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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 918271, 28 pages
Nonhomogeneous Nonlinear Dirichlet Problems with a -Superlinear Reaction
1Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza Street 6, 30-348 Kraków, Poland
2Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece
Received 9 January 2012; Revised 4 February 2012; Accepted 4 February 2012
Academic Editor: Kanishka Perera
Copyright © 2012 Leszek Gasiński and Nikolaos S. Papageorgiou. 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.
We consider a nonlinear Dirichlet elliptic equation driven by a nonhomogeneous differential operator and with a Carathéodory reaction , whose primitive is -superlinear near , but need not satisfy the usual in such cases, the Ambrosetti-Rabinowitz condition. Using a combination of variational methods with the Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions. Our result unifies the study of “superlinear” equations monitored by some differential operators of interest like the -Laplacian, the -Laplacian, and the -generalized mean curvature operator.
The motivation for this paper comes from the work of Wang  on superlinear Dirichlet equations. More precisely, let be a domain with a -boundary . Wang  studied the following Dirichlet problem: He assumes that , , , with , where and that there exist and , such that where (this is the so-called Ambrosetti-Rabinowitz condition). Under these hypotheses, Wang  proved that problem (1.1) has at least three nontrivial solutions.
The aim of this work is to establish the result of Wang  for a larger class of nonlinear Dirichlet problems driven by a nonhomogeneous nonlinear differential operator. In fact, our formulation unifies the treatment of “superlinear’’ equations driven by the -Laplacian, the -Laplacian, and the -generalized mean curvature operators. In addition, our reaction term is dependent, need not be in the -variable, and in general does not satisfy the Ambrosetti-Rabinowitz condition. Instead, we employ a weaker “superlinear’’ condition, which incorporates in our framework functions with “slower’’ growth near . An earlier extension of the result of Wang  to equations driven by the -Laplacian was obtained by Jiang [2, Theorem 12, p.1236] with a continuous “superlinear’’ reaction satisfying the Ambrosetti-Rabinowitz condition.
So, let be as above. The problem under consideration is the following: Here is a map which is strictly monotone and satisfies certain other regularity conditions. The precise conditions on are formulated in hypotheses . These hypotheses are rather general, and as we already mentioned, they unify the treatment of various differential operators of interest. The reaction is a Carathéodory function (i.e., for all , the function is measurable and for almost all , the function is continuous). We assume that the primitive exhibits -superlinear growth near . However, we do not employ the usual in such cases, the Ambrosetti-Rabinowitz condition. Instead we use a weaker condition (see hypotheses ), which permits the consideration of a broader class of reaction terms.
Our approach is variational based on the critical point theory combined with Morse theory (critical groups). In the next section for easy reference, we present the main mathematical tools that we will use in the paper. We also state the precise hypotheses on the maps and and explore some useful consequences of them.
2. Mathematical Background and Hypotheses
Let be a Banach space, and let be its topological dual. By we denote the duality brackets for the pair . Let . We say that satisfies the Cerami condition if the following is true:
‘‘every sequence , such that is bounded and
admits a strongly convergent subsequence.’’
This compactness-type condition is in general weaker than the usual Palais-Smale condition. Nevertheless, the Cerami condition suffices to have a deformation theorem, and from it the minimax theory of certain critical values of is derive (see, e.g., Gasiński and Papageorgiou ). In particular, we can state the following theorem, known in the literature as the “mountain pass theorem.’’
Theorem 2.1. If satisfies the Cerami condition, are such that , and where then and is a critical value of .
In the analysis of problem (1.5) in addition to the Sobolev space , we will also use the Banach space This is an ordered Banach space with positive cone This cone has a nonempty interior, given by where denotes the outward unit normal on .
In what follows, by we denote the first eigenvalue of , where denotes the -Laplace operator, defined by We know (see, e.g., Gasiński and Papageorgiou ) that is isolated and simple (i.e., the corresponding eigenspace is one-dimensional) and In this variational characterization of , the infimum is realized on the corresponding one-dimensional eigenspace. From (2.8), we see that the elements of the eigenspace do not change sign. In what follows, by we denote the -normalized (i.e., ) positive eigenfunction corresponding to . The nonlinear regularity theory for the -Laplacian equations (see, e.g., Gasiński and Papageorgiou [3, p. 737]) and the nonlinear maximum principle of Vázquez  imply that .
Now, let and let . We introduce the following notation:
Let be a topological pair with . For every integer , by we denote the th relative singular homology group for the pair with integer coefficients. The critical groups of at an isolated point with (i.e., ) are defined by where is a neighbourhood of , such that . The excision property of singular homology implies that this definition is independent of the particular choice of the neighbourhood .
Suppose that satisfies the Cerami condition and . Let . The critical groups of at infinity are defined by The second deformation theorem (see, e.g., Gasiński and Papageorgiou [3, p. 628]) guarantees that this definition is independent of the particular choice of the level .
