2006 International Conference in Honor of Jacqueline Fleckinger. Electronic Journal of Differential Equations, Conference 16, 2007, pp. 29–34. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

ON POSITIVE SOLUTIONS FOR A CLASS OF STRONGLY COUPLED P-LAPLACIAN SYSTEMS

JAFFAR ALI, R. SHIVAJI

Dedicated to Jacqueline Fleckinger on the occasion of an international conference in her honor

Abstract. Consider the system −Δ_{p}u = λf(u, v) in Ω

−Δ_{q}v = λg(u, v) in Ω
u = 0 = v on ∂Ω

where Δ_{s}z = div(|∇z|^{s−2}∇z), s > 1, λ is a non-negative parameter, and Ω is a bounded domain in ℝ with smooth boundary ∂Ω. We discuss the existence of a large positive solution for λ large when

lim_{x→∞} \frac{f(x, M[g(x, x)]^{1/q−1})}{x^{p−1}} = 0

for every M > 0, and lim_{x→∞} g(x, x)/x^{q−1} = 0. In particular, we do not assume any sign conditions on f(0, 0) or g(0, 0). We also discuss a multiplicity results when f(0, 0) = 0 = g(0, 0).

1. Introduction

Consider the boundary-value problem

−Δ_{p}u = λf(u, v) in Ω
−Δ_{q}v = λg(u, v) in Ω
u = 0 = v on ∂Ω

(1.1)

where Δ_{s}z = div(|∇z|^{s−2}∇z), s > 1, λ is a non-negative parameter, and Ω is a bounded domain in ℝ with smooth boundary ∂Ω.

We are interested in the study of positive solutions to (1.1) when no conditions on f(0, 0), g(0, 0) are assumed, in particular, they could be negative (semipositone systems). Semipositive problems are mathematically challenging area in the study of positive solutions (see [2] and [5]). For a review on semipositone problems, see [3]. In this paper we make the following assumptions:

2000 Mathematics Subject Classification. 35J55, 35J70. Key words and phrases. Positive solutions; p-Laplacian systems; semipositone problems. \begin{matrix}c \\ ◯ \end{matrix}2007 Texas State University - San Marcos. Published May 15, 2007.

(H1) f, g ∈ C^{1}((0, ∞) × (0, ∞)) ∩ C([0, ∞) × [0, ∞)) be monotone functions such that f_{u}, f_{v}, g_{u}, g_{v} ≥ 0 and lim_{u,v→∞} f(u, v) = lim_{u,v→∞} g(u, v) = ∞.

(H2) lim_{x→∞} \frac{f(x, M[g(x, x)]^{1/q−1})}{x^{p−1}} = 0 for every M > 0.

(H3) lim_{x→∞} \frac{g(x, x)}{x^{q−1}} = 0.

We establish the following existence and multiplicity results:

Theorem 1.1. Let (H1)–(H3) hold. Then there exists a positive number λ^{∗} such that (1.1) has a large positive solution (u, v) for λ > λ^{∗}.

Theorem 1.2. Let (H1)–(H3) hold. Further let F (s) = f(s, cs) and G(s) = g(c̃s, s) for any c, c̃ > 0 and assume that f and g be sufficiently smooth functions in the neighborhood of zero with F (0) = G(0) = 0, F^{(k)}(0) = 0 = G^{(l)}(0) for k = 1, 2, … [p − 1], l = 1, 2, … [q − 1] where [s] denotes the integer part of s. Then (1.1) has at least two positive solutions provided λ is large.

This paper extends the recent work in [1], where the authors study such systems with weaker coupling, namely systems of the form,

−Δ_{p}u = λ_{1}α(v) + μ_{1}δ(u) in Ω
−Δ_{q}v = λ_{2}β(u) + μ_{2}γ(v) in Ω
u = 0 = v on ∂Ω

(1.2)

where λ_{1}, λ_{2}, μ_{1} and μ_{2} are non-negative parameters, with the following conditions:

(C1) α, β, δ, γ ∈ C^{1}(0, ∞) ∩ C[0, ∞) be monotone functions such that

lim_{x→∞} α(x) = lim_{x→∞} β(x) = lim_{x→∞} δ(x) = lim_{x→∞} γ(x) = ∞.

