Nonlinear Differential Equations, Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 301–308 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

An elliptic problem with arbitrarily small positive solutions ^{∗}

Pierpaolo Omari & Fabio Zanolin

Dedicated to Alan Lazer on his 60th birthday

Abstract

We show that, for each λ > 0, the problem

−Δ_{p}u = λf(u) in Ω,
u = 0 on ∂Ω

has a sequence of positive solutions (u_{n})_{n} with maxΩ̄ u_{n} decreasing to

zero. We assume that lim inf_{s→0^{+}} \frac{F (s)}{s^{p}} = 0 and that lim sup_{s→0^{+}} \frac{F (s)}{s^{p}} = +∞,

where F′ = f. We stress that no condition on the sign of f is imposed.

1 Introduction

Let us consider the quasilinear elliptic problem

−Δ_{p}u = λ f(u) in Ω ,  (1.1)
u = 0 on ∂Ω ,

where Ω ⊂ ℝ^{N}) is a bounded domain, with a smooth boundary ∂Ω, Δ_{p}u = div(|∇u|^{p−2}∇u) is the p-Laplacian, with p > 1, f : [0, +∞[ → ℝ is a continuous function and λ > 0 is a real parameter.

Here, we are concerned with the existence and multiplicity of positive solutions of (1.1), where by a positive solution we mean a function u ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω), with u ≥ 0 and u ≢ 0 in Ω, such that

∫_{Ω} |∇u|^{p−2}∇u ∇w = λ ∫_{Ω} f(u) w ,

∗Mathematics Subject Classifications: 35J65, 34B15, 34C25, 47H15. Key words: Quasilinear elliptic equation, positive solution, upper and lower solutions, time-mapping estimates. \begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University. Published October 31, 2000. P.O. was supported by MURST research funds. F.Z. was supported by MURST research funds and EC grant CI1*-CT93-0323.

for every w ∈ W_{0}^{1,p}(Ω). Standard regularity results imply that u ∈ C^{1+σ}(Ω̄), for some σ > 0.

This problem has been investigated in a quite large number of papers, both in the case where p = 2 and in the case where p ≠ 2, often placing conditions on the behaviour of f(s)/s^{p−1} near 0 and near +∞ of the following types:

lim_{s→0^{+}} \frac{f(s)}{s^{p−1}} = +∞,  (1.2)
lim_{s→+∞} \frac{f(s)}{s^{p−1}} = 0,  (1.3)
lim_{s→0^{+}} \frac{f(s)}{s^{p−1}} = 0,  (1.4)
lim_{s→+∞} \frac{f(s)}{s^{p−1}} = +∞.  (1.5)

When p = 2, assumptions (1.2) and (1.3) are usually referred to as sublinearity conditions, whereas (1.4) and (1.5) as superlinearity conditions at 0 and at +∞, respectively. Just as a convention, we keep this terminology even when p ≠ 2. Note also that conditions (1.2) and (1.4) both imply, in particular,

f(0) ≥ 0.  (1.6)

The existence of (sometimes multiple) positive solutions was proved in the following cases:

• f is sublinear at 0 and at +∞;

• f is superlinear at 0 and at +∞ and has subcritical growth at +∞;

• f is sublinear at 0, superlinear at +∞, has subcritical growth at +∞ and there exists a positive strict upper solution;

• f is superlinear at 0, sublinear at +∞ and there exists a positive strict lower solution.

Classical references in this context are, for example, [1, 2, 3, 4, 7, 8, 9, 10, 11, 16, 17]. More recently, in [15] it was discussed the situation where f is eventually neither sublinear nor superlinear at +∞, in the sense that

lim inf_{s→+∞} \frac{f(s)}{s^{p−1}} = 0 and lim sup_{s→+∞} \frac{f(s)}{s^{p−1}} = +∞.  (1.7)

Yet, a counterexample given in [13] shows that, generally speaking, assumptions (1.6) and (1.7) are not sufficient to guarantee the existence of positive solutions of (1.1). Accordingly, in [15] condition (1.7) was strenghthened to

lim inf_{s→+∞} \frac{F (s)}{s^{p}} = 0 and lim sup_{s→+∞} \frac{F (s)}{s^{p}} = +∞,  (1.8)

where F : [0, +∞[→ ℝ is such that F′ = f. Then, it was proved that, under (1.6) and (1.8), problem (1.1) has, for each λ > 0, a sequence of (u_{n})_{n} of positive solutions, with maxΩ̄ u_{n} → +∞.