Suppose that is finite. We set The Morse relation says that where is a formal series in with nonnegative integer coefficients .
Now we will introduce the hypotheses on the maps and . So, let be such that for some and for some .
The hypotheses on the map are the following:
, where for all and(i),(ii)there exists , such that(iii)we have(iv)if is a function, such that for and , then there exists , such that
Remark 2.2. Let
Evidently is strictly convex and strictly increasing on . We set
Then is convex, , and
Hence the primitive function used in hypothesis is uniquely defined. Note that the convexity of implies that
Hypotheses and (2.23) lead easily to the following lemma summarizing the main properties of .
Lemma 2.3. If hypotheses hold, then(a)the map is maximal monotone and strictly monotone,(b)there exists , such that (c)we have (where is as in (2.16)).
From the above lemma and the integral form of the mean value theorem, we have the following result.
Corollary 2.4. If hypotheses hold, then there exists , such that
Example 2.5. The following maps satisfy hypotheses :(a) with .This map corresponds to the -Laplace differential operator
(b) with .This map corresponds to the -Laplace differential operator
This is an important operator occurring in quantum physics (see Benci et al. ). Recently there have been some papers dealing with the existence and multiplicity of solutions for equations driven by such operators. We mention the works of Cingolani and Degiovanni , Figueiredo , and Sun :(c), with .This map corresponds to the -generalized mean curvature operator
Such equations were investigated by Chen-Shen :(d), with ,(e), with .
Let (with ) be the nonlinear map, defined by From Gasiński and Papageorgiou [10, Proposition 3.1, p. 852], we have the following result for this map.
Proposition 2.6. If hypotheses hold, then the map defined by (2.30) is bounded, continuous, strictly monotone, hence maximal monotone too, and of type ; that is, if weakly in and then in .
Proposition 2.7. If , for almost all , , then there exists , such that
The hypotheses on the reaction are the following:
is a Carathéodory function, such that for almost all and(i)there exist , and , with such that (ii)if then (iii)there exist and , such that (iv)there exists , such that for almost all , , and (v)for every , there exists , such that
Remark 2.8. Hypothesis implies that for almost all , the primitive is -superlinear. Evidently, this condition is satisfied if that is, for almost all , the reaction is -superlinear. However, we do not use the usual for “superlinear’’ problems, the Ambrosetti-Rabinowitz condition. We recall that this condition says that there exist and , such that Integrating (2.41), we obtain the weaker condition with . Evidently from (2.42) we have the much weaker condition Here, the -superlinearity condition (2.43) is coupled with hypothesis , which is weaker than the Ambrosetti-Rabinowitz condition (2.41). Indeed, suppose that the Ambrosetti-Rabinowitz condition is satisfied. We may assume that . Then (see (2.41) and (2.42)). Therefore, hypothesis is satisfied.
Example 2.9. The following function satisfies hypotheses , but not the Ambrosetti-Rabinowitz condition. For the sake of simplicity we drop the -dependence:
In this work, for every , we set (by virtue of the Poincaré inequality). We mention that the notation will also be used to denote the -norm. However, no confusion is possible, since it is always clear from the context, whose norm is used. For every , we set and for , we define Then , and we have By we denote the Lebesgue measure on . Finally, for a given measurable function (e.g., a Carathéodory function), we define (the Nemytskii map corresponding to ).
3. Three-Solution Theorem
In this section, we prove a multiplicity theorem for problem (1.5), producing three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative).
First we produce two constant sign solutions of (1.5). For this purpose, we introduce the positive and negative truncations of , namely: Both are Carathéodory functions. We set and consider the -functionals , defined by Also, let be the -energy functional for problem (1.5), defined by
Proposition 3.1. If hypotheses and hold, then the functionals satisfy the Cerami condition.
Proof. We do the proof for , the proof for being similar.
So, let be a sequence, such that for some , and From (3.6), we have with , so In (3.8), first we choose . Then using Lemma 2.3(c), we have so Next, in (3.8) we choose . We obtain From (3.5) and (3.10), we have for some . Adding (3.11) and (3.12), we obtain for some , so (see hypothesis ).
Hypotheses and imply that we can find and , such that Using (3.15) and (3.14), we obtain with and so First suppose that . From hypothesis , it is clear that we can always assume that . So, we can find , such that Invoking the interpolation inequality (see, e.g., Gasiński and Papageorgiou [3, p. 905]), we have so for some (see (3.17) and use the Sobolev embedding theorem).
Recall that From hypothesis , we have with , . Therefore, from (3.21) and Lemma 2.3(c), we have for some and so for some (see (3.20)). The hypothesis on (see ) implies that , and so and thus (see (3.26)).
Now, suppose that . In this case, we have , while from the Sobolev embedding theorem, we have that for all . So, we need to modify the previous argument. Let . Then we choose , such that so Note that Since , we have (see ). Therefore, for large , we have that (see (3.28)). Hence, if in the previous argument, we replace with such a large , again we reach (3.26).
