Research Article | Open Access
Yu-Cheng An, Hong-Min Suo, "The Neumann Problem for a Degenerate Elliptic System Near Resonance", Advances in Mathematical Physics, vol. 2017, Article ID 2925065, 10 pages, 2017. https://doi.org/10.1155/2017/2925065
The Neumann Problem for a Degenerate Elliptic System Near Resonance
This paper studies the following system of degenerate equations , , , , , Here is a bounded domain, and is the exterior normal vector on . The coefficient function may vanish in , with . We show that the eigenvalues of the operator are discrete. Secondly, when the linear part is near resonance, we prove the existence of at least two different solutions for the above degenerate system, under suitable conditions on , and .
In recent decades, many kinds of perturbed problems were studied by many scholars, such as [1–11]. Here, we want to say that the authors in  studied the following Dirichlet boundary problem:When the parameter is close to an eigenvalue of the operator , they proved that problem (1) has two different solutions. Moreover, this result was extended to some equations and systems; see [6–10]. In particular, Massa and Rossato  studied a nondegenerate elliptic system and two solutions were obtained by using Galerkin techniques. On the other hand, we also mention that many scholars studied some elliptic equations with the Neumann or Robin boundary; see [12–17] and the references therein. Inspired by the above results, we study the following system of degenerate equations:where is a bounded domain, is the exterior normal vector on , , and . The coefficient may vanish in , with ; that is, problem (2) may be degenerate; see . As in , we will use the critical point theory and Galerkin techniques to obtain the existence of two different solutions for the above degenerate system. Now, we introduce the function set which consists of functions such thatThroughout the paper, we always assume that there exist and such thatWe also assume that belongs to with , and is a Carathéodory mapping and satisfies the following conditions: For every , there exists such that, for all , uniformly in , .Although the conditions and were introduced in , it is weaker than in  (or (1.2) in ). In fact, let ; it is easy to see that satisfies , but the function does not satisfy the condition in  (or (1.2) in ).
In Section 2, we give some preliminary lemmas and our main results. Meanwhile, we show that the eigenvalues of the operator are discrete under Neumann boundary condition. In Section 3, we prove our main results through Galerkin techniques and saddle point theorem.
2. Preliminaries and Main Results
In this section, we first collect some basic facts and then give the properties of the eigenvalues of the operator . Secondly, we define a new norm and prove it is equivalent to the usual Sobolev norm. At the end of this section, we give the main results of this paper.
Firstly, let denote the completion of with respect to the norm The inner product in is denoted by From (4), we know that the spaces and are equivalent; see . Let ; from with , one hasHence, by the Sobolev embedding theorem of , we know that is compactly embedded in . Moreover, it follows from Hölder’s inequality that
Now, we use a similar argument to that of Gasiński and Papageorgiou (see ). Let us study the following eigenvalue problem:Firstly, from (4) and the Sobolev embedding theorem of , we know thatand the first embedding is compact. Then, for any , we havefor some positive constant ; see .
Let us define It follows from (9) and (12) thatChoosing small enough, then from (14) one getsfor some positive constants . Hence, by Corollary 7.8 in , we conclude that there exists an eigenvalue sequence satisfyingas andLet be the corresponding eigenfunction sequence; then is complete in and for some ; see .
Now, let ; since the coefficient may vanish in , we need to define a new norm:and the corresponding inner product
Lemma 1. Let with ; then the norms and are equivalent.
Proof. Firstly, it follows from (17) thatWe prove that there exists such thatIn fact, if (21) is false, by mean of the 2-homogeneity of , there exists such that for all and as . Without loss of generality, we may assume thatBy the sequential weak lower semicontinuity of and the choice of , we know thatBy (17), one gets as well as for some constant . If , then in , which contradicts , ; if , by (23), one has ; this is a contradiction. Hence, (21) is true.
On the flip side, from (9), we haveHere is a positive constant. Then, by (21) and (24), one getsThis proved the norms and are equivalent.
From now on, we always assume .
Lemma 2. Under the hypotheses of Lemma 1, the embedding is continuous for , compact for .
Now, let and . For fixed , suppose that is an eigenvalue of multiplicity and denote by the eigenspace associated with the eigenvalue , . The main results are as follows.
Theorem 3. Let be the first value above and suppose that conditions and hold. Also,uniformly in , andThen for any , there exists such that ; if , then problem (2) has at least two different solutions.
3. Proof of Main Results
In this section, we firstly prove some preliminary lemmas, and then we prove our main results through variational methods and Galerkin techniques.
For the sake of simplicity, let and , with the norms the inner products and , respectively. In addition, we will always use the notation , unless otherwise specified.
Define and For every , we claim that there exist positive constants and such thatIn fact, by means of , , and (26), the arguments of (33) and (34) are quite similar to that of Lemma 3.1 in  and so is omitted.
Now, we define the functional , where , given bywhere . By means of and , one has and which implies that the critical points of are exactly weak solutions of problem (2).
Next, we need to consider the eigenvalue problem: , for ; that is,Firstly, for any , one has and for some . Now, by and using and in (38), then a straightforward calculation shows that (38) is equivalent toObviously, if and only if . Hence, we obtain two sequences of eigenvaluesand the corresponding eigenfunctionsare the corresponding eigenfunctions.
Let , for ; a simple calculation yieldswhere denotes the Kronecker symbol. Moreover, if , thenIn addition, for every , if , from (40) one getsLet us fix and defineMeanwhile, we denote by , , , and the unitary closed balls, with respect to the norm , in the spaces , , , and , respectively, and by , , , and their relative boundaries.
Lemma 7. Suppose that satisfies , , . For fixed , let and be the first eigenvalue above and , respectively.
If , then we havefor some constants and .
Further, if condition (27) also is satisfied, then there exists a positive constant such that, for , there exist , , such that (46) hold andHere, a value with index represents that depend on , and other cases are similar.
Proof. Firstly, if , then ; if , then .
For , if , then the sequence is nondecreasing, which impliesIf , then the sequence is nonincreasing, which impliesSimilarly, for , that is, , if , then the sequence is also nondecreasing, which impliesIf , then the sequence is nonincreasing, which implies for every .
In a word, for fixed , there exists such thatSecondly, because of the fact that is the first eigenvalue above and , thus , if ; , if . Proceeding as in the proof of the first step, we can also conclude that there exists such thatLet ; we have by (53) and (54)Hence, as in the proof of Lemma 4.1 in , we know thatFrom (33) and (56), we getFrom (33) and (57), we getBy (58) and (59) and choosing , we conclude that there exist and satisfying (46).
In addition, if is near enough to , in particular, if and , we claim that there exists such thatIn fact, if and , for , we haveAnd, for , we haveHence, for fixed , there exists such thatFrom this we easily get the estimates (60) and (61).
Next, we prove (47), (48), and (49). Let and . If , then .
If , it follows from (33) and (61) thatBy choosing , then there exists satisfying (47).
If , by (33) and (60), we obtainLet us choose small enough; then there exists satisfying (49) for .
Now, we prove the estimate (48). If (48) is not true, then, for any sequences and , there exist and such thatHere (or ) denotes the form (or the functional ) with and . And, denotes the eigenvalues of the bilinear form .
Let , and suppose and as . By (44), one has , which implies . So, by (60), we getfor all positive integers ; we get by (68)It follows from (33) and (69) thatWe note that ; then from (70) we obtainwhere . Note that Then, we get by (71)Let , where , . Then, by (73), there exists with , such thatLet . Then, we have From this and (74), without loss of generality, we assume