2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 249–254. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE OF NON-NEGATIVE SOLUTIONS FOR NONLINEAR EQUATIONS IN THE SEMI-POSITONE CASE

NAJI YEBARI, ABDERRAHIM ZERTITI

Abstract. Using the fibring method we prove the existence of non-negative solution of the p-Laplacian boundary value problem −Δ_{p}u = λf(u), for any λ > 0 on any regular bounded domain of ℝ^{N}, in the special case f(t) = t^{q} − 1.

1. Introduction and main results

In this paper we are interested in finding nonnegative solutions to the equation

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

for some specific f in the non positone case (f(0) < 0), under assumptions stated below.

Here Ω is a connected and bounded subset of ℝ^{N} with boundary ∂Ω in C^{1,α}. We set

Δ_{p}u = div(|∇u|^{p−2}∇u).

When p = 2, this type of problem in the nonpositone case can be studied via the shooting method. Existence of a radially symmetric nonnegative solution for λ > 0 sufficiently small have been obtained in [1, 2] and nonexistence of such a solution for λ > 0 large have been established in [1, 3], in the framework of the semi positone case and f is superlinear. Observe that, since f(0) < 0, the constant 0 is an upper solution of (1.1) and as a consequence it is not possible, in general, to apply the usual techniques (for example: the method of upper and lower solutions, etc.) and we shall work in the framework of the so-called fibration method introduced by Pohozaev in [5], and then developed in [6, 7, 8]. We shall assume that f has the form

f(t) = t^{q} − 1, with q > p − 1  (1.2)

2000 Mathematics Subject Classification. 35J65, 35J25.

Key words and phrases. Superlinear semi-positone problems; variational methods;

fibring method.

\begin{matrix}c \\ ◯ \end{matrix}2006 Texas State University - San Marcos.

Published September 20, 2006.

To avoid the noncompactness problem we shall always assume that the problem is subcritical, in the sense of the critical exponent for Ω,

p^{⋆} = \begin{cases}\frac{Np}{N−p} & if 1 < p < N, \\ +∞ & if p ≥ N. \end{cases}  (1.3)

Let

u = θv, θ = λ^{−( \frac{1}{q−p+1} )}, μ = λ^{\frac{q}{q−p+1}} > 0 .  (1.4)

and

−Δ_{p}u = u^{q} − μ in Ω,
u > 0 in Ω,
u = 0 on ∂Ω

(1.5)

It can be seen that (1.1), (1.2), (1.4) and (1.5) are equivalent. Also let p and q satisfy

0 < p − 1 < q < p^{∗} − 1  (1.6)

where p^{∗} is given by (1.3). Concerning μ, we shall assume its positivity.

By a solution of (1.5), we mean a W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω) function which is a critical point of the functional

E(v) = −\frac{1}{p} ∫_{Ω} |∇v|^{p}dx + \frac{1}{q + 1} ∫_{Ω} |v|^{q+1}dx − μ ∫_{Ω} |v(x)|dx

and therefore satisfies

∫_{Ω}(|∇u|^{p−2}∇u.∇φ − (u^{q} − μ)φ)dx = 0

for every φ ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω).

Our main result is as follows.

Theorem 1.1. Let assumptions (1.3) and (1.6) be satisfied. Then there exists a nontrivial nonnegative solution u ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω) of problem (1.1) for any λ > 0. Moreover, u ∈ C^{1,α}(\overline{Ω}) for some α > 0.

2. Proof of the main theorem

The proof is based on the fibering method and is divided into five stages. Step 1: We introduce the Euler functional associated with (1.5) as follows

E(u) = −\frac{1}{p} ∫_{Ω} |∇u|^{p}dx + \frac{1}{q + 1} ∫_{Ω} |u|^{q+1}dx − μ ∫_{Ω} |u(x)|dx

According to the fibering method, we set

u(x) = rv(x),  (2.1)

where r ∈ ℝ^{+} and v ∈ W_{0}^{1,p}(Ω). Then we obtain

Ẽ(r, v) = E(r, v) = −\frac{|r|^{p}}{p} ∫_{Ω} |∇v|^{p}dx + \frac{|r|^{q+1}}{q + 1} ∫_{Ω} |v|^{q+1}dx − μr ∫_{Ω} |v(x)|dx (2.2)

We introduce the fibering functional

∫_{Ω} |∇v|^{p}dx = 1  (2.3)

Under condition (2.3) the functional Ẽ takes the form

Ẽ(r, v) = −\frac{r^{p}}{p} + \frac{r^{q+1}}{q + 1} ∫_{Ω} |v|^{q+1}dx − μr ∫_{Ω} |v(x)|dx  (2.4)

The bifurcation equation is

0 = \frac{∂Ẽ}{∂r} = −r^{p−1} + r^{q} ∫_{Ω} |v|^{q+1}dx − μ ∫_{Ω} |v(x)|dx  (2.5)

which gives

−r^{p} + r^{q+1} ∫_{Ω} |v|^{q+1}dx − μr ∫_{Ω} |v(x)|dx = 0.  (2.6)

Let set

Ẽ(v) = E(r(v)v)  (2.7)

