
We consider semilinear elliptic equations of the form Δu+f(|x|,u)=0 on RN with f(|x|,u)=q(|x|)g(u). These type of equations arise in various problems in applied mathematics, and particularly in the study of population dynamics, solitary waves, diffusion processes, and phase transitions. We show that under suitable assumptions on the nonlinearity f, there exists an oscillating radial solution converging to a zero of the function g. We also study the oscillating and limiting behavior of this solution.
Citation: H. Al Jebawy, H. Ibrahim, Z. Salloum. On oscillating radial solutions for non-autonomous semilinear elliptic equations[J]. AIMS Mathematics, 2024, 9(6): 15190-15201. doi: 10.3934/math.2024737
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We consider semilinear elliptic equations of the form Δu+f(|x|,u)=0 on RN with f(|x|,u)=q(|x|)g(u). These type of equations arise in various problems in applied mathematics, and particularly in the study of population dynamics, solitary waves, diffusion processes, and phase transitions. We show that under suitable assumptions on the nonlinearity f, there exists an oscillating radial solution converging to a zero of the function g. We also study the oscillating and limiting behavior of this solution.
The existence and behavior of positive radial solutions of the semilinear elliptic equation
Δu+f(u)=0inRN | (1.1) |
has been studied by many authors [5,6,7,9,10,11,12]. The unknown u being radial and smooth, the study of existence shifts to the ordinary differential equation
{u″+N−1ru′+f(u)=0onR+,u(0)=α>0andu′(0)=0, | (1.2) |
where f:R+→R is a locally Lipschitz function satisfying, among other conditions,
f(ξ)=0for some ξ>0. |
It was proved (see [11]) that there exists a positive oscillating solution of (1.2) satisfying limr→∞u(r)=ξ. The proof is based on ODE methods and makes an important use of the following identity, which is derived by multiplying the first equation of (1.2) by u′ and then integrating by parts:
(u′(b))22−(u′(a))22+∫baN−1r(u′(r))2dr+F(u(b))−F(u(a))=0, | (1.3) |
where F(t)=∫t0f(s)ds and 0≤a≤b. The main advantage, remarkably and frequently taken in [11], of (1.3) is a simple observation that
F(u(b))≤F(u(a))for0≤a≤b and u′(a)=0. | (1.4) |
To our knowledge, this result of the existence of oscillating, radial, and convergent solutions of (1.1) has not been generalized to non-autonomous equations of the form
Δu+f(|x|,u)=0inRN, | (1.5) |
that appear in various problems in applied mathematics related to, for example, solitary waves for Klein-Gordon equations and the reaction-diffusion equations. Such a generalization is then worth investigating. Let us mention that the existence of radial solutions for semilinear elliptic equations that converges at infinity has attracted the attention of different authors (see for instance [1,2,3,5,6,7,10,12]). Smooth radial solutions of (1.5) satisfy the following identity, analogous to (1.3),
(u′(b))22−(u′(a))22+∫baN−1r(u′(r))2dr−∫baFr(r,u(r))dr+F(b,u(b))−F(a,u(a))=0, | (1.6) |
where
F(r,t)=∫t0f(r,s)ds. |
