On symmetric positive homoclinic solutions of semilinear p-Laplacian differential equations

In this paper we study the existence of even positive homoclinic solutions for p-Laplacian ordinary differential equations (ODEs) of the type {(u^{\mathrm{\prime }}{|u^{\mathrm{\prime }}|}^{p-2})}^{\mathrm{\prime }}-a(x)u{|u|}^{p-2}+\lambda b(x)u{|u|}^{q-2}=0, where , \lambda >0 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.

In this paper we prove the existence of positive homoclinic solutions for p-Laplacian ODEs of the type

where and \lambda >0. We assume that

(H) the functions a(x) are b(x) are continuously differentiable, strictly positive, and . Let, moreover, a(x) and b(x) be even functions on ℝ, x{a}^{\mathrm{\prime }}(x)>0 and for x\ne 0.

By a solution of (1), we mean a function u:\mathbb{R}\to \mathbb{R} such that u\in C^1(\mathbb{R}), {(u^{\mathrm{\prime }}{|u^{\mathrm{\prime }}|}^{p-2})}^{\mathrm{\prime }}\in C(\mathbb{R}) and Eq. (1) holds for every x\in \mathbb{R}. We are looking for positive solutions of (1) which are homoclinic, i.e., u(x)\to 0 and u^{\mathrm{\prime }}(x)\to 0 as |x|\to \mathrm{\infty }.

In the case p=2, q=4 and \lambda =1, similar problems are considered in [1–3] using variational methods. Note that in [2] and [3] the following second-order differential equations are considered:

and

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 [4] and Cronin [5]. Further results and the phase plane analysis of these equations with constant coefficients are given in [6]. Note that the periodic and homoclinic solutions of p-Laplacian ODEs are considered in [7, 8].

The present work is an extension of these studies to p-Laplacian ODEs. Let X_T:=W_0^{1,p}(-T,T) be the Sobolev space of p-integrable absolutely continuous functions u:[-T,T]\to \mathbb{R} such that

and u(-T)=u(T)=0.

We use a variational treatment of the problem considering the functional J_T:X_T\to \mathbb{R}

where u^{+}(x)=max\{u(x),0\}.

Using the well-known mountain-pass theorem, we conclude that the functional J_T has a nontrivial critical point u_{T,\lambda }\in X_T, which is a solution of the restricted problem

Further, we obtain uniform estimates for the solutions u_{T,\lambda }, extended by 0 outside [-T,T]. Then, a positive homoclinic solution u_{\lambda } of (1) is found as a limit of u_{T,\lambda }, as T\to \mathrm{\infty } in C_{\mathrm{loc}}^1(\mathbb{R}). The function u_{\lambda } is also an even function.

To obtain the property, we extend the symmetry lemma of Korman and Ouyang [9] to the p-Laplacian equations. The result is formulated and proved in Section 2.

Our main result is:

Theorem 1 Suppose that , \lambda >0 and assumptions (H) hold. Then Eq. (1) has a positive solution u_{\lambda } such that u_{\lambda }(x)\to 0 and u_{\lambda }^{\mathrm{\prime }}(x)\to 0 as |x|\to \mathrm{\infty }. Moreover, the solution u_{\lambda } is an even function, max\{u_{\lambda }(x):x\in \mathbb{R}\}=u_{\lambda }(0)\to +\mathrm{\infty } as \lambda \to 0 and for x>0.

Theorem 1 is proved in Section 3. From its proof we have

from which it follows that u_{\lambda }(0)\to +\mathrm{\infty } as \lambda \to 0. Observe that if \lambda =0, the problem

has a unique solution u=0. Indeed, multiplying the equation by u and integrating by parts over ℝ, we obtain

which implies that u\equiv 0.

A simplified method can be applied to the equations

under assumptions (H) and , \lambda >0. Note that in this case, the even homoclinic solution u_{\lambda } of Eq. (3) satisfies

and again u_{\lambda }(0)\to +\mathrm{\infty } as \lambda \to 0. If a and b are constants, Eq. (3) is a conservative system and one can plot the phase curves {(\frac{v}{2})}^2-a\frac{{|u|}^p}{p}+\lambda b\frac{{|u|}^q}{q}=C in the phase plane (u,v)=(u,u^{\mathrm{\prime }}). An example is given at the end of Section 3.