(C2) lim_{x→∞} \frac{α(M[β(x)]^{1/q−1})}{x^{p−1}} = 0 for every M > 0.

(C3) lim_{x→∞} \frac{δ(x)}{x^{p−1}} = lim_{x→∞} \frac{γ(x)}{x^{q−1}} = 0.

In [1], authors establish an existence result for the system (1.2) when λ_{1} +μ_{1} and λ_{2} + μ_{2} are large. In addition, for the case when f(0) = h(0) = g(0) = γ(0) = 0, authors discuss a multiplicity result for λ_{1} + μ_{1} and λ_{2} + μ_{2} large. Here we extend this study to classes of systems with much stronger coupling. Our approach is based on the method of sub-and supersolutions (see e.g. [4]). In Section 2, we will prove Theorem 1.1, in Section 3, we will prove Theorem 1.2 and in Section 4, we discuss some examples with strong coupling.

2. Proof of Theorem 1.1

We extend f(u, v) and g(u, v) for all (u, v) ∈ ℝ^{2} smoothly such that there exists a constant k_{0} > 0 such that f(u, v), g(u, v) ≥ −k_{0} for all (u, v) ∈ ℝ^{2}. We shall establish Theorem 1.1 by constructing a positive weak subsolution (ψ_{1}, ψ_{2}) ∈ W^{1,p}(Ω) ∩ C(\overline{Ω}) × W^{1,q}(Ω) ∩ C(\overline{Ω}) and a supersolution (z_{1}, z_{2}) ∈ W^{1,p}(Ω) ∩ C(\overline{Ω}) × W^{1,q}(Ω) ∩ C(\overline{Ω}) of (1.1) such that ψ_{i} ≤ z_{i} for i = 1, 2. That is, ψ_{i}, z_{i} satisfies (ψ_{1}, ψ_{2}) = (0, 0) = (z_{1}, z_{2}) on ∂Ω,

∫_{Ω} |∇ψ_{1}|^{p−2}∇ψ_{1} · ∇ξ dx ≤ λ ∫_{Ω} f(ψ_{1}, ψ_{2})ξ dx,
∫_{Ω} |∇ψ_{2}|^{p−2}∇ψ_{2} · ∇ξ dx ≤ λ ∫_{Ω} g(ψ_{1}, ψ_{2})ξ dx,
∫_{Ω} |∇z_{1}|^{p−2}∇z_{1} · ∇ξ dx ≥ λ ∫_{Ω} f(z_{1}, z_{2})ξ dx,
∫_{Ω} |∇z_{2}|^{p−2}∇z_{2} · ∇ξ dx ≥ λ ∫_{Ω} g(z_{1}, z_{2})ξ dx

for all ξ ∈ W := {η ∈ C_{0}^{∞}(Ω) : η ≥ 0 in Ω}.

Let λ_{1}^{(r)} the first eigenvalue of −Δ_{r} with Dirichlet boundary conditions and ϕ_{r} the corresponding eigenfunction with ϕ_{r} > 0; Ω and ∥ϕ_{r}∥_{∞} = 1 for r = p, q. Let m, δ > 0 be such that |∇ϕ_{r}|^{r} − λ_{1}^{(r)}ϕ_{r}^{r} ≥ m on \overline{Ω}_{δ} = \{x ∈ Ω|d(x, ∂Ω) ≤ δ\} for r = p, q. (This is possible since |∇ϕ_{r}| ≠ 0 on ∂Ω while ϕ_{r} = 0 on ∂Ω for r = p, q). We shall verify that

