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.
Keywords:p-Laplacian ODEs; homoclinic solution; weak solution; Palais-Smale condition; mountain-pass theorem
1 Introduction and main results
In this paper we prove the existence of positive homoclinic solutions for p-Laplacian ODEs of the type
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].
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 is proved in Section 3. From its proof we have
A simplified method can be applied to the equations
2 Preliminary results
The statement of Lemma 3 follows simply from the identity
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 .
The term satisfies (5) in the interval , but the solution of the problem is negative in and not an even function. Its graph is presented in Figure 1. It would be more interesting to show an example for the case and f satisfying the additional assumption .
Figure 1. Graph of the functions.
Assume that the function has a finite number of local minima in the interval , and let be the largest local minimum. Let be the local maximum and be such that . Denote and , and let and be the inverse functions of the function in the intervals and , respectively. Multiplying the equation in (4) by and integrating in , we obtain by Lemma 3 and (5):
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
We need an extension to the p-case of the following proposition by Rabinowitz .
where . It is easy to see that solutions of the problem () are positive solutions of the problem (). Indeed, if is a solution of () and has negative minimum at , since for , , by the equation , we reach a contradiction
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)
which implies that the sequence is bounded in . By the compact embedding , there exist and the subsequence of , still denoted by , such that weakly in and strongly in . We will show that strongly in using Lemma 2. By uniform convergence of to u in , it follows that
and by Lemma 2,
Step 2. Geometric conditions.
for μ large enough.
Moreover, using the variational characterization (11), we have
Step 3. Uniform estimates.
Then by (12),
Then, as a consequence of (13), we obtain
To prove this statement, we follow the method given by Tang and Xiao . For completeness, we present it in details.
Let . By Claim 1 and Claim 2 and the Arzelà-Ascoli theorem, there is a subsequence of , still denoted by , and functions and of such that and . Trivially, it follows that , and . Repeating this procedure as in , we obtain that there is a subsequence of , still denoted by , and such that in . The function satisfies Eq. (1). Indeed, let be an interval of ℝ and such that . By the above considerations, taking a limit as in the equation
By assumption (H)
From (16) and Proposition 5, it follows
which contradicts (16).
Remark 2 A simplified method can be applied to the equations
Since is a Hilbert space, compactly embedded in the proof of the (PS)-condition is easier. Similar considerations are made in  and . Then, the even homoclinic solution is obtained as a limit of the sequence . Note that in this case, the even homoclinic solution of Eq. (3) satisfies
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 . Consider the equation . The phase portrait in a plane, for in the rectangle , is plotted on Figure 2.
Figure 2. Phase portrait of, in.
The author declares that he has no competing interests.
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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