Let {\phi }_p(t)=t{|t|}^{p-2}, p\ge 2 and {\mathrm{\Phi }}_p(t)=\frac{{|t|}^p}{p}. It is clear that {\mathrm{\Phi }}_p(t) is a differentiable function and {\mathrm{\Phi }}_p^{\mathrm{\prime }}(t)={\phi }_p(t). Moreover, {\phi }_p^{\mathrm{\prime }}(t) exists and {\phi }_p^{\mathrm{\prime }}(t)=(p-1){|t|}^{p-2} for p\ge 2.

Let L^p(a,b), be the space of Lebesgue measurable functions u:(a,b)\to \mathbb{R} such that the norm .

The dual space of L^p(a,b) is L^{p^{\mathrm{\prime }}}(a,b), where \frac{1}{p}+\frac{1}{p^{\mathrm{\prime }}}=1. Let 〈\cdot ,\cdot 〉 be the duality pairing between L^{p^{\mathrm{\prime }}}(a,b) and L^p(a,b). By the Hölder inequality, |〈v,u〉|\le {|v|}_{p^{\mathrm{\prime }}}{|u|}_p for any v\in L^{p^{\mathrm{\prime }}}(a,b) and u\in L^p(a,b). We will use the following lemmata in further considerations.

Lemma 2 For any u,v\in L^p(a,b), the following inequality holds:

Proof of Lemma 2. Note that for u\in L^p(a,b), {\phi }_p(u)\in L^{p^{\mathrm{\prime }}}(a,b). From the Hölder inequality, we have

 □

Lemma 3 Let p\ge 2, u\in C^1([a,b]) and {(u^{\mathrm{\prime }}{|u^{\mathrm{\prime }}|}^{p-2})}^{\mathrm{\prime }}\in C([a,b]). Then

The statement of Lemma 3 follows simply from the identity

The one-dimensional p-Laplacian operator L_p for a differentiable function u on the interval I is introduced as L_p(u):={({\phi }_p(u^{\mathrm{\prime }}))}^{\mathrm{\prime }}. Let us consider the problem

where f\in C^1([-T,T]\times {\mathbb{R}}^{+}) and satisfies

A function u:[-T,T]\to \mathbb{R} is said to be a solution of the problem (4) if u\in C^1([-T,T]) with u(-T)=u(T)=0 is such that u^{\mathrm{\prime }}{|u^{\mathrm{\prime }}|}^{p-2} is absolutely continuous and L_{p}u(x)+f(x,u(x))=0 holds a.e. in (-T,T).

We formulate an extension of Lemma 1 of [9] for p-Laplacian nonlinear equations. The result of Korman and Ouyang is one-dimensional analogue of the result of Gidas, Ni and Nirenberg [10] 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 [11].

Theorem 4 Assume that f\in C^1([-T,T]\times {\mathbb{R}}^{+}) satisfies (5). Then any positive solution u of (4) is an even function such that max\{u(x):-T\le x\le T\}=u(0) and for x\in (0,T].

Remark 1 Let us note that if the function f satisfies (5), but u is not a positive solution of (4), then u is not necessarily an even function. A simple counter example in the case p=2 is the problem

The term f(x,u)=u-x^2+{\pi }^2-2 satisfies (5) in the interval (-\pi ,\pi ), but the solution of the problem u(x)=x^2-{\pi }^2+sinx is negative in (-\pi ,\pi ) and not an even function. Its graph is presented in Figure 1. It would be more interesting to show an example for the case p>2 and f satisfying the additional assumption f(x,0)=0.

Graph of the functions \mathit{u}\mathbf{(}\mathit{x}\mathbf{)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{-}{\mathit{\pi }}^{\mathbf{2}}\mathbf{+}\mathbf{sin}\mathit{x} .

Full size image

Sketch of Proof of Theorem 4 Suppose that the function u has only one global maximum on [-T,T].