Because of (3.26), we may assume that In (3.8), we choose , pass to the limit as , and use (3.30). Then so (see Proposition 2.6). This proves that satisfies the Cerami condition.
Similarly we show that satisfies the Cerami condition.
With some obvious minor modifications in the above proof, we can also have the following result.
Proposition 3.2. If hypotheses and hold, then the functional satisfies the Cerami condition.
Next we determine the structure of the trivial critical point for the functionals and .
Proposition 3.3. If hypotheses and hold, then is a local minimizer for the functionals and .
Proof. By virtue of hypotheses and , for a given we can find , such that Then for all , we have for some (see Corollary 2.4, (2.8), (3.34), and Proposition 2.7). Choosing , we have for some . Since , from (3.36), it follows that we can find small , such that so Similarly, we show that is a local minimizer for the functionals and .
We may assume that is an isolated critical point of (resp., ). Otherwise, we already have a sequence of distinct positive (resp., negative) solutions of (1.5) and so we are done. Moreover, as in Gasiński and Papageorgiou [12, proof of Theorem 3.4,] we can find small , such that By virtue of hypothesis (the -superlinear condition), we have the next result, which completes the mountain pass geometry for problem (1.1).
Proposition 3.4. If hypotheses and hold and , then as .
Proof. By virtue of hypotheses and , for a given , we can find , such that
Then for and , we have
for some and with (see Corollary 2.4 and (3.40)).
Choosing , from (3.41), it follows that Similarly, we show that
Now we are ready to produce two constant sign smooth solutions of (1.5).
Proposition 3.5. If hypotheses and hold, then problem (1.5) has at least two nontrivial constant sign smooth solutions
Proof. From (3.39), we have
Moreover, according to Proposition 3.4, for , we can find large , such that
Then because of (3.45), (3.46), and Proposition 3.1, we can apply the mountain pass theorem (see Theorem 2.1) and find , such that
From (3.47) we see that . From (3.48), we have
On (3.49) we act with and obtain
(see Lemma 2.3(c)), so
Then, from (3.49), we have
Theorem 7.1 of Ladyzhenskaya and Uraltseva [13, p. 286] implies that . Then from Lieberman [14, p. 320], we have that for some . Let , and let be as postulated by hypothesis . Then
(see (3.52) and hypothesis ), so
Then, from Theorem 5.5.1 of Pucci and Serrin [15, p. 120], we have that .
Similarly, working with , we obtain another constant sign smooth solution .
Proposition 3.6. If hypotheses and hold, then
Proof. We do the proof for , the proof for being similar.
By virtue of hypotheses and , for a given , we can find , such that Let where For and , we have for some (see Corollary 2.4, (3.56) and recall that ).
Choosing , we see that Hypothesis implies that we can find and , such that Then for all , we have for some (see (3.61)). Let (see hypothesis (iv)) and choose . Because of (3.60), for and for large , we have Since , we can find , such that Also, for large , we have (see hypothesis , (3.62), (3.63), and recall that ). Hence, by the implicit function theorem, is unique and in fact there is a unique function , such that Let We set Then and Moreover, if , then . We set Evidently . Let be defined by Clearly is continuous and (see (3.70)) and so It is easy to see that is contractible in itself. Hence (see Granas and Dugundji [16, p. 389]), so (see (3.73)) and thus (choosing negative enough).
The same applies for , using this time the sets
With suitable changes in the above proof, we can have the following result.
Proposition 3.7. If hypotheses and hold, then
Proof. As before, hypotheses and imply that for a given , we can find , such that
Let and . Then
for some (see Corollary 2.4, (3.79) and recall that ). Choosing , we see that
Hypothesis implies that we can find and , such that
Then, for any , we have
for some (see hypothesis .
Let (see hypothesis ) and choose . Because of (3.81), for a given and for large , we have We also have (see hypothesis , (3.83), (3.84) and note that ).
The implicit function theorem implies that there exists unique , such that We define Then and Moreover, if , then . We introduce Evidently . Let be defined by Then so Via the radial retraction, we see that is a retract of and is deformable into . Invoking Theorem 6.5 of Dugundji [17, p. 325], we have that so Thus and finally But is an absolute retract of (see, e.g., Gasiński and Papageorgiou [3, p. 691]), hence contractible in itself. Therefore so
Now we are ready to produce the third nontrivial smooth solution for problem (1.5).
Theorem 3.8. If hypotheses and hold, then problem (1.5) has at least three nontrivial smooth solutions
Proof. From Proposition 3.5, we already have two nontrivial constant sign and smooth solutions
We assume that
Otherwise we already have a third nontrivial solution , and by the nonlinear regularity theory, , so we completed the proof.
Claim 1. for all .
We do the proof for the pair , the proof for the pair being similar.
We start by noting that . Indeed, suppose that . Then