The aim of this paper is to show that the above considered conditions at +∞ can be replaced by similar ones at 0, in order to produce arbitrarily small positive solutions of (1.1). Namely, the following holds.

Theorem Assume

lim inf_{s→0^{+}} \frac{F (s)}{s^{p}} = 0 and lim sup_{s→0^{+}} \frac{F (s)}{s^{p}} = +∞.  (1.9)

Then, problem (1.1) has, for each λ > 0, a sequence (u_{n})_{n} of positive solutions, satisfying maxΩ̄ u_{n} ↘ 0 and \frac{1}{p} ∫_{Ω} |∇u_{n}|^{p} − λ ∫_{Ω} F (u_{n}) ↗ 0.

Remark 1 The assumptions

lim sup_{s→0^{+}} \frac{F (s)}{s^{p}} = +∞ and lim inf_{s→+∞} \frac{F (s)}{s^{p}} = 0

and, respectively,

lim inf_{s→0^{+}} \frac{F (s)}{s^{p}} = 0 and lim sup_{s→+∞} \frac{F (s)}{s^{p}} = +∞,

together with some other technical conditions, have been also considered in [14], [12], [6] and [5], for proving the existence of at least one positive radial solution of (1.1), in the case where Ω is an annular domain.

Remark 2 No condition on the sign of f is required in our result; yet, if f(s) ≥ 0 in a neighbourhood of 0, the strong maximum principle implies that every (small) positive solution u of (1.1) is actually strictly positive, i.e. u(x) > 0 in Ω and \frac{∂u}{∂ν}(x) < 0 on ∂Ω. The same conclusion still holds in the case where f changes sign near 0, provided that the nondecreasing regularization f̂^{−} of f^{−}, defined by f̂^{−}(s) = max_{t∈[0,s]} f^{−}(t), satisfies ∫_{0}^{1}(sf̂^{−}(s))^{−1/p}ds = +∞. To verify this, it is sufficient to observe that −Δ_{p}u ≥ −λf̂^{−}(u) in Ω and to apply Theorem 5 in [18].

Remark 3 It is quite easy to find continuous functions f : [0, +∞[→ ℝ which change sign in any neighbourhood of 0 and for which condition (1.9) is fulfilled. For instance, one can take f = F′, with

F (s) = s^{q} sin(s^{−γ}) + s^{r} cos(s^{−γ}) for s > 0 and F (0) = 0,

where q, r, γ satisfy q > p > r > 1 + γ and γ > 0. On the contrary, it seems less immediate to exhibit positive functions f, for which (1.9) holds. We produce here the example of a continuous (even nondecreasing) function f : [0, +∞[→ ℝ, with f(s) > 0 in ]0, +∞[, such that F satisfies (1.9). Let (s_{n})_{n}, (t_{n})_{n} and (δ_{n})_{n} be sequences defined by

s_{n} = 2^{− \frac{1}{2} n!}, t_{n} = 2^{−2n!} and δ_{n} = 2^{−(n!)^{2}}.

Observe that, for all large n,

s_{n+1} < t_{n} < s_{n} − δ_{n}.

Fix p > 1. Let f : [0, +∞[→ ℝ be a continuous nondecreasing function such that f(0) = 0, f(s) > 0 for s > 0 and, for all large n,

f(s) = 2^{−(p−1)n!} for s ∈ [s_{n+1}, s_{n} − δ_{n}].

Let us set F (s) = ∫_{0}^{s} f(t)dt for s ≥ 0. Then, it is not difficult to verify that

F (s_{n})/s_{n}^{p} ≤ (f(s_{n+1})s_{n} + f(s_{n})δ_{n}) /s_{n}^{p} → 0

and

F (t_{n})/t_{n}^{p} ≥ (f(s_{n+1})(t_{n} − s_{n+1})) /t_{n}^{p} → +∞,

as n → +∞. Since F (s) > 0 for s > 0, we can conclude that condition (1.9) holds.