Assume that the function u(x) has a finite number of local minima in the interval [0,T], and let x_1 be the largest local minimum. Let \overline{x}\in [x_1,T] be the local maximum and \tilde{x}\in [\overline{x},T] be such that u(x_1)=u(\tilde{x}). Denote u_1=u(x_1)=u(\tilde{x}) and u_2=u(\overline{x}), and let x=\alpha (u) and x=\beta (u) be the inverse functions of the function u=u(x) in the intervals [x_1,\overline{x}] and [\overline{x},T], respectively. Multiplying the equation in (4) by u^{\mathrm{\prime }} and integrating in [x_1,\tilde{x}], 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 [12]. Suppose now that u has infinitely many local minima in [-T,x^{\ast }]. Further, we can follow the steps of the proof of Lemma 1 of [9] with corresponding modifications based on Lemma 3. □

Let X_T=W_0^{1,p}(-T,T) be the Sobolev space of p-integrable absolutely continuous functions u:[-T,T]\to \mathbb{R} such that

and u(-T)=u(T)=0. Note that if a(x) is strictly positive and bounded, i.e., there exist a and A such that , then {\parallel u\parallel }_{a,T}^p={\int }_{-T}^T({|u^{\mathrm{\prime }}(x)|}^p+a(x){|u(x)|}^p)dx is an equivalent norm in X_T.

We need an extension to the p-case of the following proposition by Rabinowitz [13].

Proposition 5 Let u\in W_{\mathrm{loc}}^{1,p}(\mathbb{R}). Then:

1. (i)

   If T\ge 1, for x\in [T-1/2,T+1/2],

   \underset{x\in [T-1/2,T+1/2]}{max}|u(x)|\le 2^{\frac{p-1}{p}}{\left({\int }_{T-1/2}^{T+1/2}({|u^{\mathrm{\prime }}(t)|}^p+{|u(t)|}^p)dt\right)}^{1/p}.

   (6)
2. (ii)

   For every u\in W_0^{1,p}(-T,T),

   {\parallel u\parallel }_{L^{\mathrm{\infty }}(-T,T)}\le 2^{\frac{p-1}{p}}{\parallel u\parallel }_T.

   (7)

Proof of Proposition 5 Let x,t\in [T-1/2,T+1/2]. It follows

Integrating with respect to t\in [T-1/2,T+1/2] and using the Hölder and Jensen inequalities, we obtain

1. (ii)

   Take u\in W_0^{1,p}(-T,T). Since W_0^{1,p}(-T,T)\subset C[-T,T], there exists \tau \in [-T,T] such that by (i)

   \begin{array}{rcl}{\parallel u\parallel }_{L^{\mathrm{\infty }}(-T,T)}& =& {\parallel u\parallel }_{C[\tau -1/2,\tau +1/2]}\le 2^{\frac{p-1}{p}}{\left({\int }_{\tau -1/2}^{\tau +1/2}({|u^{\mathrm{\prime }}(t)|}^p+{|u(t)|}^p)dt\right)}^{1/p}\\ \le & 2\parallel u\parallel .\end{array}

 □

We are looking for positive solutions of (1), which are homoclinic, i.e., u(x)\to 0 and u^{\mathrm{\prime }}(x)\to 0 as |x|\to \mathrm{\infty }. Firstly, we look for positive solutions of the problem

A function u:[-T,T]\to \mathbb{R} is said to be a solution of the problem (P_T) if u\in C^1([-T,T]) with u(-T)=u(T)=0 is such that {\phi }_p(u^{\mathrm{\prime }}) is absolutely continuous and {({\phi }_p(u^{\mathrm{\prime }}))}^{\mathrm{\prime }}(x)-a(x){\phi }_p(u)(x)+\lambda b(x){\phi }_q(u)(x)=0 holds a.e. in (-T,T).

A function u:[-T,T]\to \mathbb{R} is said to be a weak solution of the problem (P_T) if

Standard arguments show that a weak solution of the problem (P_T) is a solution of (P_T) (see [14] and [15]). Consider the modified problem

where u^{+}=max(u,0). It is easy to see that solutions of the problem (P_T^{+}) are positive solutions of the problem (P_T). Indeed, if u(x) is a solution of (P_T^{+}) and u(x) has negative minimum at x_0\in (-T,T), since for p\ge 2, {({\phi }_p(u^{\mathrm{\prime }}))}^{\mathrm{\prime }}(x_0)\ge 0, by the equation {({\phi }_p(u^{\mathrm{\prime }}))}^{\mathrm{\prime }}-a(x){\phi }_p(u)+\lambda b(x){(u^{+})}^{q-1}=0, we reach a contradiction

Then u(x)\ge 0 and u is a solution of (P_T). We use a variational treatment of the problem (P_T^{+}), considering the functional J_T:X_T\to \mathbb{R}

Critical points of J_T are weak solutions of (P_T^{+}), i.e.,

and, by a standard way, they are solutions of (P_T^{+}). We show that J_T satisfies the assumptions of the mountain-pass theorem of Ambrosetti and Rabinowitz [16].

