Nonlinear Differential Equations, Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 1–12 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Explicit construction, uniqueness, and bifurcation curves of solutions for a nonlinear Dirichlet problem in a ball ^{∗}

Horacio Arango & Jorge Cossio

Dedicated to Alan Lazer on his 60th birthday

Abstract This paper presents a method for the explicit construction of radially

symmetric solutions to the semilinear elliptic problem

Δv + f(v) = 0 in B
v = 0 on ∂B ,

where B is a ball in ℝ^{N} and f is a continuous piecewise linear function. Our construction method is inspired on a result by E. Deumens and H. Warchall [8], and uses spline of Bessel’s functions. We prove uniqueness of solutions for this problem, with a given number of nodal regions and different sign at the origin. In addition, we give a bifurcation diagram when f is multiplied by a parameter.

1 Introduction

The purpose of this paper is to explicitly construct radially symmetric solutions v : B → ℝ to the nonlinear Dirichlet problem

\begin{matrix}Δv + f(v) = 0 in B \\ v = 0 on ∂B, \end{matrix}  (1.1)

where B is the ball in ℝ^{N} centered at the origin with radius π, Δ is the Laplacian operator, and f : ℝ → ℝ is a continous piecewise-linear function such that f(0) = 0, f has a positive zero, and f′(0) = f′(∞).

∗Mathematics Subject Classifications: 35B32, 35J60, 65D07, 65N99. Key words: Nonlinear Dirichlet problem, radially symmetric solutions, bifurcation, explicit solutions, spline. \begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University. Published Ocotber 24, 2000. Partially supported by Colciencias-BID grant 381-97. Part of this research was done while the second author was visiting The University of Texas at San Antonio

We construct solutions to (1.1) with a given number of zeros in their radial profiles. Our method provides an explicit calculation rather than the existence result presented in [3, 4, 5, 6, 9, 10, 11, 12]. Our constructions further develops the authors’ work in [2] and the paper by E. Deumens and H. Warchall [8].

Let λ_{1} < λ_{2} < ⋯ < λ_{k} < ⋯ be the eigenvalues of −Δ acting on radial functions of H_{0}^{1}(B) (see [1]) and {φ_{1}, φ_{2}, ⋯, φ_{k}, ⋯ } be the corresponding complete set of eigenfunctions.

Let λ_{j+1} > α^{2} > λ_{j}, β > 0, and

f(t) = \begin{cases}α^{2} t & if t ≤ \frac{β}{2} , \\ −α^{2} t + α^{2} β & if \frac{β}{2} ≤ t ≤ β, \\ α^{2} t − α^{2} β & if t ≥ β. \end{cases}  (1.2)

In Section 2, we shall construct radially symmetric solutions to (1.1) with the above nonlinear function f.

We recall that the radial solutions to (1.1) are the solutions to the ordinary differential equation

\begin{matrix}v′′ + \frac{N − 1}{r}v′ + f(v) = 0 (0 < r ≤ π) \\ v′(0) = 0 , v(π) = 0 . \end{matrix}  (1.3)

We give a method for finding the initial data v(0) corresponding to a radially symmetric solution with i nodes in (0, π), 0 ≤ i ≤ j − 1, (see Table 1).

We observe that Deumens and Warchall [8] studied a nonlinear wave equation in ℝ^{N+1}. Derrick et al [7] studied problem (1.1) in unbounded domains. It is worth remarking here that our construction is made in a bounded domain. Castro and Cossio [4] dealt with a type of nonlinearity similar to the nonlinearity found in (1.2). They use bifurcation theory to show the existence of solutions but they do not give a method for the explicit construction of solutions.

In Section 3, we prove uniqueness of the solution constructed in Section 2. More precisely, we show the following theorem.

Theorem 1.1 Let f be as in (1.2). For each 0 ≤ i ≤ j − 1 there exist unique solutions v_{i} and u_{i} to (1.1) with i nodes in (0, π) such that v_{i}(0) > β > 0 and 0 < u_{i}(0) < β.

In Section 4, we obtain a description of the graph of the set of radial solutions to

\begin{matrix}Δv + λ f(v) = 0 in B \\ v = 0 on ∂B, \end{matrix}  (1.4)

where λ ∈ ℝ is a parameter (see Figures 4 and 5). Figures 6, 7, and 8 were generated with software, written by the authors, following the method of construction given in Section 2.

