We consider a nonlinear parametric equation driven by the sum of a p-Laplacian () and a Laplacian (a -equation) with a Carathéodory reaction, which is strictly -sublinear near +∞. Using variational methods coupled with truncation and comparison techniques, we prove a bifurcation-type theorem for the nonlinear eigenvalue problem. So, we show that there is a critical parameter value such that for the problem has at least two positive solutions, if , then the problem has at least one positive solution and for , it has no positive solutions.

MSC: 35J25, 35J92.

Let be a bounded domain with a -boundary ∂Ω. In this paper, we study the following nonlinear Dirichlet eigenvalue problem:

Here, by we denote the p-Laplace differential operator defined by

(with ). In , is a parameter and is a Carathéodory function (i.e., for all , the function is measurable and for almost all , the function is continuous), which exhibits strictly -sublinear growth in the ζ-variable near +∞. The aim of this paper is to determine the precise dependence of the set of positive solutions on the parameter . So, we prove a bifurcation-type theorem, which establishes the existence of a critical parameter value such that for all , problem has at least two nontrivial positive smooth solutions, for , problem has at least one nontrivial positive smooth solution and for , problem has no positive solution. Similar nonlinear eigenvalue problems with -sublinear reaction were studied by Maya and Shivaji [1] and Rabinowitz [2] for problems driven by the Laplacian and by Guo [3], Hu and Papageorgiou [4] and Perera [5] for problems driven by the p-Laplacian. However, none of the aforementioned works produces the precise dependence of the set of positive solutions on the parameter (i.e., they do not prove a bifurcation-type theorem). We mention that in problem the differential operator is not homogeneous in contrast to the case of the Laplacian and p-Laplacian. This fact is the source of difficulties in the study of problem which lead to new tools and methods.

We point out that -equations (i.e., equations in which the differential operator is the sum of a p-Laplacian and a Laplacian) are important in quantum physics in the search for solitions. We refer to the work of Benci, D’Avenia-Fortunato and Pisani [6]. More recently, there have been some existence and multiplicity results for such problems; see Cingolani and Degiovanni [7], Sun [8]. Finally, we should mention the recent papers of Marano and Papageorgiou [9,10]. In [9] the authors deal with parametric p-Laplacian equations in which the reaction exhibits competing nonlinearities (concave-convex nonlinearity). In [10], they study a nonparametric -equation with a reaction that has different behavior both at ±∞ and at 0 from those considered in the present paper, and so the geometry of the problem is different.

Out approach is variational based on the critical point theory, combined with suitable truncation and comparison techniques. In the next section, for the convenience of the reader, we briefly recall the main mathematical tools that we use in this paper.

Let X be a Banach space and let be its topological dual. By we denote the duality brackets for the pair . Let . A point is a critical point of φ if . A number is a critical value of φ if there exists a critical point such that .

We say that satisfies the Palais-Smale condition if the following is true:

‘Every sequence , such that is bounded and

admits a strongly convergent subsequence.’

This compactness-type condition is crucial in proving a deformation theorem which in turn leads to the minimax theory of certain critical values of (see, e.g., Gasinski and Papageorgiou [11]). A well-written discussion of this compactness condition and its role in critical point theory can be found in Mawhin and Willem [12]. One of the minimax theorems needed in the sequel is the well-known ‘mountain pass theorem’.

Theorem 2.1Ifsatisfies the Palais-Smale condition, , ,

and

where

thenandcis a critical value ofφ.

In the analysis of problem , in addition to the Sobolev space , we will also use the Banach space

This is an ordered Banach space with a positive cone:

This cone has a nonempty interior given by

where by we denote the outward unit normal on ∂Ω.

Let be a Carathéodory function with subcritical growth in , i.e.,

with , and , where

(the critical Sobolev exponent).

We set

and consider the -functional defined by

The next proposition is a special case of a more general result proved by Gasinski and Papageorgiou [13]. We mention that the first result of this type was proved by Brezis and Nirenberg [14].

Proposition 2.2Ifis defined by (2.1) andis a local-minimizer of, i.e., there existssuch that

thenfor someandis also a local-minimizer of, i.e., there existssuch that

Let . We say that if for all compact subsets , we can find such that

Clearly, if and for all , then . A slight modification of the proof of Proposition 2.6 of Arcoya and Ruiz [15] in order to accommodate the presence of the extra linear term leads to the following strong comparison principle.