Theorem 6 (Mountain-pass theorem)

Let X be a Banach space with norm \parallel \cdot \parallel, I\in C^1(X,\mathbf{R}), I(0)=0 and I satisfy the (PS) condition. Suppose that there exist r>0, \alpha >0 and e\in X such that \parallel e\parallel >r

1. (i)

   I(x)\ge \alpha if \parallel x\parallel =r,

2. (ii)

   . Let c={inf}_{\gamma \in \mathrm{\Gamma }}\{{max}_{0\le t\le 1}I(\gamma (t))\}\ge \alpha, where

   \mathrm{\Gamma }=\{\gamma \in C([0,1],X):\gamma (0)=0,\gamma (1)=e\}.

Then c is a critical value of I, i.e., there exists x_0 such that I(x_0)=c and I^{\mathrm{\prime }}(x_0)=0.

Next, denote by C_j several positive constants.

Lemma 7 Let , \lambda >0 and assumptions (H) hold. Then for every T>0, the problem (P_T) has a positive solution u_{T,\lambda }. Moreover, there is a constant K>0, independent of T, such that

Proof Step 1. J_T satisfies the (PS) condition.

Let {(u_k)}_k\subset X_T be a sequence, and suppose there exist C_1 and k_0 such that for k\ge k_0

and

Let us denote \hat{a}=min(1,a). From (9) and (10), it follows that

and

Then

and

We have

which implies that the sequence {(u_k)}_k is bounded in X_T. By the compact embedding X_T\subset C([-T,T]), there exist u\in X_T and the subsequence of {(u_k)}_k, still denoted by {(u_k)}_k, such that u_k\rightharpoonup u weakly in X_T and u_k\to u strongly in C([-T,T]). We will show that u_k\to u strongly in X_T using Lemma 2. By uniform convergence of u_k to u in C([-T,T]), it follows that

and

Then

and by Lemma 2,

which implies that {|u_k^{\mathrm{\prime }}|}_p\to {|u^{\mathrm{\prime }}|}_p. Then {\parallel u_k\parallel }_T\to {\parallel u\parallel }_T and by the uniform convexity of the space X_T, it follows that {\parallel u_k-u\parallel }_T\to 0, as k\to \mathrm{\infty }.

Step 2. Geometric conditions.

Obviously, J_T(0)=0. By assumption (H) it follows

if {\parallel u\parallel }_T=\rho :={(\frac{aq}{2\lambda p^2})}^{1/(q-p)}>0. Then J_T(u)\ge \frac{\hat{a}{\rho }^p}{2p}>0.

Let u_0(x)\in X_T be such that u_0(x)>0 if x\in (-T,T) and also u_0(-T)=u_0(T)=0. Consider the function

Then

for μ large enough.

By the mountain-pass theorem, there exists a solution u_{T,\lambda }\in X_T such that

where

Moreover, using the variational characterization (11), we have

Therefore, u_{T,\lambda } is a nontrivial and positive solution of (P_T). By Theorem 4, max\{u_{T,\lambda }:-T\le x\le T\}=u_{T,\lambda }(0) and for x\in (0,T].

Step 3. Uniform estimates.

Let T_1\ge T\ge 1. By continuation with zero of a function u\in X_T to [-T_1,T_1], we have X_T\subset X_{T_1}and {\mathrm{\Gamma }}_T\subset {\mathrm{\Gamma }}_{T_1}. Using the variational characterization (11), we infer that c_{T_1}\le c_T\le c_1 and then

Multiplying the equation of (P_T) by u_T and integrating by parts, we have

Then by (12),

We get (8) with K=\frac{pq{c}_1}{\hat{a}(q-p)}, which completes the proof. □

Proof of Theorem 1 Take T_n\to \mathrm{\infty } and let u_n be the solution of the problem (P_{T_n}) given by Lemma 2. Consider the extension of u_n to ℝ with zero outside [-T_n,T_n] and denote it by the same symbol.

Claim 1. The sequence of functions {(u_n)}_n is uniformly bounded and equicontinuous.

