Multiplicity of positive solutions for eigenvalue problems of -equations
© Gasiński and Papageorgiou; licensee Springer 2012
Received: 13 September 2012
Accepted: 7 December 2012
Published: 28 December 2012
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.
(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  and Rabinowitz  for problems driven by the Laplacian and by Guo , Hu and Papageorgiou  and Perera  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 . More recently, there have been some existence and multiplicity results for such problems; see Cingolani and Degiovanni , Sun . Finally, we should mention the recent papers of Marano and Papageorgiou [9, 10]. In  the authors deal with parametric p-Laplacian equations in which the reaction exhibits competing nonlinearities (concave-convex nonlinearity). In , 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.
2 Mathematical background
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:
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 ). A well-written discussion of this compactness condition and its role in critical point theory can be found in Mawhin and Willem . One of the minimax theorems needed in the sequel is the well-known ‘mountain pass theorem’.
then and c is a critical value of φ.
where by we denote the outward unit normal on ∂ Ω.
(the critical Sobolev exponent).
Clearly, if and for all , then . A slight modification of the proof of Proposition 2.6 of Arcoya and Ruiz  in order to accommodate the presence of the extra linear term leads to the following strong comparison principle.
then in .
If , then we write .
By we denote the Lebesgue measure on .
3 Positive solutions
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
(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 ;
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 .
with . Clearly is not monotone.
(if , then ).
So, we can apply the boundary point theorem of Pucci and Serrin [, p.120] and have that . Therefore, .
(see (3.1) and recall that ), which contradicts (2.3). Therefore, . □
Proposition 3.4 If hypotheses H hold, then .
(see (3.2) and (2.3)).
hence , .
so (see Proposition 3.3).
So, for big, we have and so . □
Proposition 3.5 If hypotheses H hold and , then .
and hence .
hence . This proves that . □
(see (3.12) and Proposition 2.3).
for some .
for some (see (3.17) and (2.3)).
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 3.7 If hypotheses H hold, then .
so , which contradicts (3.27).
(see Proposition 2.4).
and so is a solution of problem .
Let to get a contradiction. This proves that and so , hence . □
The bifurcation-type theorem summarizes the situation for problem .
Theorem 3.8 If hypotheses H hold, then there exists such that
(b) for problem has at least one positive solution ;
(c) for problem has 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  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 , which studies the -equations and produces nodal solutions for them.
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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