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

where $2\le p<q$ and $\lambda >0$. We assume that

(H) the functions $a(x)$ are $b(x)$ are continuously differentiable, strictly positive, $0<a\le a(x)\le A$ and $0<b\le b(x)\le B$. Let, moreover, $a(x)$ and $b(x)$ be even functions on ℝ, $x{a}^{\mathrm{\prime }}(x)>0$ and $x{b}^{\mathrm{\prime }}(x)<0$ 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 $|x|\to \mathrm{\infty }$.

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].

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

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

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 $2\le p<q$, $\lambda >0$ 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 $u_{\lambda }^{\mathrm{\prime }}(x)<0$ for $x>0$.

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

which implies that $u\equiv 0$.

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 $L^p(a,b)$, $1<p<\mathrm{\infty }$ be the space of Lebesgue measurable functions $u:(a,b)\to \mathbb{R}$ such that the norm ${|u|}_p^p={\int }_a^b{|u(x)|}^{p}dx<\mathrm{\infty }$.

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.

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

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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 $u^{\mathrm{\prime }}(x)<0$ for $x\in (0,T]$.

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 $f(x,0)=0$.

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

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. □

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 $0<a\le a(x)\le A$, 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].

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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 $(-T,T)$.

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

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)

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.

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

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

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.

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

for μ large enough.

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 $u_{T,\lambda }^{\mathrm{\prime }}(x)<0$ for $x\in (0,T]$.

Step 3. Uniform estimates.

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.

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.

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.