Figure 1: Radial profile of a solution of (1.3) with 2 nodes

2 Explicit construction of radially symmetric solutions

In each r-interval where v(r) lies between −∞ and \frac{β}{2} , or between \frac{β}{2} and β, or between β and +∞, the equation (1.3) has the form

v′′ + \frac{N − 1}{r} v′ + K_{1} v + K_{2} = 0,  (2.1)

with K_{1} and K_{2} constants depending only on f′(0) = α^{2} and β. The solution to this equation is

N ≥ 2 : v(r) = Ar^{−ν}J_{ν}(kr) + Br^{−ν}N_{ν}(kr) − \frac{K_{2}}{K_{1}} ,

where k^{2} = K_{1}, ν = \frac{N−2}{2}, and J_{ν} and N_{ν} are the Bessel and Neumann functions (see [2]).

To build solutions to (1.3) we put together several of the above pieces, subject to continuity conditions for v and its first two derivatives, and subject to the boundary conditions v′(0) = 0 and v(π) = 0. For the sake of clarity and easy of manipulations, we henceforth deal with the three-dimensional case.

We discuss the construction of a solution v to problem (1.3) under assumption (1.2) with i nodes in (0, π) (0 ≤ i ≤ j − 1) and v(0) = d > β. The construction of a solution u with i nodes in (0, π) (0 ≤ i ≤ j − 1) and 0 < u(0) = d < β follows a similar pattern.

For 0 ≤ r ≤ p we take v(r) ≥ β. Thus f(v) = α^{2} v − α^{2} β, and the solution to (1.3) is

v_{1}(r) = β + \frac{p_{1}}{r} sin α (r − P_{1}).

For p ≤ r ≤ q, \frac{β}{2} ≤ v(r) ≤ β. Thus f(v) = −α^{2} v + α^{2} β, and the solution is

v_{2}(r) = β + \frac{p_{2}}{r} sinh α (r − P_{2}).

For q ≤ r ≤ π, 0 ≤ v(r) ≤ \frac{β}{2} . Thus f(v) = α^{2} v, and the solution is

v_{3}(r) = \frac{p_{3}}{r} sin α (r − P_{3}).

This ansatz specifies the solution in terms of 3 coefficients p_{1}, p_{2}, p_{3} and 2 welding points p and q, and 3 unknowns P_{1}, P_{2}, P_{3}. These 8 unknowns are to be found from the equations stating that v, v′, and v′′ are continuous at the 2 welding points and the boundary conditions.

The weld point p and the 3 unknowns P_{1}, P_{2}, and P_{3} are determined by the conditions v′_{1}(0) = 0, v_{1}(p) = v_{2}(p) = β, and v_{3}(π) = 0, and we find

p = \frac{π}{α}, P_{1} = 0, P_{2} = \frac{π}{α}, P_{3} = \frac{π}{α} (α − k),

where k ∈ ℤ − {0}.

Remark 1: Note that all solutions of (1.3) with v(0) > β satisfy v(\frac{π}{α}) = β.

Since v_{3}(r) has (k−1) nodes in (\frac{π}{α}(α−k), π), in order to construct a solution with i nodes in (0, π) to problem (1.3) we take k = i + 1. Let z = α q. Since v′_{2}(q) = v′_{3}(q) it follows that z must be a solution of the equation

g(z) := \frac{z}{tan(z − π α)} + \frac{z}{tanh(z − π)} − 2 = 0.  (2.2)

Equation (2.2) has a unique solution z over the interval (π (α − k), π (α − k + 1)) (see Figure 2), which can be found by using Newton’s method with initial condition z_{0} ∈ (π (α − k), π (α − k + 1)) and z_{0} ≃ π (α − k + 1). Using the solution z we get the weld point q = \frac{z}{α}.

The remaining continuity conditions yield

p_{1} = −p_{2} = \frac{β z}{2α sinh(z − π)}

and

p_{3} = \frac{β z}{2α sin(z − π(α − i − 1))}.

Since lim_{r→0^{+}} v_{1}(r) = d, it follows that

d = β + \frac{β z}{2 sinh(z − π)} (z > π).  (2.3)

Thus, we have constructed a solution with i nodes in (0, π) and initial condition d = v(0) > β.

Figure 2: Solutions to (2.2) with α = 4.9

Remark 2: For each positive integer m with 1 ≤ m ≤ j, let α_{m} = α − j + m. Since j < α < j + 1, it follows that

m < α_{m} < m + 1.

Therefore, using our method of construction we can obtain solutions with i nodes in (0, π) (0 ≤ i ≤ m − 1) to (1.3) with nonlinearity f given by (1.2) with α = α_{m}. Let us call d_{mi} the initial data corresponding to this solution, which can be found by using (2.3).