The difficulties here are in fact twofold: the determination of the exact limit ξ=limr→∞u(r) strongly depends on the behavior of f(r,t) when r→∞, so we may directly get into a limiting problem of u due to wild limiting behavior of f. The second difficulty is to obtain a practical inequality (useful in various technical situations) like (1.4) due to the presence of the term ∫baFr(r,u(r))dr in (1.6). Indeed, maintaining a negative sign of this term is mainly subjected to the radial variation of f, and to the location of the unknown function u. Since our aim is to understand how to generalize the existence result of [11], we see that the consideration of all these conditions for the general nonlinearity f is not our best starting point. For this reason, we hereby consider functions f:R+×R+→R of the form
f(r,t)=q(r)g(t), | (1.7) |
where q:R+⟶]0,∞[ is a positive, increasing C1 function with limr→∞q(r)=q∞<∞, and g:R+→R is a locally Lipschitz function satisfying the following conditions:
g<0 in (0,ξ) with g(0)=g(ξ)=0 for some ξ>0, | (1.8) |
∃ η>ξ such that ∫η0g(t)dt=0 and g>0 in (ξ,η), | (1.9) |
g′(ξ)>0. | (1.10) |
As an immediate consequence, we deduce that f(r,t) is decreasing in r for 0<t<ξ, increasing in r for ξ<t<η, and limr→∞f(r,t)=f∞(t)=q∞g(t). Moreover, for every r≥0, we have
∫η0f(r,t)dt=∫η0q(r)g(t)dt=q(r)∫η0g(t)dt=0. |
Since we are interested in radial solutions of (1.5) with f given by (1.7), we consider the following initial value problem on [0,∞[:
{u″+N−1ru′+q(r)g(u)=0,u(0)=α>0andu′(0)=0, | (2.1) |
where g satisfies (1.8)–(1.10). Then, for every α∈(0,η) with α≠ξ, (2.1) admits a solution u that remains positive for all r>0 (see for instance [8]). Furthermore, we prove the following result:
Theorem 2.1. If f satisfies (1.7)–(1.10), then for every α∈(0,η) with α≠ξ, the solution u of (2.1) oscillates (has infinitely many local maxima and local minima) with limr→∞u(r)=ξ in such a way that the local maxima of u are strictly decreasing to ξ at ∞ and the local minima are strictly increasing to ξ at ∞, and the distance between two consecutive zeros of u−ξ tends to π√q∞g′(ξ).
We adopt the shooting method used in [4], which consists of varying α in (0,η) to obtain a radial oscillating solution of (2.1). The main ingredient of our proof is the energy Eq (1.6) that now reads
(u′(b))22−(u′(a))22+∫baN−1r(u′(r))2dr−∫baq′(r)(∫u(r)0g(s)ds)dr+q(b)∫u(b)0g(s)ds−q(a)∫u(a)0g(s)ds=0. | (2.2) |
Also, multiplying (2.1) by u′ and integrating between 0≤a≤b with u′(a)=u′(b)=0 gives
∫baq(r)g(u(r))u′(r)dr≤0. | (2.3) |
This inequality plays a crucial role in regards to the monotonicity of the local extrema of u. Finally, a direct integration of (2.1) between 0≤a≤b leads to
u′(b)−u′(a)+∫baN−1ru′(r)dr+∫baq(r)g(u(r))dr=0. | (2.4) |
Finally, if v(r)=rN−12(u(r)−ξ), then
v″+{q(r)g(u(r))u(r)−ξ−(N−1)(N−3)4r2}v=0, | (2.5) |
where we use the convention that g(u)u−ξ=g′(ξ) when u=ξ.
Proof of Theorem 2.1. We only consider the case α∈]ξ,η[. The case α∈]0,ξ[ is treated similarly. The proof is divided into several steps.
Step 1. (0<u(r)<η for all r≥0)
Let us show that if 0<u(0)=α<η, then 0<u(r)<η for all r≥0. This inequality satisfied by u ensures a negative sign for the term ∫u(r)0g(s)ds appearing in (2.2), and thus leads to useful results later on. Let
R=inf{r>0:u(r)=0oru(r)=η}, |
and assume that R<∞. Since u(0)=α with α≠0 and α≠η, then there exists δ>0 such that u(r)≠0 and u(r)≠η for all 0<r<δ. Hence, R>δ>0. Again, using the continuity of u, we get that
u(R)=0oru(R)=η. |
The important point is that 0<u(r)<η for 0≤r<R, and so by using (2.2) with a=0 and b=R, and owing to the fact that ∫u(r)0g(s)ds≤0 for 0≤r<R, q′≥0, u′(0)=0, and ∫u(R)0g(s)ds=0, we obtain
q(0)∫α0g(s)ds≥0. |
But, q(0)>0 and ∫α0g(s)ds<0, and hence there is a contradiction. This proves R=∞.