Remark 4 It will be clear from the proof that condition (1.9) can be replaced by

−∞ < λ lim inf_{s→0^{+}} \frac{F (s)}{s^{p}} < μ_{∗} ≤ μ^{∗} < λ lim sup_{s→0^{+}} \frac{F (s)}{s^{p}},

where μ_{∗}, μ^{∗} are suitable positive constants, depending only on Ω and p.

Remark 5 Our result extends to equations involving a more general class of quasilinear operators of the type div A(x, ∇u), where A satisfies suitable ellipticity and growth conditions of Leray-Lions type, and nonlinearities f also depending on the x-variable. The existence of positive periodic solutions for some classes of quasilinear parabolic equations can be proved along the same lines too.

2 Proof

We will exploit some arguments similar to those introduced in [15], therefore only the main steps of the proof will be produced.

At first, we notice that condition (1.9) implies that F (0) = 0 and f(0) = 0. Hence, we have, in particular, that the function 0 is a (lower) solution of problem (1.1). It is also convenient for the sequel to extend f and F to the whole of ℝ, as an odd and as an even function, respectively. Throughout this proof, we further suppose that the coefficient λ > 0 is fixed.

Then, using the former condition in (1.9), we prove the existence of a sequence (β_{n})_{n} ⊂ C^{1}(Ω̄) of upper solutions of (1.1), satisfying

min_{Ω̄} β_{n} > 0 and max_{Ω̄} β_{n} → 0.  (2.1)

It is obvious that, if inf{s > 0 | f(s) ≤ 0} = 0, then there exists a sequence (β_{n})_{n} of constant upper solutions satisfying (2.1). Therefore, let us suppose that there is a number s_{0} > 0 such that

f(s) > 0 for s ∈]0, s_{0}]  (2.2)

and therefore

F (s) > 0 for s ∈]0, s_{0}].  (2.3)

By the former condition in (1.9), we can find a sequence (c_{n})_{n} ⊂]0, s_{0}[ such that c_{n} ↘ 0 and

\frac{F (c_{n})}{c_{n}^{p}} → 0.  (2.4)

Let ]a, b[ be the projection of Ω onto, say, the x_{1}-axis and consider, for each n, the initial-value problem

−(|v′|^{p−2}v′)′ = λf(v) in [a, b[,  (2.5)
v(a) = c_{n},
v′(a) = 0 .

By a local solution of (2.5) we mean a function v defined on some interval I ⊂ [a, b[, with a ∈ I, which is of class C^{1} in I, together with |v′|^{p−2}v′, and satisfies the equation in I and the initial conditions. It is known that (2.5) admits local solutions, which can be extended to a right maximal interval of existence [a, ω[⊂ [a, b[. Let v be a noncontinuable solution of (2.5) and define

σ = sup\{t ∈]a, ω[ | \frac{1}{2}c_{n} < v(s) < s_{0} in [a, t]\}.

We want to prove that σ = b. By (2.2), we immediately realize that |v′|^{p−2}v′ and, hence, v′ are decreasing in [a, σ[. Hence, we have v′(t) < 0 in ]a, σ[. Multiplying the equation in (2.5) by v′ and integrating between a and t, with t ∈]a, σ[, we obtain

\frac{p − 1}{p}|v′(t)|^{p} = λ(F (c_{n}) − F (v(t)))

and then, by (2.3),

−v′(t) ≤ (\frac{p}{p − 1})^{1/p} (λF (c_{n}))^{1/p}.  (2.6)

Now, assume, by contradiction, that σ < b and set v(σ) = lim_{t→σ^{−}} v(t) = \frac{1}{2}c_{n}. Integrating (2.6) between a and σ, we get

\frac{1}{2}c_{n} = ∫_{a}^{σ} −v′(t) ≤ (\frac{p}{p − 1})^{1/p} (λF (c_{n}))^{1/p} diam(Ω).