(ψ_{1}, ψ_{2}) := ([\frac{λk_{0}}{m} ]^{1/p−1}(\frac{p − 1}{p})ϕ_{p}^{p/p−1}, [\frac{λk_{0}}{m} ]^{1/q−1}(\frac{q − 1}{q})ϕ_{q}^{q/q−1}),

is a subsolution of (1.1) for λ large. Let ξ ∈ W. Then

∫_{Ω} |∇ψ_{1}|^{p−2}∇ψ_{1} · ∇ξ dx = (\frac{λk_{0}}{m} ) ∫_{Ω} ϕ_{p}|∇ϕ_{p}|^{p−2}∇ϕ_{p} · ∇ξ dx

= (\frac{λk_{0}}{m} )\{ ∫_{Ω} |∇ϕ_{p}|^{p−2}∇ϕ_{p} · ∇(ϕ_{p}ξ) dx − ∫_{Ω} |∇ϕ_{p}|^{p}ξ dx\}

= (\frac{λk_{0}}{m} )\{ ∫_{Ω}[λ_{1}^{(p)}ϕ_{p}^{p} − |∇ϕ_{p}|^{p}]ξ dx\}.

Similarly

∫_{Ω} |∇ψ_{2}|^{q−2}∇ψ_{2} · ∇ξ dx = (\frac{λk_{0}}{m} )\{ ∫_{Ω}[λ_{1}^{(q)}ϕ_{q}^{q} − |∇ϕ_{q}|^{q}]ξ dx\}.

Now on \overline{Ω}_{δ} we have |∇ϕ_{r}|^{r} − λ_{1}^{(s)}ϕ_{r}^{r} ≥ m for r = p, q. Which implies that

\frac{k_{0}}{m} (λ_{1}^{(p)}ϕ_{p}^{p} − |∇ϕ_{p}|^{p}) − f(ψ_{1}, ψ_{2}) ≤ 0,

\frac{k_{0}}{m} (λ_{1}^{(q)}ϕ_{q}^{q} − |∇ϕ_{q}|^{q}) − g(ψ_{1}, ψ_{2}) ≤ 0.

Next on Ω − \overline{Ω}_{δ} we have ϕ_{p} ≥ μ, ϕ_{q} ≥ μ for some μ > 0, and therefore for λ large

f(ψ_{1}, ψ_{2}) ≥ \frac{k_{0}}{m} λ_{1}^{(p)} ≥ \frac{k_{0}}{m} λ_{1}^{(p)}ϕ_{p}^{p} − |∇ϕ_{p}|^{p},

g(ψ_{1}, ψ_{2}) ≥ \frac{k_{0}}{m} λ_{1}^{(q)} ≥ \frac{k_{0}}{m} λ_{1}^{(q)}ϕ_{q}^{q} − |∇ϕ_{q}|^{q}.

Hence

∫_{Ω} |∇ψ_{1}|^{p−2}∇ψ_{1} · ∇ξ dx ≤ λ ∫_{Ω} f(ψ_{1}, ψ_{2})ξ dx,
∫_{Ω} |∇ψ_{2}|^{q−2}∇ψ_{2} · ∇ξ dx ≤ λ ∫_{Ω} g(ψ_{1}, ψ_{2})ξ dx;

i.e., (ψ_{1}, ψ_{2}) is a subsolution of (1.1) for λ large.

Next let e_{r} be the solution of −Δ_{r}e_{r} = 1 in Ω, e_{r} = 0 on ∂Ω for r = p, q. Let (z_{1}, z_{2}) := (\frac{c}{μ_{p}} λ^{1/p−1}e_{p}, [g(cλ^{1/p−1}, cλ^{1/p−1})]^{1/q−1}λ^{1/q−1}e_{q}) where μ_{r} = ∥e_{r}∥_{∞}; r = p, q. Then

∫_{Ω} |∇z_{1}|^{p−2}∇z_{1} · ∇ξ dx = λ(\frac{c}{μ_{p}} )^{p−1} ∫_{Ω} |∇e_{p}|^{p−2}∇e_{p} · ∇ξ dx

= \frac{1}{(μ_{p})^{p−1}} (cλ^{1/p−1})^{p−1} ∫_{Ω} ξ dx.