Step 2. (lim infr→∞u(r)>0 and lim supr→∞u(r)<η)
Since u>0, then lim infr→∞u(r)≥0. Assume that lim infr→∞u(r)=0, then there exists a sequence (rn) of positive numbers such that
limn→∞rn=∞andlimn→∞u(rn)=0. | (2.6) |
Applying (2.2) for a=0 and b=rn, we get
(u′(rn))22+∫rn0N−1r(u′(r))2dr−∫rn0q′(r)(∫u(r)0g(s)ds)dr+q(rn)∫u(rn)0g(s)ds−q(0)∫α0g(s)ds=0. |
The first three terms of this equation are nonnegative, so
q(rn)∫u(rn)0g(s)ds≤q(0)∫α0g(s)ds, | (2.7) |
and using (2.6), we get that
limn→∞q(rn)=q∞>0andlimn→∞∫u(rn)0g(s)ds=0, |
therefore, by taking the limit n→∞ in (2.7), we finally obtain
q(0)∫α0g(s)ds≥0. |
This is a contradiction since q(0)>0 and ∫α0g(s)ds<0. The proof that lim supr→∞u(r)<η is done in a similar manner.
Step 3. (u is an oscillating function)
Let us show that u oscillates on [0,∞[. First, note that
u″(0)=−q(0)g(α)<0, |
and then, by the regularity of u, there exists δ>0 such that u decreases on ]0,δ[. Let
r1=sup{δ>0:u is decreasing on ]0,δ[}, |
then r1<∞. Suppose this is not true, i.e., r1=∞, then u decreases to a limit 0<ℓ≤α. We observe that ℓ>0 since lim infr→∞u(r)>0 by step 2. This is an essential observation to ensure that g(ℓ)≠0 if ℓ≠ξ. Two cases can be considered:
● Case ℓ≠ξ. Without loss of generality, we assume ℓ>ξ. Applying the mean value theorem between n∈N and n+1 we get u(n+1)−u(n)=u′(bn), n<bn<n+1, and hence the existence of a sequence (bn) such that
limn→∞bn=∞andlimn→∞u′(bn)=0. |
Applying inequality (2.4) between 1 and bn, we get
u′(bn)−u′(1)+∫bn1N−1ru′(r)dr+∫bn1q(r)g(u(r))dr=0. | (2.8) |
Straightforward computations give
0≥∫bn1N−1ru′(r)dr≥(N−1)(u(bn)−u(1))≥(N−1)(ℓ−u(1)), |
and so, as limn→∞u′(bn)=0, the first three terms of (2.8) are bounded. On the other hand, g(u(r))>0 since ξ<u(r)≤α and then q(r)g(u(r))>0 with limr→∞q(r)g(u(r))=q∞g(ℓ)>0. Consequently,
limn→∞∫bn1q(r)g(u(r))dr=∞, |
which is in contradiction with (2.8). The case ℓ<ξ is treated similarly, possibly with the application of (2.4) with a large enough to ensure a negative sign of the term q(r)g(u(r)) that converges, as r→∞, to q∞g(ℓ)<0, leading to the same kind of contradiction as above.
● Case ℓ=ξ. Since limr→∞u(r)=ξ, then, using (1.10),
limr→∞q(r)g(u(r))u(r)−ξ=q∞g′(ξ)>0. |
Hence, for r large enough, say r>R0, we have
q(r)g(u(r))u(r)−ξ−(N−1)(N−3)4r2>ϵ2>0, | (2.9) |
for some ϵ>0. Therefore, by the Sturm comparison principle applied to ODE (2.5), we deduce that v must vanish infinitely many times in (R0,+∞), which leads to a contradiction.
Another approach to see this contradiction is as follows. Since v is a solution of (2.5), then we deduce from (2.9) that v″<0 for r>R0, which implies that v′(r)↘L∈[−∞,+∞[ as r→∞. If L<0, then v(r)→−∞ and this is impossible by the positivity of v. Otherwise, if L≥0, then v′>0 and v increases on [R0,∞[, and thus v(r)≥v(R0)>0 for r≥R0. Again, by (2.9) we get v″(r)≤−ϵ2v(R0)<0, and consequently v′(r)→−∞ as r→∞, and this is also impossible by the positivity of v′.