Proposition 2.3If, , and, are solutions of the problems

then.

Let and let (where ) be a nonlinear map defined by

The next proposition can be found in Dinca, Jebelean and Mawhin [16] and Gasiński and Papageorgiou [11].

Proposition 2.4If (where) is defined by (2.2), thenis continuous, strictly monotone (hence maximal monotone too), bounded and of type, i.e., ifweakly inand

thenin.

If , then we write .

In what follows, by we denote the first eigenvalue of the negative Dirichlet p-Laplacian . We know that and it admits the following variational characterization:

Finally, throughout this work, by we denote the norm of the Sobolev space . By virtue of the Poincaré inequality, we have

The notation will also be used to denote the norm of . No confusion is possible since it will always be clear from the context which norm is used. For , we set . Then for , we define . We know that

If is superpositionally measurable (for example, a Carathéodory function), then we set

By we denote the Lebesgue measure on .

The hypotheses on the reaction f are the following.

H: is a Carathéodory function such that for almost all , for almost all and all and

(i) for every , there exists such that

(ii) uniformly for almost all ;

(iii) uniformly for almost all ;

(iv) for every , there exists such that for almost all , the map is nondecreasing on ;

(v) if

then there exists such that

Remark 3.1 Since we are looking for positive solutions and hypotheses H concern only the positive semiaxis , we may and will assume that for almost all and all . Hypothesis H(ii) implies that for almost all , the map is strictly -sublinear near +∞. Hypothesis H(iv) is much weaker than assuming the monotonicity of for almost all .

Example 3.2 The following functions satisfy hypotheses H (for the sake of simplicity, we drop the z-dependence):

with . Clearly is not monotone.

Let

and let be the set of solutions of . We set

(if , then ).

Proposition 3.3If hypotheses H hold, then

Proof Clearly, the result is true if . So, suppose that and let . So, we can find such that

From Ladyzhenskaya and Uraltseva [[17], p.286], we have that . Then we can apply Theorem 1 of Lieberman [18] and have that . Let and let be as postulated by hypothesis H(iv). Then

so

From the strong maximum principle of Pucci and Serrin [[19], p.34], we have that

So, we can apply the boundary point theorem of Pucci and Serrin [[19], p.120] and have that . Therefore, .

By virtue of hypotheses H(ii) and (iii), we see that we can find such that

Let and . Suppose that . Then from the first part of the proof, we know that we can find . We have

so

(see (3.1) and recall that ), which contradicts (2.3). Therefore, . □

For , let be the energy functional for problem defined by

Evidently, .

Proposition 3.4If hypotheses H hold, then.

Proof By virtue of hypotheses H(i) and (ii), for a given , we can find such that

Then for and , we have

(see (3.2) and (2.3)).

Let . Then from (3.3) it follows that is coercive. Also, exploiting the compactness of the embedding (by the Sobolev embedding theorem), we see that is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find such that

Consider the integral functional defined by

Hypothesis H(v) implies that and since for almost all , all , we may assume that . Since is dense in and , we can find , , such that . Then for large, we have

so

and thus

(see (3.4)), hence . From (3.4), we have

so

On (3.5), we act with . Then

hence , .

From (3.5), we have

so (see Proposition 3.3).

So, for big, we have and so . □

Proposition 3.5If hypotheses H hold and, then.

Proof Since by hypothesis , we can find a solution of (see Proposition 3.3). Let and consider the following truncation of the reaction in problem :

This is a Carathéodory function. Let

and consider the -functional , defined by

As in the proof of Proposition 3.4, using hypotheses H(i) and (ii), we see that is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find such that

so

and thus

On (3.7) we act with . Then

(see (3.6) and use the facts that and ), so

thus

and hence .

