hypothesis (1.2), one gets the following:

(i)for , given the constant , there exists satisfying (3.5);(ii)for :

(a)there exists satisfying (3.14),(b)for a given , there exists satisfying (3.13),(c)for a given , there exists satisfying (3.18).

Moreover, given the values , , one may always choose , as claimed in Propositions 3.1 and 3.2.

Proof. Let , by estimates (4.2) and (2.13) we get

For , letting , then one obtains (3.5) for suitably large .

For , letting , one obtains, for suitable and , (3.14) and (3.18).

Finally, let and set , we get

Letting , it is clear that (once that is fixed) this goes to and then we may find the claimed such that (3.13) holds.

Observe that and can be chosen uniformly for , while , in fact depend on .

4.2. Estimating the Effect of the Nontrivial Perturbation

In this section we will prove the remaining inequalities in Propositions 3.1 and 3.2, which rely on the hypotheses , or , or , or , which, roughly speaking, say that the perturbation is nontrivial in such a way that a new solution arises when it is sufficiently near to the eigenvalue . The proof is simpler for Theorem 2.4, since we need to estimate the functional in the compact set , while for Theorem 2.5 the same kind of estimate is required in the noncompact set .

4.2.1. Estimating in

For the next estimates, we will need the following lemma.

Lemma 4.3. Hypotheses implies that there exists a nondecreasing function such that

Proof. First we claim that there exists a constant such that the sets have measure , for all .

Actually, is a finite-dimensional subspace and the functions are smooth, they are uniformly bounded, that is, there exists such that for all . Suppose that for there exists such that .

On one hand, by (2.13), one obtains

On the other hand,

where , , . That is a contradiction.

Now for any , we will show that we can find an large enough so that for any and , which means that

Actually, letting , by we have that there exists such that for .

For , one has , and then one gets

For , by and , there exists such that , for all .

One finally obtains

it is elementary that

is well defined and satisfies the claim.

Now we may prove the following.

Lemma 4.4. Consider Theorem 2.4 with one of the sets of hypotheses or . Given the constant , there exist , such that, for any , (3.7) and (3.8) hold.

Proof. We consider the two sets of hypotheses separately.

(i) In case , assuming (1.2) and hold, we claim that there exists such that

uniformly in , in particular we set .

In fact, by , there exits such that for , uniformly in . For any , from (4.1) (letting ) we have

where .

Let , for being in a finite-dimensional subspace, all the norms are equivalent, so that (set and uses estimates (4.15) and (2.13))

In case , let be as in Lemma 4.3, for , let with and ,

Assume that , it is easy to see that for some constant , we estimate

Considering (4.17) and (4.19), we see that since by Lemma 4.3, we may fix so that (or for the case ) and then for one gets (3.7).

To obtain (3.8), we observe that (since ) if , that is, if , then in estimates (4.17) and (4.19) we may avoid the term so that (remember that is nondecreasing) for .

4.2.2. Estimating in

We consider the corresponding of the previous lemma, for Theorem 2.5.

Lemma 4.5. Considering Theorem 2.5 with one of the sets of hypotheses or . Given the constant , there exists , such that, for any , (3.15) and (3.16) hold.

Proof. Letting , we see from (4.3), that property (3.16) will be satisfied provided that is large enough (say ) and observing that this value can be made independent from once that is small enough.

Now we consider the two sets of hypotheses separately.

(i) In case , suppose , we can assume that , with and . Since is a finite dimension subspace, all the norms are equivalent, so that there exists such that for all we have . By , from the proof of (4.15), we also have the similar inequality: there exists such that

uniformly in . So by (2.14) and (4.20),

where , . Since

suppose , (4.21) becomes

since , so is bounded below for all , that is, there exists such that

by (4.23) one gets

In case , first we give some conclusions which are similar to Lemma  3 of [16]. Under the property of , there exists a constant , and which is subadditive, that is,

for all , and coercive, that is,

as , and satisfies that

for all , such that

for all and .

In fact, since as uniformly for all , there exists a sequence of positive integers with for all positive integers such that

for all and all . Let and define

for , where .

By the definition of we have

for all . By and , there exists such that

It follows that

where . In fact, when for some , one has, by (4.30) and (4.32),

for all . When , we have, by (4.32) and (4.33),

for all .

It is obvious that is continuous and coercive. Moreover one has

for all . In fact, for every there exists such that

which implies that

for all by (4.32) and the fact that for all integers .

Now we only need to prove the subadditivity of . Let

and . Then we have

Hence we obtain, by (4.32),

which shows that is subadditive.

For , assuming that , with and , and letting , by (2.13), (4.29), (4.26), and (4.28), one gets

where , . Since , is a finite-dimensional subspace, and is coercive, from the proof of (4.8), one can get

that is, is coercive on . Since , so is coercive on , and and is bounded below, it is obvious that

for all .

Considering (4.25), (4.43), and (4.45), we can choose large enough such that for all one gets

(or for the case ) and property (3.16) holds, then for and one gets , that is, the property (3.15) holds.

5. Proof of the Geometry in Propositions 3.1 and 3.2

We finally give the proof of Propositions 3.1 and 3.2, which is nothing but a resume of the lemmata above, verifying that all the constants can be chosen sequentially without contradictions.

Proof of Proposition 3.1. Under hypothesis (1.2), if we fix a value , then we obtain the constant from Lemma 4.1 and with this we get from Lemma 4.2. If we also consider one of the two sets of hypotheses or , then we proceed as follows: first of all, we determine (once for ever) the constant from Lemma 4.1, with this we obtain from Lemma 4.4 the values and . Then, for any (now fixed) , we obtain from Lemma 4.1 the value . Finally, we can get from Lemma 4.2 the corresponding value of .

Proof of Proposition 3.2. Under hypothesis (1.2), if we fix a value , then we obtain the constant from Lemma 4.1 and with this we get from Lemma 4.2. If we also consider one of the two sets of hypotheses or , then we proceed as follows: first of all, we determine (once for ever) the constant from Lemma 4.2, with this we obtain from Lemma 4.5 the values and . Since we have , we can get from Lemma 4.1 the constant and with this obtain from Lemma 4.2.

Finally, for any (now fixed) , we obtain from Lemma 4.1 the constant and with this we get from Lemma 4.2 the corresponding value of .

Acknowledgments

This paper was supported by National Commission of Ethnic Affairs, Science Foundation of China (no. [2011]02) and by Natural Science Foundation of the Education Department of Guizhou Province (no. [2010]305).

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