By (H2) we can choose c large enough so that

\frac{1}{(μ_{p})^{p−1}} (cλ^{1/p−1})^{p−1} ∫_{Ω} ξ dx

≥ λ ∫_{Ω} f(cλ^{1/p−1}, [g(cλ^{1/p−1}, cλ^{1/p−1})]^{1/q−1}λ^{1/q−1}μ_{q})ξ dx

≥ λ ∫_{Ω} f(cλ^{1/p−1} \frac{e_{p}}{μ_{p}} , [g(cλ^{1/p−1}, cλ^{1/p−1})]^{1/q−1}λ^{1/q−1}e_{q})ξ dx

= λ ∫_{Ω} f(z_{1}, z_{2})ξ dx.

Next

∫_{Ω} |∇z_{2}|^{q−2}∇z_{2} · ∇ξ dx = λ[g(cλ^{1/p−1}, cλ^{1/p−1})] ∫_{Ω} |∇e_{q}|^{q−2}∇e_{q} · ∇ξ dx
= λ[g(cλ^{1/p−1}, cλ^{1/p−1})] ∫_{Ω} ξ dx

By (H3) choose c large so that \frac{1}{λ^{1/q−1}} μ_{q} ≥ \frac{[g(cλ^{1/p−1}, cλ^{1/p−1})]^{1/q−1}}{cλ^{1/p−1}} , then

λ[g(cλ^{1/p−1}, cλ^{1/p−1})] ∫_{Ω} ξ dx

≥ λ ∫_{Ω} g(cλ^{1/p−1}, [g(cλ^{1/p−1}, cλ^{1/p−1})]^{1/q−1}λ^{1/q−1}μ_{q})ξ dx

≥ λ ∫_{Ω} g(cλ^{1/p−1} \frac{e_{p}}{μ_{p}} , [g(cλ^{1/p−1}, cλ^{1/p−1})]^{1/q−1}λ^{1/q−1}e_{q})ξ dx

= λ ∫_{Ω} g(z_{1}, z_{2})ξ dx;

i.e., (z_{1}, z_{2}) is a supersolution of (1.1) with z_{i} ≥ ψ_{i} for c large, i = 1, 2. (Note |∇e_{r}| ≠ 0; ∂Ω for r = p, q).

Thus, there exists a solution (u, v) of (1.1) with ψ_{1} ≤ u ≤ z_{1}, ψ_{2} ≤ v ≤ z_{2}. This completes the proof of Theorem 1.1.

3. Proof of Theorem 1.2

To prove Theorem 1.2, we will construct a subsolution (ψ_{1}, ψ_{2}), a strict supersolution (ζ_{1}, ζ_{2}), a strict subsolution (w_{1}, w_{2}), and a supersolution (z_{1}, z_{2}) for (1.1) such that (ψ_{1}, ψ_{2}) ≤ (ζ_{1}, ζ_{2}) ≤ (z_{1}, z_{2}), (ψ_{1}, ψ_{2}) ≤ (w_{1}, w_{2}) ≤ (z_{1}, z_{2}), and (w_{1}, w_{2}) ≰ (ζ_{1}, ζ_{2}). Then (1.1) has at least three distinct solutions (u_{i}, v_{i}), i = 1, 2, 3, such that (u_{1}, v_{1}) ∈ [(ψ_{1}, ψ_{2}), (ζ_{1}, ζ_{2})], (u_{2}, v_{2}) ∈ [(w_{1}, w_{2}), (z_{1}, z_{2})], and

(u_{3}, v_{3}) ∈ [(ψ_{1}, ψ_{2}), (z_{1}, z_{2})] \ ([(ψ_{1}, ψ_{2}), (ζ_{1}, ζ_{2})] ∪ [(w_{1}, w_{2}), (z_{1}, z_{2})]).

