2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 41–47. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Strongly nonlinear elliptic problem without growth condition ^{∗}

Aomar Anane & Omar Chakrone

Abstract We study a boundary-value problem for the p-Laplacian with a nonlinear term. We assume only coercivity conditions on the potential and do not assume growth condition on the nonlinearity. The coercivity is obtained by using similar non-resonance conditions as those in [1].

1 Introduction

Consider the boundary-value problem

\begin{matrix}−Δ_{p}u = f(x, u) + h in Ω, \\ u = 0 on ∂Ω, \end{matrix}  (1.1)

where Ω is a bounded domain of ℝ^{N}, −Δ_{p}: W_{0}^{1,p}(Ω) → W^{−1,p′}(Ω) is the p-Laplacian operator defined by

Δ_{p}u ≡ div(|∇u|^{p−2}∇u), 1 < p < ∞.

The p-Laplacian is a degenerated quasilinear elliptic operator that reduces to the classical Laplacian when p = 2. The notation ⟨., .⟩ stands hereafter for the duality pairing between W^{−1,p′}(Ω) and W_{0}^{1,p}(Ω). While f: Ω × ℝ → ℝ is a Carathéodory function and h ∈ W^{−1,p′}(Ω).

Consider the energy functional Φ: W_{0}^{1,p}(Ω) → \overline{ℝ} associated with the problem

Φ(u) = \frac{1}{p} ∫_{Ω} |∇u|^{p} dx − ∫_{Ω} F (x, u) dx − ⟨h, u⟩,

where F (x, s) = ∫_{0}^{s} f(x, t) dt. We are interested in conditions to be imposed on the nonlinearity f in order that problem (1.1) admits at least one solution u(x) for any given h. Such conditions are usually called non-resonance conditions.

When the nonlinearity satisfies a growth condition of the type

|f(x, s)| ≤ a|s|^{q−1} + b(x) for all s ∈ ℝ, and a.e. in Ω,  (1.2)

∗Mathematics Subject Classifications: 49R50, 74G65, 35D05.

Key words: p-Laplacian, growth condition.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published December 28, 2002.

with q < p^{∗} where the Sobolev exponent p^{∗} = \frac{Np}{N−p} when p < N and p^{∗} = +∞ when p ≥ N and b(x) ∈ L^{(p^{∗})′}(Ω), the functional Φ is well defined and is of class C^{1}, l.s.c. and its critical points are weak solutions of (1.1) in the usual sense.

However, when this growth condition is not satisfied, Φ is not necessarily of class C^{1} on W_{0}^{1,p}(Ω) and may take infinite values. The first eigenvalue of the p-Laplacian characterized by the variational formulation

λ_{1} = λ_{1}(−Δ_{p}) = min \{\frac{∫_{Ω} |∇u|^{p}}{∫_{Ω} |u|^{p}} dx; u ∈ W_{0}^{1,p}(Ω) \ \{0\}\}

is known to be associated to a simple eigenfunction that does not change sign [4].

A procedure used to treat (1.1) when the nonlinearity lies asymptotically on the left of λ_{1} consists in supposing a “coercivity” condition on F of the type

lim sup_{s→±∞} \frac{pF (x, s)}{|s|^{p}} < λ_{1} for almost every x ∈ Ω  (1.3)

and minimizing Φ on W_{0}^{1,p}(Ω). The minimum being a weak solution of (1.1) in an appropriate sense [1, 2, 3]. Another way is to obtain a priori estimates on the solutions of some equations approximating (1.1) and to show that their weak limit is indeed a weak solution.

Note that with the help of the conditions (1.2) and (1.3), we know since the work of Hammerstein (1930) that (1.1) admits a weak solution that minimizes the functional Φ on W_{0}^{1,p}(Ω). The condition (1.3) does not imply a growth condition on f unless f(x, u) is convex in u (see for example [5]).