By (8) and the embedding of X_{T_n} in C([-T_n,T_n]), there is K_1 such that {\parallel u_n\parallel }_{L^{\mathrm{\infty }}([-T_n,T_n])}\le K_1. Then by the equation of (P_{T_n}), it follows that

By the mean value theorem for every natural n and every t\in \mathbb{R}, there exists {\xi }_n\in [t-1,t] such that

Then, as a consequence of (13), we obtain

from which it follows {\parallel u_n^{\mathrm{\prime }}\parallel }_{L^{\mathrm{\infty }}([-T_n,T_n])}\le K_3 and the sequence of functions (u_n) is equicontinuous. Further, we claim that the sequence {(u_n^{\mathrm{\prime }})}_n is also equicontinuous.

Claim 2. The sequence of functions {(u_n^{\mathrm{\prime }})}_n is equicontinuous.

To prove this statement, we follow the method given by Tang and Xiao [7]. For completeness, we present it in details.

Suppose that {(u_n^{\mathrm{\prime }})}_n is not an equicontinuous sequence in C_{\mathrm{loc}}(\mathbb{R}). Then there exist an {\epsilon }_0 and sequences (t_k^1) and (t_k^2) such that and

By (14), there are numbers w^1 and w^2 and the subsequence (u_{n_k}^{\mathrm{\prime }}) such that u_{n_k}^{\mathrm{\prime }}(t_k^1)\to w^1 and u_{n_k}^{\mathrm{\prime }}(t_k^2)\to w^2 as k\to \mathrm{\infty }. By (15), |w^1-w^2|\ge {\epsilon }_0. On the other hand, by (13) we have

Then passing to a limit as k\to \mathrm{\infty }, we obtain {\phi }_p(w^1)={\phi }_p(w^2). Hence, w^1=w^2 which contradicts |w^1-w^2|\ge {\epsilon }_0. Thus, the sequence {(u_n^{\mathrm{\prime }})}_n is equicontinuous.

Let T>0. By Claim 1 and Claim 2 and the Arzelà-Ascoli theorem, there is a subsequence of {(u_n)}_n, still denoted by {(u_n)}_n, and functions u_{\lambda 1} and v_{\lambda 1} of C([-T,T]) such that {\parallel u_n-u_{\lambda 1}\parallel }_{C([-T,T])}\to 0 and {\parallel u_n^{\mathrm{\prime }}-v_{\lambda 1}\parallel }_{C([-T,T])}\to 0. Trivially, it follows that u_{\lambda 1}\in C^1([-T,T]), u_{\lambda 1}^{\mathrm{\prime }}=v_{\lambda 1} and {\parallel u_n-v_{\lambda 1}\parallel }_{C^1([-T,T])}\to 0. Repeating this procedure as in [7], we obtain that there is a subsequence of {(u_n)}_n, still denoted by {(u_n)}_n, and u_{\lambda } such that u_n\to u_{\lambda } in C_{\mathrm{loc}}^1(\mathbb{R}). The function u_{\lambda } satisfies Eq. (1). Indeed, let [x_1,x_2] be an interval of ℝ and T_n>0 such that [x_1,x_2]\subset [-T_n,T_n]. By the above considerations, taking a limit as n\to \mathrm{\infty } in the equation

equivalent to

we obtain

and hence

Since x_1 and x_2 are arbitrary, u_{\lambda } is a solution of (1). Moreover, we have

It remains to show that u_{\lambda } is nonzero and u_{\lambda }(\pm \mathrm{\infty })=0 and u_{\lambda }^{\mathrm{\prime }}(\pm \mathrm{\infty })=0.

By Theorem 4, u_n is an even function and attains its maximum at 0. Then by Eq. (1),

By assumption (H)

independently of n. Hence, passing to a limit as n\to \mathrm{\infty }, we obtain

Note, that this implies max\{u_{\lambda }(x):x\in R\}=u_{\lambda }(0)\to +\mathrm{\infty } as \lambda \to 0.

From (16) and Proposition 5, it follows

(17)

so u_{\lambda }(\pm \mathrm{\infty })=0.

Now, we will show that u_{\lambda }^{\mathrm{\prime }}(\mathrm{\infty })=0. The arguments for u_{\lambda }^{\mathrm{\prime }}(-\mathrm{\infty })=0 are similar.

If u_{\lambda }^{\mathrm{\prime }}(\mathrm{\infty })\ne 0, there exist {\epsilon }_1>0 and a monotone increasing sequence x_k\to \mathrm{\infty } such that |u_{\lambda }^{\mathrm{\prime }}(x_k)|\ge {(2{\epsilon }_1)}^{1/(p-1)}. Then for x\in [x_k,x_k+\frac{{\epsilon }_1}{K_2}],