We first note that (ψ_{1}, ψ_{2}) = (0, 0) is a solution (hence a subsolution). As in Section 2, we can always construct a large supersolution (z_{1}, z_{2}). We next consider

−Δ_{p}w_{1} = λf̃(w_{1}, w_{2}) in Ω
−Δ_{q}w_{2} = λg̃(w_{1}, w_{2}) in Ω
w_{1} = 0 = w_{2} on ∂Ω

(3.1)

where f̃(u, v) = f(u, v) − 1 and g̃(u, v) = g(u, v) − 1. Then by Theorem 1.1, (3.1) has a positive solution (w_{1}, w_{2}) when λ is large. Clearly this (w_{1}, w_{2}) is a strict subsolution of (1.1). Finally we construct the strict supersolution (ζ_{1}, ζ_{2}).

To do so, we let ϕ_{p}, ϕ_{q} as described in Section 2. We note that there exists positive constants c_{1} and c_{2} such that

ϕ_{p} ≤ c_{1}ϕ_{q} and ϕ_{q} ≤ c_{2}ϕ_{p}.  (3.2)

Let (ζ_{1}, ζ_{2}) = (ϵϕ_{p}, ϵϕ_{q}) where ϵ > 0. Let H_{p}(s) := λ_{1}^{(p)}s^{p−1} − λf(s, c_{2}s) and H_{q}(s) := λ_{1}^{(q)}s^{q−1} − λg(c_{1}s, s). Observe that H_{p}(0) = H_{q}(0) = 0, H_{p}^{(k)}(0) = 0 = H_{q}^{(l)}(0) for k = 1, 2, … [p − 2] and l = 1, 2, … [q − 2]. H_{p}^{(p−1)}(0) > 0 and H_{q}^{(q−1)}(0) > 0 if p, q are integers, while lim_{r→0} H^{([p])}(r) = +∞ = lim_{r→0} H^{([q])}(r) if p, q are not integers. Thus there exists θ such that H_{p}(s) > 0 and H_{q}(s) > 0 for s ∈ (0, θ]. Hence for 0 < ϵ ≤ θ we have

\begin{aligned}λ_{1}^{(p)}(ζ_{1})^{p−1} = λ_{1}^{(p)}(ϵϕ_{p})^{p−1} & > λf(ϵϕ_{p}, c_{2}ϵϕ_{p}) \\ & ≥ λf(ϵϕ_{p}, ϵϕ_{q}) \\ & = λf(ζ_{1}, ζ_{2}) x ∈ Ω, \end{aligned}  (3.3)

and similarly we get

\begin{aligned}λ_{1}^{(q)}(ζ_{2})^{q−1} = λ_{1}^{(q)}(ϵϕ_{q})^{q−1} & > λg(c_{1}ϵϕ_{q}, ϵϕ_{q}) \\ & ≥ λg(ϵϕ_{p}, ϵϕ_{q}) \\ & = λg(ζ_{1}, ζ_{2}), x ∈ Ω. \end{aligned}  (3.4)

Using the inequalities (3.3) and (3.4) we have,

∫_{Ω} |∇ζ_{1}|^{p−2}∇ζ_{1} · ∇ξ dx = ϵ^{p−1} ∫_{Ω} |∇ϕ_{p}|^{p−2}∇ϕ_{p} · ∇ξ
= ∫_{Ω} λ_{1}^{(p)}(ϵϕ_{p})^{p−1}ξ dx
> λ ∫_{Ω} f(ζ_{1}, ζ_{2})ξ dx.

Similarly we have

∫_{Ω} |∇ζ_{2}|^{q−2}∇ζ_{2} · ∇ξ dx > λ ∫_{Ω} g(ζ_{1}, ζ_{2})ξ dx

Thus (ζ_{1}, ζ_{2}) is a strict supersolution. Here we can choose ϵ small so that (w_{1}, w_{2}) ≰ (ζ_{1}, ζ_{2}).