In [1], Anane and Gossez supposed only a one-sided growth condition with respect to the Sobolev (conjugate) exponent that do not suffice to guarantee the differentiability of Φ, which may even take infinite values. Nevertheless, they showed that any minimum of Φ solves (1.1) in a suitable sense.

Here, we assume 1 < p < ∞ and only that f maps L^{∞}(Ω) into L^{1}(Ω); i.e.,

sup_{|s|≤R} |f(., s)| ∈ L_{loc}^{1}(Ω), ∀R > 0  (1.4)

and a coercivity condition of the type (1.3). We prove that any minimum u of Φ, which is not of class C^{1} on W_{0}^{1,p}(Ω) and may take infinite values too, is a weak solution of (1.1) in the sense

∫_{Ω} |∇u|^{p−2}∇u∇v dx = ∫_{Ω} f(x, u)v dx + ⟨h, v⟩,

for v in a dense subspace of W_{0}^{1,p}(Ω). This result is proved by Degiovanni-Zani [2] in the case p = 2.

In the autonomous case f(x, s) = f(s), De Figueiredo and Gossez [6] have proved the existence of solutions for any h ∈ L^{∞}(Ω) by a topological method. They supposed only a coercivity condition and established that

∫_{Ω} |∇u|^{p−2}∇u∇v dx = ∫_{Ω} f(x, u)v dx + ⟨h, v⟩

for all v ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω) ∪ {u} but the solution obtained may not minimize Φ. Indeed, an example is given in [6] in the case p = 2 and an other one is given in [3] where p may be different from 2.

Note that in our case, the condition (1.4) implies no growth condition on f as it may be seen in the following example.

Example Consider the function

f(x, s) = \begin{cases}d(x)( sin(\frac{πs}{2}) − \frac{sign(s)}{2}) exp (\frac{2 cos(\frac{πs}{2})}{π} + \frac{|s|−1}{2}) & if |s| ≥ 1 \\ d(x)\frac{s}{2}(10s^{2} − 9) & if |s| ≤ 1 , \end{cases}

where d(x) ∈ L_{loc}^{1}(Ω) and d(x) ≥ 0 almost everywhere in Ω, so that

F (x, s) = \begin{cases}−d(x) exp (\frac{2 cos(\frac{πs}{2})}{π}) exp (\frac{|s|−1}{2}) & if |s| ≥ 1 \\ −d(x)\frac{s^{2}}{4} (−5s^{2} + 9) & if |s| ≤ 1 . \end{cases}

Then F (x, s) ≤ 0 for all s ∈ ℝ almost everywhere in Ω. So, Φ is coercive. Nevertheless, as we can check easily, f satisfies no growth condition.

2 Theoretical approach

We will show that when (1.4) is fulfilled, any minimum u of ϕ is a weak solution of (1.1) in an acceptable sense.

Definition The space L_{0}^{∞}(Ω) is defined by

L_{0}^{∞}(Ω) = {v ∈ L^{∞}(Ω); v(x) = 0 a.e. outside a compact subset of Ω}.

For u ∈ W_{0}^{1,p}(Ω), we set

V_{u} = {v ∈ W_{0}^{1,p}(Ω) ∩ L_{0}^{∞}(Ω); u ∈ L^{∞}({x ∈ Ω; v(x) ≠ 0})}.

Proposition 2.1 (Brezis-Browder [7]) If u ∈ W_{0}^{1,p}(Ω), there exists a sequence (u_{n})_{n} ⊂ W_{0}^{1,p}(Ω) such that:

(i) (u_{n})_{n} ⊂ W_{0}^{1,p}(Ω) ∩ L_{0}^{∞}(Ω).

(ii) |u_{n}(x)| ≤ |u(x)| and u_{n}(x).u(x) ≥ 0 a.e. in Ω.

(iii) u_{n} → u in W_{0}^{1,p}(Ω), as n → ∞.

The linear space V_{u} enjoys some nice properties.