The oscillation. From all that precedes, we deduce that r1<∞, u′(r1)=0 and u is increasing on ]r1,r1+δ1[ for some δ1>0. This, together with the equation u″(r1)=−q(r1)g(u(r1)) and the fact that q>0 and g(r)>0 for ξ<r≤α, show that u(r1)≤ξ. However, if u(r1)=ξ, then, by the uniqueness of the ODE, we get u≡ξ, which leads to a contradiction. Finally,
u(r1)<ξ. |
By essentially repeating the same arguments of this step, we are lead to the existence of r2∈]r1,∞[ such that u is increasing on ]r1,r2[, u′(r2)=0 and u is decreasing on ]r2,r2+δ2[ for some δ2>0. Here, it is very important to remark that a part of the method of showing r2≠∞ will essentially depend on the fact that lim supr→∞u(r)<η, as proved in step 2.
Again, incidentally,
u(r2)>ξ. |
We redo the same analysis to conclude that u has infinitely many local maxima and local minima. More precisely, there exists a sequence (rn)n≥1 such that
r1<r2<⋯<rk<⋯→∞, |
u(r2k),k≥1, are local maxima with u(r2k)>ξ, |
and
u(r2k−1),k≥1, are local minima with u(r2k−1)<ξ. |
For the simplicity of notation we set
ui:=u(ri)fori∈N. |
Step 4. ({u2k−1}k≥1 is increasing and {u2k}k≥1 is decreasing)
We only show that the sequence {u2k−1}k≥1 is increasing. To show that {u2k}k≥1 is decreasing we follow the exact same arguments. First note that, since
u2k−1<ξ<u2k | (2.10) |
and u is increasing on ]r2k−1,r2k[, then there exists a point ˉr∈]r2k−1,r2k[ such that
u(ˉr)=ξ,u≤ξ on ]r2k−1,ˉr[ and u≥ξ on ]ˉr,r2k[. | (2.11) |
Owing to (2.10) and the regularity of u, we infer that u′≠0 on some nonempty open subinterval of ]r2k−1,r2k[. Therefore, using (2.3) with a=r2k−1 and b=r2k, we get
∫r2kr2k−1q(r)g(u(r))u′(r)dr<0, |
and so
∫ˉrr2k−1q(r)g(u(r))u′(r)dr+∫r2kˉrq(r)g(u(r))u′(r)dr<0. | (2.12) |
Now, using (1.8), (1.9), (2.11), the non-negativity of u′, and the monotonicity of q in (2.12), we obtain
q(ˉr)∫ˉrr2k−1g(u(r))u′(r)dr+q(ˉr)∫r2kˉrg(u(r))u′(r)dr<0, |
and thus
q(ˉr)∫r2kr2k−1g(u(r))u′(r)dr<0. |
But, q>0, and therefore
∫u2ku2k−1g(s)ds<0. | (2.13) |
We reuse (2.3) with a=r2k and b=r2k+1 to get
∫r2k+1r2kq(r)g(u(r))u′(r)dr<0. |
Following a similar approach, we also note that
u2k+1<ξ<u2k |
leading to the existence of r_∈]r2k,r2k+1[ with u(r_)=ξ, and thanks here to the non-positivity of u′, the monotonicity of q, and the sign of g(u) on ]r2k,r2k+1[,
q(r_)∫r_r2kg(u(r))u′(r)dr+q(r_)∫r2k+1r_g(u(r))u′(r)dr<0. |
Consequently,
∫u2k+1u2kg(s)ds<0. | (2.14) |
Combining (2.13) and (2.14), we deduce that
∫u2k+1u2k−1g(s)ds<0. |
Finally, as u2k−1,u2k+1∈]0,ξ[ and since g<0 on ]0,ξ[, the previous inequality asserts that
u2k−1<u2k+1, |
and therefore the sequence {u2k−1}k≥1 is increasing. Having u2k−1≤ξ for all k, we also deduce that
limk→∞u2k−1=γ≤ξ. |
Similarly, {u2k}k≥1 is decreasing; u2k≥ξ for all k, and therefore
limk→∞u2k=β≥ξ. |
A particular case. Assume that ξ=η2 and
g(s)=s(s−ξ)(η−s), |
then g is antisymmetric with respect to s=ξ. In such a situation we may show the monotonicity of {u2k−1}k≥1 and {u2k}k≥1 by a different approach. We only give an idea of the proof by showing
u1<u3. | (2.15) |
We first show that u1<η−u2. Assume to the contrary that u1≥η−u2 (see Figure 1).
