On symmetric positive homoclinic solutions of semilinear p-Laplacian differential equations
© Tersian; licensee Springer 2012
Received: 31 July 2012
Accepted: 8 October 2012
Published: 24 October 2012
In this paper we study the existence of even positive homoclinic solutions for p-Laplacian ordinary differential equations (ODEs) of the type , where , and the functions a and b are strictly positive and even. First, we prove a result on symmetry of positive solutions of p-Laplacian ODEs. Then, using the mountain-pass theorem, we prove the existence of symmetric positive homoclinic solutions of the considered equations. Some examples and additional comments are given.
MSC:34B18, 34B40, 49J40.
Keywordsp-Laplacian ODEs homoclinic solution weak solution Palais-Smale condition mountain-pass theorem
1 Introduction and main results
where and . We assume that
(H) the functions are are continuously differentiable, strictly positive, and . Let, moreover, and be even functions on ℝ, and for .
By a solution of (1), we mean a function such that , and Eq. (1) holds for every . We are looking for positive solutions of (1) which are homoclinic, i.e., and as .
where a, b and c are periodic, bounded functions and a and c are positive. These equations come from a biomathematics model suggested by Austin  and Cronin . Further results and the phase plane analysis of these equations with constant coefficients are given in . Note that the periodic and homoclinic solutions of p-Laplacian ODEs are considered in [7, 8].
Further, we obtain uniform estimates for the solutions , extended by 0 outside . Then, a positive homoclinic solution of (1) is found as a limit of , as in . The function is also an even function.
To obtain the property, we extend the symmetry lemma of Korman and Ouyang  to the p-Laplacian equations. The result is formulated and proved in Section 2.
Our main result is:
Theorem 1 Suppose that , and assumptions (H) hold. Then Eq. (1) has a positive solution such that and as . Moreover, the solution is an even function, as and for .
which implies that .
and again as . If a and b are constants, Eq. (3) is a conservative system and one can plot the phase curves in the phase plane . An example is given at the end of Section 3.
2 Preliminary results
Let , and . It is clear that is a differentiable function and . Moreover, exists and for .
Let , be the space of Lebesgue measurable functions such that the norm .
The dual space of is , where . Let be the duality pairing between and . By the Hölder inequality, for any and . We will use the following lemmata in further considerations.
A function is said to be a solution of the problem (4) if with is such that is absolutely continuous and holds a.e. in .
We formulate an extension of Lemma 1 of  for p-Laplacian nonlinear equations. The result of Korman and Ouyang is one-dimensional analogue of the result of Gidas, Ni and Nirenberg  for symmetry of positive solutions of semilinear Laplace equations. In the case of p-Laplacian equations, the symmetry of solutions in higher dimensions is discussed by Reihel and Walter .
Theorem 4 Assume that satisfies (5). Then any positive solution u of (4) is an even function such that and for .
Sketch of Proof of Theorem 4 Suppose that the function u has only one global maximum on .
which leads to contradiction. One can prove the last fact using other arguments; see, for instance, Theorem 2.1 of . Suppose now that u has infinitely many local minima in . Further, we can follow the steps of the proof of Lemma 1 of  with corresponding modifications based on Lemma 3. □
3 Proof of the main result
and . Note that if is strictly positive and bounded, i.e., there exist a and A such that , then is an equivalent norm in .
We need an extension to the p-case of the following proposition by Rabinowitz .
- (i)If , for ,(6)
- (ii)For every ,(7)
- (ii)Take . Since , there exists such that by (i)
A function is said to be a solution of the problem () if with is such that is absolutely continuous and holds a.e. in .
and, by a standard way, they are solutions of (). We show that satisfies the assumptions of the mountain-pass theorem of Ambrosetti and Rabinowitz .
Theorem 6 (Mountain-pass theorem)
- (ii). Let , where
Then c is a critical value of I, i.e., there exists such that and .
Next, denote by several positive constants.
Proof Step 1. satisfies the (PS) condition.
which implies that . Then and by the uniform convexity of the space , it follows that , as .
Step 2. Geometric conditions.
if . Then .
for μ large enough.
Therefore, is a nontrivial and positive solution of (). By Theorem 4, and for .
Step 3. Uniform estimates.
We get (8) with , which completes the proof. □
Proof of Theorem 1 Take and let be the solution of the problem () given by Lemma 2. Consider the extension of to ℝ with zero outside and denote it by the same symbol.
Claim 1. The sequence of functions is uniformly bounded and equicontinuous.
from which it follows and the sequence of functions is equicontinuous. Further, we claim that the sequence is also equicontinuous.
Claim 2. The sequence of functions is equicontinuous.
To prove this statement, we follow the method given by Tang and Xiao . For completeness, we present it in details.
Then passing to a limit as , we obtain . Hence, which contradicts . Thus, the sequence is equicontinuous.
It remains to show that is nonzero and and .
Note, that this implies as .
Now, we will show that . The arguments for are similar.
which contradicts (16).
Moreover, u is an even function that attains its only maximum at 0, since the same holds for the functions . Arguing as in the proof of Theorem 4, we easily obtain that if . □
Dedicated to Professor Jean Mawhin on the occasion of his 70th anniversary.
The author thanks Prof. Alberto Cabada and Prof. Luis Sanchez for helpful remarks concerning Theorem 4. The author would like to thank the Department of Mathematics and Theoretical Informatics at the Technical University of Kosice, Slovakia, where the paper was prepared during his visit on the SAIA Fellowship programme. The author is thankful to the editor and anonymous referee for their comments and suggestions on the article.
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