Proposition 2.2 The space V_{u} is dense in W_{0}^{1,p}(Ω). And if we assume that (1.4) holds, then

A_{u} = {φ ∈ W_{0}^{1,p}(Ω); f(x, u)φ ∈ L^{1}(Ω)}

is a dense subspace of W_{0}^{1,p}(Ω) as V_{u} ⊂ A_{u}. More precisely, Brezis-Browder’s result holds true if we replace W_{0}^{1,p}(Ω) ∩ L_{0}^{∞}(Ω) by V_{u}.

Proof It suffices to show that V_{u} is dense in W_{0}^{1,p}(Ω) and that V_{u} ⊂ A_{u} when (1.4) holds. The density of V_{u} in W_{0}^{1,p}(Ω): We have to show that for any φ ∈ W_{0}^{1,p}(Ω), there exists a sequence (φ_{n})_{n} ⊂ V_{u} satisfying (ii) and (iii). This is done in two steps. First, we show it is true for all φ ∈ W_{0}^{1,p}(Ω) ∩ L_{0}^{∞}(Ω). Then, using Proposition 2.1, we show it is true in W_{0}^{1,p}(Ω). First Step: Suppose φ ∈ W_{0}^{1,p}(Ω) ∩ L_{0}^{∞}(Ω) and consider a sequence (Θ_{n})_{n} ⊂ C_{0}^{∞}(ℝ) such that:

(1) supp Θ_{n} ⊂ [−n, n],

(2) Θ_{n} ≡ 1 on [−n + 1, n − 1],

(3) 0 ≤ Θ_{n} ≤ 1 on ℝ and

(4) |Θ′_{n}(s)| ≤ 2.

The sequence we are looking for is obtained by setting

φ_{n}(x) = (Θ_{n} ∘ u)(x)φ(x) for a.e. x in Ω.

Indeed, let’s check the following three statements

(a) φ_{n} ∈ V_{u},

(b) |φ_{n}(x)| ≤ |φ(x)| and φ_{n}(x)φ(x) ≥ 0 a.e. in Ω and

(c) φ_{n} → φ in W_{0}^{1,p}(Ω). For (a), since φ ∈ L_{0}^{∞}(Ω), we have that φ_{n} ∈ L_{0}^{∞}(Ω) and it’s clear by (4) that φ_{n} ∈ W_{0}^{1,p}(Ω). Finally, by (1), u(x) ∈ [−n, n] for a.e. x in {x ∈ Ω; φ_{n}(x) ≠ 0}. The assumption (b) is a consequence of (3). For (c), by (2), φ_{n}(x) → φ(x) a.e. in Ω and

\frac{∂φ_{n}}{∂x_{i}} (x) = Θ′_{n}(u(x))\frac{∂u}{x_{i}}φ(x) + Θ_{n}(u(x))\frac{∂φ}{∂x_{i}} → \frac{∂φ}{∂x_{i}} in Ω.

And by (4),

|\frac{∂φ_{n}}{∂x_{i}} (x)| ≤ 2|\frac{∂u}{∂x_{i}} (x)||φ(x)| + |\frac{∂φ}{∂x_{i}} (x)| ∈ L^{p}(Ω).

Finally, by the dominated convergence theorem we get (c). Second Step: Suppose that φ ∈ W_{0}^{1,p}(Ω). By Proposition 2.1, there is a sequence (ψ_{n})_{n} ⊂ W_{0}^{1,p}(Ω) satisfying (i), (ii) and (iii).