Therefore, (3.7) becomes

so

hence . This proves that . □

Proposition 3.6If hypotheses H hold, then for everyproblemhas at least two positive solutions

Proof Note that Proposition 3.5 implies that . Let . Then we can find and . We have

(recall that and ). As in the proof of Proposition 3.5, we can show that . We introduce the following truncation of the reaction in problem :

This is a Carathéodory function. We set

and consider the -functional defined by

It is clear from (3.10) that is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find such that

so

and thus

Acting on (3.11) with and next with (similarly as in the proof of Proposition 3.5), we get

Hence, we have

where .

Then (3.11) becomes

(see (3.10)), so

Let

Then (recall that ) and

so

Note that

So, we can apply the tangency principle of Pucci and Serrin [[19], p.35] and infer that

Let and let be as postulated by hypothesis H(iv). Then

(see hypothesis H(iv) and use the facts that and ), so

(see (3.12) and Proposition 2.3).

In a similar fashion, we show that

From (3.13) and (3.14), it follows that

From (3.10), we see that

for some .

So, (3.15) implies that is a local -minimizer of . Invoking Proposition 2.3, we have that

Hypotheses H(i), (ii) and (iii) imply that for given and , we can find such that

Then for all , we have

for some (see (3.17) and (2.3)).

Choose . Then, from (3.18) and since , we infer that is a local minimizer of . Without any loss of generality, we may assume that (the analysis is similar if the opposite inequality holds). By virtue of (3.16), as in Gasinski and Papageorgiou [20] (see the proof of Theorem 2.12), we can find such that

Recall that is coercive, hence it satisfies the Palais-Smale condition. This fact and (3.19) permit the use of the mountain pass theorem (see Theorem 2.1). So, we can find such that

and

From (3.20) and (3.19), we have that , . From (3.21), it follows that . □

Next, we examine what happens at the critical parameter .

Proposition 3.7If hypotheses H hold, then.

Proof Let be a sequence such that

and

For every , we can find , such that

We claim that the sequence is bounded. Arguing indirectly, suppose that the sequence is unbounded. By passing to a suitable subsequence if necessary, we may assume that . Let

Then and for all . From (3.22), we have

Recall that

(see (3.1)), so the sequence is bounded. This fact and hypothesis H(ii) imply that at least for a subsequence, we have

(see Gasinski and Papageorgiou [20]). Also, passing to a subsequence if necessary, we may assume that

On (3.23) we act with , pass to the limit as and use (3.24) and (3.26). Then

so

Using Proposition 2.4, we have that

and so

Passing to the limit as in (3.23) and using (3.24), (3.27) and the fact that , we obtain

so , which contradicts (3.27).

This proves that the sequence is bounded. So, passing to a subsequence if necessary, we may assume that

On (3.22) we act with , pass to the limit as and use (3.28) and (3.29). Then

so

(since A is monotone) and thus

(see Proposition 2.4).

Therefore, if in (3.22) we pass to the limit as and use (3.30), then

and so is a solution of problem .

We need to show that . From (3.22), we have

From Ladyzhenskaya and Uraltseva [[17], p.286], we know that we can find such that

Then applying Theorem 1 of Lieberman [18], we can find and such that

Recall that is embedded compactly in . So, by virtue of (3.28), we have

Suppose that . Then

Hypothesis H(iii) implies that for a given , we can find such that

From (3.31), it follows that we can find such that

Therefore, for almost all and all , we have

(see (3.32) and (3.33)), so

(see (2.3)), thus

and so

Let to get a contradiction. This proves that and so , hence . □

The bifurcation-type theorem summarizes the situation for problem .

Theorem 3.8If hypotheses H hold, then there existssuch that

(a) for everyproblemhas at least two positive solutions:

(b) forproblemhas at least one positive solution;

(c) forproblemhas no positive solution.

Remark 3.9 As the referee pointed out, it is an interesting problem to produce an example in which, at the bifurcation point , the equation has exactly one solution. We believe that the recent paper of Gasiński and Papageorgiou [21] on the existence and uniqueness of positive solutions will be helpful. Concerning the existence of nodal solutions for , we mention the recent paper of Gasiński and Papageorgiou [22], which studies the -equations and produces nodal solutions for them.

The authors declare that they have no competing interests.

The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.

The authors would like to express their gratitude to both knowledgeable referees for their corrections and remarks. This research has been partially supported by the Ministry of Science and Higher Education of Poland under Grants no. N201 542438 and N201 604640.

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