Let l be a positive integer less than or equal to i. Since

(π (α_{m} − (i + 1)), π (α_{m} − i)) = (π (α_{m} − l − (i − l + 1)), π (α_{m} − l − (i − l))),

we see that finding a solution of (2.3) on (π (α_{m} − (i + 1)), π (α_{m} − i)) it is equivalent to find a solution of (2.3) over the interval (π (α_{m} − l − (i − l + 1)), π (α_{m} − l − (i − l))). Therefore,

d_{mi} = d_{(m−l)(i−l)}, (1 ≤ m ≤ j, 0 ≤ i ≤ m − 1, 0 ≤ l ≤ i).

We summarize the above discussion in Table 2 which will be useful for constructing bifurcation diagrams in Section 4.

3 Proof of Theorem 1.1

In this section, we prove uniqueness for the solution to (1.3) with i nodes in (0, π) and initial data v(0) > β.

As we mentioned in Remark 1, solutions to (1.3) satisfy the equation v(p) = v(\frac{π}{α}) = β. Next we derive a basic lemma about the solutions of (1.3).

6 Construction of solutions for a nonlinear Dirichlet problem

α \ nodes 0 1 2 … m-1
1 < α_{1} < 2  d_{10}
2 < α_{2} < 3  d_{20}  d_{21} = d_{10}
3 < α_{3} < 4  d_{30}  d_{31} = d_{20}  d_{32} = d_{10}
⋮  ⋮  ⋮  ⋮
m < α_{m} < m + 1 d_{m0} d_{m1} = d_{m−1,0} d_{m2} = d_{m−2,0}  … d_{m,m1} = d_{10}

Table 1: Initial data v(0) = d corresponding to solutions of (1.3) Figure 3: Radial profile of a solution v(r) to problem (1.3)

Lemma 3.1 Let v_{1} and v_{2} be two solutions of (1.3) such that v_{1}(q) = v_{2}(q). Then

v_{1} = v_{2} on [p, q] .

Proof. Let

w(r) = v_{1}(r) − v_{2}(r), r ∈ [p, q].

Because v_{1} and v_{2} are solutions of (1.3), w satisfies

\begin{matrix}w′′ + \frac{2}{r}w′ + f(v_{1}) − f(v_{2}) = 0 p ≤ r ≤ q \\ w(p) = w(q) = 0 . \end{matrix}

Using the Mean Value Theorem, we see that there exists ξ such that

w′′ + \frac{2}{r} w′ + f′(ξ) w(r) = 0 r ∈ [p, q].  (3.1)

We multiply (3.1) by r^{2}. This yields

(r^{2} w′)′ + r^{2} f′(ξ) w = 0, r ∈ [p, q].

Now we multiply by w and integrate by parts over [p, q], we obtain

− ∫_{p}^{q} r^{2} (w′)^{2} + ∫_{p}^{q} r^{2} f′(ξ) w^{2} = 0 .  (3.2)

To prove the lemma we proceed by contradiction. Suppose w ≠ 0 on [p, q]. Since r ∈ (p, q) we know that v, ξ ∈ (\frac{β}{2} , β) so that f′(ξ) < 0 on [p, q], we see that

− ∫_{p}^{q} r^{2} (w′)^{2} + ∫_{p}^{q} r^{2} f′(ξ) w^{2} < 0 .  (3.3)

This contradicts (3.2). The contradiction shows that w ≡ 0 on [p, q]. The proof of the lemma follows.

Proof of Theorem 1.1. Let v_{1} and v_{2} be solutions to (1.3), with v_{1}(0) = d_{1} and v_{2}(0) = d_{2}. Since v_{1}(p) = v_{2}(p) = β, by uniqueness of the initial value problem for ordinary differential equations applied to (1.3) on [0, p], we see that

d_{1} ≠ d_{2} ⟹ v′_{1}(p) ≠ v′_{2}(p).

Using Lemma 3.1 we obtain

v′_{1}(p) ≠ v′_{2}(p) ⟹ v_{1}(q) ≠ v_{2}(q).

Finally, using again the uniqueness of the initial value problem for ordinary differential equations, we obtain

v_{1}(q) ≠ v_{2}(q) ⟹ v_{1}(π) ≠ v_{2}(π).

Therefore, if d_{1} ≠ d_{2} we infer that

v_{1}(π) ≠ v_{2}(π),

which is a contradiction because v_{1}(π) = 0 = v_{2}(π). Hence d_{1} = d_{2}. This proves uniqueness of solutions to (1.3). Thus, we have proved Theorem 1.1.