For k = 1, 2, …, there is n_{k} ∈ ℕ such that ||ψ_{n_{k}} − φ||_{1,p} ≤ 1/k. Since ψ_{n_{k}} ∈ W_{0}^{1,p}(Ω)∩L_{0}^{∞}(Ω), by the first step, there is φ_{k} ∈ V_{u} such that |φ_{k}(x)| ≤ |ψ_{n_{k}}(x)| and φ_{k}(x)ψ_{n_{k}}(x) ≥ 0 almost everywhere in Ω and ||φ_{k} − ψ_{n_{k}}||_{1,p} ≤ 1/k, so that (φ_{k})_{k} is the sequence we are seeking. Indeed, |φ_{k}(x)| ≤ |ψ_{n_{k}}(x)| ≤ |φ(x)|, φ_{k}(x)φ(x) ≥ 0 a.e. in Ω and ||φ_{k}−φ(x)||_{1,p} ≤ ||φ_{k}−ψ_{n_{k}}||_{1,p}+||ψ_{n_{k}} −φ(x)||_{1,p} ≤ 2/k. The inclusion V_{u} ⊂ A_{u}: Indeed, for φ ∈ V_{u}, set E = {x ∈ Ω; φ(x) ≠ 0} so that

\begin{aligned}|f(x, u)φ| & = |f(x, u)χ_{E}φ(x)| \\ & ≤ max \{|f(x, s)φ(x)|; |s| ≤ ||u||_{L^{∞}(E)}\} \end{aligned}

where χ_{E} is the characteristic function of the set E. By (1.4), the last term lies to L^{1}(Ω), so that φ ∈ A_{u}. □

Theorem 2.3 Assume (1.4). If u ∈ W_{0}^{1,p}(Ω) is a minimum of Φ such that F (x, u) ∈ L^{1}(Ω), then

(i) ∫_{Ω} |∇u|^{p−2}∇u∇ϕ dx = ∫_{Ω} f(x, u)ϕ dx + ⟨h, ϕ⟩ for all ϕ ∈ A_{u}.

(ii) f(x, u) ∈ W^{−1,p′}(Ω) in the sense that the mapping T : V_{u} → ℝ : T (ϕ) = ∫_{Ω} f(x, u)ϕ dx is linear, continuous and admits an unique extension T̃ to the whole space W_{0}^{1,p}(Ω).

(iii) ⟨f(x, u), ϕ⟩ = ∫_{Ω} f(x, u)ϕ dx ∀ϕ ∈ A_{u}.

(iv) −Δ_{p}u = f(x, u) + h in W^{−1,p′}(Ω).

Remark There are in In [1] some conditions that guarantee the existence of a minimum u of Φ in W_{0}^{1,p}(Ω) and consequently F (x, u) ∈ L^{1}(Ω).

Proof of Theorem 2.3 We will prove that the assertion (i) holds for all ϕ ∈ V_{u} as a first step, then prove (iii), (iv) and (i). Let ϕ ∈ V_{u} and s ∈ ℝ such that 0 < s < 1. There exists β = β(x, s, ϕ, u) ∈ [−1, 1] such that

|\frac{F (x, u + sϕ) − F (x, u)}{s}| = |f(x, u + βϕ)ϕ|

≤ max {|f(x, t)ϕ(x)|; |t| ≤ ∥u∥_{L^{∞}(E)} + ∥ϕ∥_{L^{∞}(Ω)}},

where E = {x ∈ Ω; ϕ(x) ≠ 0 a.e. }. Since F (x, u) ∈ L^{1}(Ω), by (1.4), we have F (x, u + sϕ) ∈ L^{1}(Ω) for all 0 < s < 1. On the other hand

lim_{s→0} \frac{F (x, u(x) + sϕ(x)) − F (x, u(x))}{s} = f(x, u(x))ϕ a.e. in Ω.

It follows from Lebesgue’s dominated convergence that

lim_{s→0} \frac{F (x, u + sϕ) − F (x, u)}{s} = f(x, u)ϕ strongly in L^{1}(Ω).

Since u ∈ W_{0}^{1,p}(Ω) is a minimum point of Φ, we get

\frac{Φ(u + sϕ) − Φ(u)}{s} ≥ 0 for all 0 < s < 1,

then, we get (i) for all ϕ ∈ V_{u}.