By applying (2.2) with a=r1 and b=r2, we get
q(r2)∫u20g(s)ds−q(r1)∫u10g(s)ds<∫r2r1∫u(r)0q′(r)g(s)dsdr, |
and as g is antisymmetric with respect to s=ξ, then
∫r2r1∫u(r)0q′(r)g(s)dsdr=∬ |
and therefore
\begin{equation} q(r_{2})\int_{0}^{u_{2}}g(s)ds - q(r_{1})\int_{0}^{u_{1}}g(s)ds < \iint_{\mathcal{R}}q'(r)g(s)dsdr, \end{equation} | (2.16) |
where \mathcal{R} is the shaded area in Figure 1. Notice that, since u_{1} \geq \eta - u_{2} and q'(r)g(s) \leq 0 on \mathcal{R} , then
\iint_{\mathcal{R}}q'(r)g(s)dsdr \leq \int_{r_{1}}^{r_{2}}\int_{0}^{\eta - u_{2}}q'(r)g(s)dsdr = q(r_{2})\int_{0}^{\eta - u_{2}}g(s)ds - q(r_{1})\int_{0}^{\eta - u_{2}}g(s)ds. |
Using this inequality in (2.16), we finally get
q(r_{2})\int_{0}^{u_{2}}g(s)ds - q(r_{1})\int_{0}^{u_{1}}g(s)ds < q(r_{2})\int_{0}^{\eta - u_{2}}g(s)ds - q(r_{1})\int_{0}^{\eta - u_{2}}g(s)ds. |
Again, the antisymmetry of g implies
\int_{0}^{\eta - u_{2}}g(s)ds = \int_{0}^{u_{2}}g(s)ds, |
and then
q(r_{1})\int_{0}^{u_{1}}g(s)ds > q(r_{1})\int_{0}^{u_{2}}g(s)ds, |
hence
\int_{u_{1}}^{u_{2}} g(s)ds < 0, |
which is in contradiction with the fact that u_{1} \geq \eta - u_{2} and the definition of g . Consequently,
\begin{equation} u_{1} < \eta - u_{2}. \end{equation} | (2.17) |
We now show that u_{1} < u_{3} . Assume to the contrary that u_{1}\geq u_{3} . Using this inequality together with (2.17), we draw Figure 2 below.
By applying (2.2) with a = r_{1} and b = r_{3} , we get
q(r_{3})\int_{0}^{u_{3}}g(s)ds - q(r_{1})\int_{0}^{u_{1}}g(s)ds < \int_{r_{1}}^{r_{3}}\int_{0}^{u(r)}q'(r)g(s)dsdr. |
Similar arguments as above yield
q(r_{3})\int_{0}^{u_{3}}g(s)ds - q(r_{1})\int_{0}^{u_{1}}g(s)ds < \iint_{\mathcal{R}}q'(r)g(s)dsdr, |
where \mathcal{R} is the shaded area in Figure 2, and then
q(r_{3})\int_{0}^{u_{3}}g(s)ds - q(r_{1})\int_{0}^{u_{1}}g(s)ds < \int_{r_{1}}^{r_{3}}\int_{0}^{u_{3}}q'(r)g(s)dsdr = q(r_{3})\int_{0}^{u_{3}}g(s)ds - q(r_{1})\int_{0}^{u_{3}}g(s)ds. |
As a result, we obtain
\int_{u_{3}}^{u_{1}}g(s)ds > 0, |
which leads to a contradiction, and inequality (2.15) is therefore valid.