Dividing by c_{n} and passing to the limit, condition (2.4) yields a contradiction. Hence, we can conclude that there is a sequence (v_{n})_{n} of solutions of (2.5), defined on [a, b] and satisfying \frac{1}{2}c_{n} ≤ v_{n}(t) ≤ c_{n} in [a, b]. Therefore, setting, for each n,

β_{n}(x_{1}, …, x_{n}) = v_{n}(x_{1}) for (x_{1}, …, x_{n}) = x ∈ Ω̄,

we define a sequence (β_{n})_{n} ⊂ C^{1}(Ω̄) of upper solutions of problem (1.1), such that, for every n,

\frac{1}{2}c_{n} ≤ β_{n}(x) ≤ c_{n} in Ω̄.  (2.7)

Now, let us introduce the functional ϕ : W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω) → ℝ, defined by ϕ(u) = \frac{1}{p} ∫_{Ω} |∇u|^{p} − λ ∫_{Ω} F (u).

Let ζ ∈ C^{1}(Ω̄) be a function such that ζ(x) = 1 in some closed ball B ⊂ Ω, ζ(x) > 0 in Ω and ζ(x) = 0 on ∂Ω. By the former condition in (1.9), we can find a number s_{λ} > 0 such that F (s) ≥ \frac{−s^{p}}{λ} in [0, s_{λ}]. On the other hand, the latter condition in (1.9) yields the existence of a sequence (d_{n})_{n} ⊂]0, s_{λ}[, such that d_{n} ↘ 0 and \frac{F (d_{n})}{d_{n}^{p}} → +∞. Hence, we get

ϕ(d_{n}ζ) = \frac{1}{p}d_{n}^{p} ∫_{Ω} |∇ζ|^{p} − λ ∫_{Ω\B} F (d_{n}ζ) − λ ∫_{B} F (d_{n})
≤ d_{n}^{p} (\frac{1}{p} ∫_{Ω} |∇ζ|^{p} + ∫_{Ω} ζ^{p} − λ meas(B) \frac{F (d_{n})}{d_{n}^{p}}) < 0,

for all n large enough.

Now, we are in position of constructing a sequence (u_{n})_{n} of positive solutions of problem (1.1), with maxΩ̄ u_{n} → 0. Since 0 is a lower solution and β_{1} is an upper solution of (1.1), with minΩ̄ β_{1} > 0, there exists a solution u_{1} of (1.1), satisfying 0 ≤ u_{1} ≤ β_{1} in Ω and ϕ(u_{1}) = min{ϕ(u) | u ∈ W_{0}^{1,p}(Ω), 0 ≤ u ≤ β_{1}}. Since we can find a positive number, say d_{n_{1}}, such that d_{n_{1}}ζ ≤ minΩ̄ β_{1} in Ω and ϕ(d_{n_{1}}ζ) < 0, it follows that ϕ(u_{1}) < 0 and therefore u_{1} ≢ 0. Hence, u_{1} is a positive solution of (1.1), which, by (2.7), satisfies maxΩ̄ u_{1} ≤ c_{1}. Next, we pick an upper solution, say β_{2}, such that maxΩ̄ β_{2} < maxΩ̄ u_{1}. Proceeding as above, we find a solution u_{2} of (1.1) such that 0 ≤ u_{2} ≤ β_{2} in Ω and ϕ(u_{2}) < 0. Hence, u_{2} is a positive solution of (1.1), which satisfies maxΩ̄ u_{2} < maxΩ̄ u_{1} and, by (2.7), maxΩ̄ u_{2} ≤ c_{2}. Iterating this argument, we build a sequence (u_{n})_{n} of distinct positive solutions of (1.1) satisfying maxΩ̄ u_{n} ≤ c_{n} → 0. Thus, the proof is concluded.

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Pierpaolo Omari Dipartimento di Scienze Matematiche Università, Via A. Valerio 12/1 I–34127 Trieste, Italy e-mail: omari@univ.trieste.it

Fabio Zanolin Dipartimento di Matematica e Informatica Università, Via delle Scienze 206 I–33100 Udine, Italy e-mail : zanolin@dimi.uniud.it