The linear mapping defined by T (ϕ) = ∫_{Ω} f(x, u)ϕ is continuous, because for all ϕ ∈ V_{u},

|T (ϕ)| = | ∫_{Ω} |∇u|^{p−2}∇u∇ϕ − ⟨h, ϕ⟩| ≤ (∥u∥_{1,p}^{p/p′} + ∥h∥_{W^{−1,p′} (Ω)})∥ϕ∥_{1,p}.

By Proposition 2.2, T admits an unique extension T̃ to the whole space W_{0}^{1,p}(Ω). Henceforth, we will make the identification f(x, u) = T̃. Since

⟨−Δ_{p}u, ϕ⟩ = ⟨f(x, u), ϕ⟩ − ⟨h, ϕ⟩ ∀ϕ ∈ V_{u},

we conclude (iv). Let ϕ ∈ W_{0}^{1,p}(Ω) such that f(x, u)ϕ ∈ L^{1}(Ω), i.e. ϕ ∈ A_{u}. By Proposition 2.2 there exists (ϕ_{n}) ⊂ V_{u}. We can suppose that ϕ_{n} → ϕ almost everywhere, |f(x, u)ϕ_{n}| ≤ |f(x, u)ϕ| and f(x, u)ϕ_{n} → f(x, u)ϕ a.e.. By the dominated convergence theorem,

f(x, u)ϕ_{n} → f(x, u)ϕ in L^{1}(Ω).

Since ⟨f(x, u), ϕ_{n}⟩ = ∫_{Ω} f(x, u)ϕ_{n} for all n ∈ ℕ and f(x, u) ∈ W^{−1,p′}(Ω) we get (iii). Finally, (i) is an immediate consequence of (iii) and (iv). □

3 Description of the space A_{u}

Now, we will see some condition that guarantee some properties of A_{u}.

Proposition 3.1 Assume (1.4). Let u be a minimum of Φ in W_{0}^{1,p}(Ω) with F (x, u) ∈ L^{1}(Ω). And let ϕ ∈ W_{0}^{1,p}(Ω), v ∈ L^{1}(Ω) such that f(x, u(x))ϕ(x) ≥ v(x) or f(x, u(x))ϕ(x) ≤ v(x) a.e. in Ω, then ϕ ∈ A_{u}.

Proof Suppose f(x, u(x))ϕ(x) ≥ v(x) a.e. in Ω (the same argument works if f(x, u(x))ϕ(x) ≤ v(x) a.e. in Ω). By Proposition 2.2, there exists (ϕ_{n}) ⊂ V_{u} such that ϕ_{n} → ϕ in W_{0}^{1,p}(Ω), |ϕ_{n}| ≤ |ϕ| and ϕ_{n}(x)ϕ(x) ≥ 0 a.e. in Ω. We have

f(x, u(x))ϕ_{n}(x) = f^{+}(x, u(x))ϕ_{n}(x) − f^{−}(x, u(x))ϕ_{n}(x)
≥ −f^{+}(x, u(x))ϕ^{−}(x) − f^{−}(x, u(x))ϕ^{+}(x)
≥ −v^{−}(x).

By Fatou lemma, we have

−∞ < ∫_{Ω} f(x, u(x))ϕ(x) ≤ lim inf_{n} ∫_{Ω} f(x, u(x))ϕ_{n}(x)

= lim inf_{n}⟨f(x, u), ϕ_{n}⟩ < +∞,

which implies f(x, u)ϕ ∈ L^{1}(Ω), i.e. u ∈ A_{u}. □

Corollary 3.2 If η, η_{1} and η_{2} in L_{loc}^{1}(Ω), such that one of the following conditions is satisfied:

(1) f(x, u(x)) ≥ η(x) a.e. in Ω

(2) f(x, u(x)) ≤ η(x) a.e. in Ω

(3) f(x, u(x)) ≤ η_{1}(x) a.e. in {x ∈ Ω; u(x) < 0} and f(x, u(x)) ≥ η_{2}(x) a.e. in {x ∈ Ω; u(x) > 0},

(4) f(x, u(x)) ≥ η_{1}(x) a.e. in {x ∈ Ω; u(x) < 0} and f(x, u(x)) ≤ η_{2}(x) a.e. in {x ∈ Ω; u(x) > 0}.