Step 5. ( \gamma = \beta = \xi )
We know that \underset{r\geq 0}{\sup}\ u(r) = {\alpha} < \eta and \underset{r\geq 0}{\inf}\ u(r) = u(r_1) > 0 . Since g'(\xi) > 0 , and by the boundedness of q , we deduce that there exist c_1, c_2 > 0 such that for every r\geq 0 we have
c_1 < \frac{q(r)g(u(r))}{u(r)-\xi} < c_2. |
Therefore, for r large enough (say r\geq R_1 ), we deduce that there exist \epsilon_1, \epsilon_2 > 0 such that
\epsilon_1^2 < \frac{q(r)g(u(r))}{u(r)-\xi}-\frac{(N-1)(N-3)}{4r^2} < \epsilon_2^2. |
Recall that v(r) = r^{\frac{N-1}{2}}(u(r)-\xi) solves (2.5). Then, using the Sturm comparison theorem, we deduce that for r\geq R_1 we have
\frac{\pi}{\epsilon_2} < \text{ distance between two consecutive zeros of }u(r)-\xi < \frac{\pi}{\epsilon_1}. |
Consequently, there exists c > 0 such that
\underset{k\geq 0}{\sup}\ (r_k-r_{k-1})\leq c. |
Then, applying Schwarz's inequality, we get
\beta -\gamma < |u(r_k)-u(r_{k-1})| < c^{{}^{1}\!\!\diagup\!\!{}_{2}\;}\left(\int_{r_{k-1}}^{r_k}|u'(r)|^2dr\right)^{{}^{1}\!\!\diagup\!\!{}_{2}\;}. |
Therefore, for k large enough (say k\geq k_0 ), we have
\begin{align*} \int_{r_{k-1}}^{r_k}\frac{(u'(r))^2}{r}dr\geq\frac{1}{r_k}\frac{(\beta-\gamma)^2}{c}\geq \frac{c'}{r_{k-1}}\frac{(\beta-\gamma)^2}{c}\geq \frac{c'(\beta-\gamma)^2}{c^2}\int_{r_{k-1}}^{r_k}\frac{dr}{r}, \end{align*} |
where c' is a positive constant that depends only on k_0 . Summing over all k\geq k_0 , we get
\begin{equation} \int_{r_{k_0}}^{\infty}\frac{(u'(r))^2}{r}dr\geq \frac{c'(\beta-\gamma)^2}{c^2}\int_{r_{k_0}}^{\infty}\frac{dr}{r}. \end{equation} | (2.18) |
Moreover, by taking a = 0 and b = r_k\rightarrow \infty in (2.2), we note that
\int_{r_{k_0}}^{\infty}\frac{(u'(r))^2}{r}dr < \infty. |
Therefore, we deduce from (2.18) that \beta = \gamma . Moreover, since \gamma < \xi < \beta , we finally get \beta = \gamma = \xi .
Step 6. (conclusion)
Finally, we claim that the distance between two consecutive zeros of u(r)-\xi tends to \frac{\pi}{\sqrt{q_\infty g'(\xi)}} as r\rightarrow \infty . In fact, since u(r)\rightarrow \xi as r\rightarrow \infty , then
h(r) = \frac{q(r)g(r)}{u(r)-\xi}-\frac{(N-1)(N-3)}{4r^2}\underset{r\rightarrow \infty}{\longrightarrow}q_\infty g'(\xi). |
Therefore, for \epsilon > 0 , one can find R large enough such that for every r\geq R we have
q_\infty g'(\xi)-\epsilon < h(r) < q_\infty g'(\xi)+\epsilon. |
Therefore, applying the Sturm comparison theorem again on (2.5), we deduce that
\frac{\pi}{\sqrt{q_\infty g'(\xi)+\epsilon}} < \text{distance between two consecutive zeros of }u(r)-\xi < \frac{\pi}{\sqrt{q_\infty g'(\xi)-\epsilon}}. |
Taking the limit as \epsilon\rightarrow 0 we get the desired result.
To summarize, we were finally able to generalize the existence of an oscillating radial solution that converges to a root of f in the non-autonomous case despite the difficulties that rise from the presence of the terms related to q(r) in the energy Eq (2.2). Furthermore, inequality (2.3) allows us to prove the monotonicity of the local extrema. The question that arises now is whether we can generalize these results for f having a singularity at 0 ; more precisely, for f(r, u) = q(r)g(u) with g(u) = u^{-{\alpha}} for some {\alpha} < 1 .
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors declare no conflict of interest.
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