4 Construction of bifurcation curves and graphs of solutions

In this section we give a description of the graph of the set of radial solutions to

\begin{matrix}Δv + λ f(v) = 0 in B \\ v = 0 on ∂B, \end{matrix}  (4.1)

Figure 4: Bifurcation diagram for (4.1) with initial data d_{mi} > β

where λ ∈ ℝ^{+} is a parameter.

Let λ ∈ ℝ^{+}, m ∈ ℕ be such that m < λ α < m + 1, and i = 0, 1, ⋯, m − 1. Now, as we have seen in Section 2, we can find a unique solution z = z(λ) to the equation

\frac{z}{tan(z − π (λ α))} + \frac{z}{tanh(z − π)} − 2 = 0, on (π (λ α − (i + 1)), π (λ α − i)).

With this solution and (2.3) we find the initial data d_{mi} > β corresponding to the solution with i nodes in (0, π). Since

d = β + \frac{β z}{2 sinh(z − π)} (z > π),

we see that d′(z) < 0 (z > π),

lim_{z→∞} d(z) = β, and

d′(λ) < 0.

The sequence {d_{mi}}_{i=m−1}^{0} = {d_{j0}}_{j=1}^{m} is decreasing. Thus, using Table 1 and the previous information, we obtain the following bifurcation diagram

Similarly, we can construct the bifurcation diagram for solutions with initial data 0 < d_{mi} < β (see Figure 5). In this case, since

d = β − \frac{β z}{2 sinh(z)} (z > 0),

Figure 5: Bifurcation diagram for (4.1) with initial data 0 < d_{mi} < β

we see that d′(z) > 0 (z > 0),

lim_{z→∞} d(z) = β, and

d′(λ) > 0 .

The sequence {d_{mi}}_{i=m−1}^{0} = {d_{j0}}_{j=1}^{m} is increasing.

Figures 6-8 of radially symmetric solutions to problem (1.1) were generated with software, written by the authors, following the method of construction given in Section 2.

Acknowledgment. The authors want to express their gratitude to Professor Alfonso Castro for his comments about Theorem 1.1.

References

[1] R. Adams, Sobolev Spaces, New York, Academic Press (1975).

[2] H. Arango and J. Cossio, Construcción de soluciones radialmente simétricas para un problema elíptico semilineal, Rev. Colombiana Mat., Vol. 30 (1996), pp. 77–92.

[3] A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem, SIAM J. Math. Anal., Vol. 25 (1994), No. 6 , pp. 1554–1561.

[4] A. Castro and J. Cossio, Multiple radial solutions for a semilinear Dirichlet problem in a ball, Rev. Colombiana Mat. Vol. XXVII (1993), pp. 15–24.

10 Construction of solutions for a nonlinear Dirichlet problem Figure 6: Radial solution in three dimensions with α = 5.1, β = 2.0, and i = 4

Figure 7: Radial profile of the solution with α = 8.9, β = 3.0, and i = 7 Figure 8: Radial profile of the solution with α = 40.3, β = 3.0, and i = 25

[5] A. Castro and S. Gadam, The Lazer-Mckenna conjecture for radial solutions in the ℝ^{N} ball, Electronic Journal of Differential Equations, Vol 1993 (1993), No. 07, pp. 1–6

[6] D. Costa and D. G. De Figueiredo, Radial solutions for a Dirichlet problem in a ball, J. Differential Equations, Vol. 60 (1985), pp. 80–89.

[7] S. Chen, J. Cima, and W. Derrick, Positive and oscilatory radial solutions of semilinear elliptic equations, to appear in J. Applied Math. Stochastic Analysis.

[8] E. Deumens and H. Warchall, Explicit construction of all spherically symmetric solitary waves for a nonlinear wave equation in multiple dimensions, Nonlinear Analysis, Theory, Methods and Applications, Vol. 12 (1988), No. 4, pp. 419–447.

[9] M. Esteban, Multiple solutions of semilinear elliptic problems in a ball, J. Differential Equations, Vol. 57 (1985), pp. 112–137.

[10] M. Grillakis Existence of nodal solutions of semilinear equations in ℝ^{N}, J. Differential Equations, Vol. 85 (1990), pp. 367–400.

[11] S. Kichenassamy and J. Smoller, On the existence of radial solutions of quasi-linear elliptic equations, Nonlinearity, Vol. 3 (1990), pp. 677–694.

[12] P. L. Lions, On the existence of positive solutions in semilinear elliptic equations, SIAM Review, Vol. 24 (1982), pp. 441-467.

Horacio Arango (e-mail: harango@perseus.unalmed.edu.co) Jorge Cossio (e-mail: jcossio@perseus.unalmed.edu.co) Departamento de Matemáticas Universidad Nacional de Colombia Apartado Aéreo 3840 Medellín, Colombia