Then f(x, u) ∈ L_{loc}^{1}(Ω) and consequently L_{c}^{∞}(Ω) ∩ W_{0}^{1,p}(Ω) ⊂ A_{u}.

Proof Assume (3) (the same argument works for (4)). Let ϕ ∈ C_{c}^{∞}(Ω). We set Ω_{1} = {x ∈ Ω; u(x) ≤ −1 a.e.}, Ω_{2} = {x ∈ Ω; |u(x)| ≤ 1 a.e.} and Ω_{3} = {x ∈ Ω; u(x) ≥ 1 a.e.}. It suffices to prove that f(x, u)|ϕ|χ_{Ω_{i}} ∈ L^{1}(Ω) for i = 1, 2, 3. By (1.4) we have f(x, u)ϕχ_{Ω_{2}} ∈ L^{1}(Ω). Let θ ∈ C^{∞}(ℝ) :

θ(s) = \begin{cases}1 & if s ≥ 1, \\ 0 ≤ θ(s) ≤ 1 & if 0 ≤ s ≤ 1, \\ 0 & if s ≤ 0. \end{cases}

It is clear that (θ ∘ u)|ϕ| ∈ W_{0}^{1,p}(Ω) and that

f(x, u(x))(θ ∘ u(x))|ϕ(x)| ≥ (θ ∘ u(x))|ϕ(x)|η_{2}(x) ∈ L^{1}(Ω).

By Proposition 3.1, we have f(x, u)(θ ∘ u)|ϕ| ∈ L^{1}(Ω), then f(x, u)ϕχ_{Ω_{3}} ∈ L^{1}(Ω) (the same argument to prove f(x, u)ϕχ_{Ω_{1}} ∈ L^{1}(Ω)). We conclude that f(x, u)ϕ ∈ L^{1}(Ω) for all ϕ ∈ C_{c}^{∞}(Ω), which implies f(x, u) ∈ L_{loc}^{1}(Ω).

Now assume (1) (the same argument works for (2)). For all ϕ ∈ C_{c}^{∞}(Ω) we have f(x, u)|ϕ| ≥ η(x)|ϕ| ∈ L^{1}(Ω), then f(x, u)|ϕ| ∈ L^{1}(Ω); therefore, f(x, u)ϕ ∈ L^{1}(Ω). Then we conclude that f(x, u) ∈ L_{loc}^{1}(Ω). □

References

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[2] M. Degiovanni and S. Zani, Euler equations involving nonlinearities without growth conditions. Potential Anal., 5, 505–512, (1996).

[3] O. Chakrone, Sur certains problèmes non-linéaires à la résonance. D.E.S. Thesis, Faculty of sciences, Oujda, (1995).

[4] A. Anane, Simplicité et isolation de la première valeur propre du p-Laplacien avec poids. C. R. Acad. Sci. Paris, 305, 725–728, (1987).

[5] Y. Jabri and M. Moussaoui, A saddle point theorem without compactness and applications to semilinear problems. Nonlinear Analysis, TMA, 32, No. 3, 363–380, (1997).

[6] D.G. de Figueiredo and J.-P. Gossez, Un problème elliptique semi-linéaire sans conditions de croissance. C.R. Acad. Sci. Paris, 308, 277–280, (1989).

[7] H. Brezis and F.E. Browder, A property of Sobolev spaces. Comm. Partial Differential equations, 4, 1077–1083, (1979).

Aomar Anane (e-mail: anane@sciences.univ-oujda.ac.ma) Omar Chakrone (e-mail: chakrone@sciences.univ-oujda.ac.ma) University Mohamed I, Department of Mathematics, Faculty of Sciences, Box 524, 60000 Oujda, Morocco