Hence there exists solutions (u_{1}, v_{1}) ∈ [(ψ_{1}, ψ_{2}), (ζ_{1}, ζ_{2})], (u_{2}, v_{2}) ∈ [(w_{1}, w_{2}), (z_{1}, z_{2})], and (u_{3}, v_{3}) ∈ [(ψ_{1}, ψ_{2}), (z_{1}, z_{2})] \ ([(ψ_{1}, ψ_{2}), (ζ_{1}, ζ_{2})] ∪ [(w_{1}, w_{2}), (z_{1}, z_{2})]). Since (ψ_{1}, ψ_{2}) ≡ (0, 0) is a solution it may turn out that (u_{1}, v_{1}) ≡ (ψ_{1}, ψ_{2}) ≡ (0, 0). In any case we have two positive solutions (u_{2}, v_{2}) and (u_{3}, v_{3}). Hence Theorem 1.2 holds.

Remark 3.1. Note that in the construction of the supersolution (ζ_{1}, ζ_{2}) we require the conditions at zero on F and G only for the constants c = c_{2} and c̃ = c_{1}.

4. Examples

Example 4.1. Consider the problem

−Δ_{p}u = λ[v^{α} + (uv)^{β} − 1] in Ω
−Δ_{q}v = λ[u^{σ} + (uv)^{γ/2} − 1] in Ω
u = 0 = v on ∂Ω

(4.1)

where α, β, σ, γ are positive parameters. Then it is easy to see that (4.1) satisfies the hypotheses of Theorem 1.1 if max\{σ, γ\}\frac{α}{q−1} < p − 1, (max\{σ, γ\}\frac{1}{q−1} + 1)β < p − 1 and max{σ, γ} < q − 1.

Example 4.2. Let

h(x) = \begin{cases}x^{α}; & x ≤ 1 \\ \frac{α}{σ}x^{σ} + (1 − \frac{α}{σ}); & x > 1, \end{cases} and γ(x) = \begin{cases}x^{μ}; & x ≤ 1 \\ \frac{μ}{δ}x^{δ} + (1 − \frac{μ}{δ}); & x > 1, \end{cases}

where α, σ, μ, δ are positive parameters. Here we assume α > p−1 if p is an integer, α > [p] if p is not an integer, μ > q − 1 if q is an integer and μ > [q] if q is not an integer.

Consider the problem

−Δu = λ[1 + u^{β}]h(v) in Ω
−Δv = λγ(u) in Ω
u = 0 = v on ∂Ω

(4.2)

where 0 ≤ β < p − 1. Then it is easy to see that (4.2) satisfies the hypotheses of Theorem 1.2 if δσ < [p − 1 − β](q − 1) and δ < q − 1.

References

[1] Jaffar Ali, R. Shivaji. Positive solutions for a class of p-laplacian systems with multiple parameters. To appear in the Journal of Mathematical Analysis and Applications.

[2] H. Berestycki, L. A. Caffarelli and L. Nirenberg. Inequalities for second order elliptic equations with applications to unbounded domains. A Celebration of John F. Nash Jr., Duke Math. J. 81 (1996), 467-494.

[3] A. Castro, C. Maya and R. Shivaji. Nonlinear eigenvalue problems with semipositone structure. Electron. Jour. of Diff. Eqns, Conf 05 (2000), pp. 33-49.

[4] P. Drábek and J. Hernandez. Existence and uniqueness of positive solutions for some quasilinear elliptic problem. Nonlin. Anal. 44 (2001), 189-204.

[5] P. L Lions. On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 24 (1982), 441-467.

Department of Mathematics, Mississippi State University, Mississippi State, MS 39759, USA

E-mail address, Jaffar Ali: js415@ra.msstate.edu

E-mail address, R. Shivaji: shivaji@ra.msstate.edu