Step 2: Let us consider the variational problem

M_{0} = sup {Ẽ(v); v ∈ W_{0}^{1,p}(Ω)/ ∫_{Ω} |∇v|^{p}dx = 1}.  (2.8)

It follows that

Ẽ(v) = min_{r≥0} Ẽ(r, v) = min_{r≥0}\{−\frac{r^{p}}{p} + \frac{r^{q+1}}{q + 1} ∫_{Ω} |v|^{q+1}dx − μr ∫_{Ω} |v(x)|dx\} < 0, (2.9)

as a matter of fact, (2.6) gives

−\frac{r^{p}(v)}{p} = −\frac{r^{q+1}(v)}{p} ∫_{Ω} |v|^{q+1}dx + μ\frac{r(v)}{p} ∫_{Ω} |v(x)|dx,

On the other hand,

Ẽ(v) = E(r(v)v)

= −\frac{r^{q+1}(v)}{p} ∫_{Ω} |v|^{q+1}dx + μ\frac{r(v)}{p} ∫_{Ω} |v(x)|dx

+ \frac{r^{q+1}(v)}{p} ∫_{Ω} |v|^{q+1}dx − μr(v) ∫_{Ω} |v(x)|dx,

which gives

Ẽ(v) = \frac{(p − q − 1)}{(q + 1)p}r^{q+1}(v) ∫_{Ω} |v|^{q+1}dx − μr(v)(1 − \frac{1}{p}) ∫_{Ω} |v(x)|dx  (2.10)

By (1.6), Ẽ(v) < 0.

Let us prove the following Lemma.

Lemma 2.1. The sequence maximizing problem (2.8) is bounded in W_{0}^{1,p}(Ω).

Proof. Let (v_{n}) be a maximizing sequence for (2.8). We set

v_{n}(x) = c_{n} + \overline{v}_{n}(x)  (2.11)

with

∫_{Ω} \overline{v}_{n}(x)dx = 0 .  (2.12)

Since

∫_{Ω} |∇v_{n}|^{p}dx = ∫_{Ω} |∇\overline{v}_{n}|^{p}dx = 1  (2.13)

and by the Sobolev embedding theorems (the Poincare-Wirtinger inequality), the sequence (\overline{v}_{n}) is bounded in W^{1,p}(Ω). From the bifurcation equation (2.5), we obtain

r_{n}^{p} = r_{n}^{q+1} ∫_{Ω} |c_{n} + \overline{v}_{n}|^{q+1}dx − μr_{n} ∫_{Ω} |c_{n} + \overline{v}_{n}|dx.  (2.14)

Let us assume that

c_{n} → +∞, as n → +∞ .  (2.15)

Then

∫_{Ω} |1 + \frac{\overline{v}_{n}}{c_{n}} |^{q+1}dx = \frac{1}{c_{n}^{q+1}r_{n}^{q−p+1}} + \frac{μ}{c_{n}^{q}r_{n}^{q}} ∫_{Ω} |1 + \frac{\overline{v}_{n}}{c_{n}} |dx .  (2.16)

By embedding results, there exists C > 0 such that

∥\overline{v}_{n}∥_{W^{1,p}(Ω)} ≤ C, ∀n ∈ ℕ

Using (2.15) and since by assumption (1.6) the space W^{1,p}(Ω) is compactly embedded in L^{q+1}(Ω). We may assume that (\overline{v}_{n}) converges strongly in latter space. Then from (2.16) we have

∫_{Ω} |1 + \frac{\overline{v}_{n}}{c_{n}} |^{q+1}dx → |Ω| > 0, as n → +∞.  (2.17)

The proof is complete. □

Hence, we can assume that the sequence (v_{n}) converges weakly in W_{0}^{1,p}(Ω). By assumption (1.6), it follows that v_{n} → \overline{v} in L^{q+1}(Ω). This implies that

∥∇v_{0}∥_{p} ≤ lim inf_{n→+∞} ∥∇v_{n}∥_{p} .

Since

∥∇v_{n}∥_{p}^{p} = ∫_{Ω} |∇v_{n}|^{p}dx = 1,

we obtain

0 ≤ ∥∇v_{0}∥_{p}^{p} = ∫_{Ω} |∇v_{0}|^{p}dx ≤ 1.  (2.18)

Now we shall prove the equality

∫_{Ω} |∇v_{0}|^{p}dx = 1.  (2.19)

We assume the contrary; i.e, that

∫_{Ω} |∇v_{0}|^{p}dx < 1.  (2.20)

Note that

0 < ∫_{Ω} |∇v_{0}|^{p}dx .  (2.21)

Otherwise, if ∫_{Ω} |∇v_{0}|^{p}dx = 0, v_{0} = c_{0} is a constant, and from (2.8), we have for all ϵ > 0 there exist n_{0} ∈ ℕ such that for all n ≥ n_{0} we have

M_{0} − ϵ < Ẽ(v_{n}) < M_{0}.

Let θ ∈]0, 1[. Then

Ẽ(θv_{n}) − ϵ ≤ M_{0} − ϵ < Ẽ(v_{n}) < M_{0}  (2.22)

by (2.10). We recall that

Ẽ(v_{n}) = \frac{(p − q − 1)}{(q + 1)p}r_{n}^{q+1} ∫_{Ω} |v_{n}|^{q+1}dx − μr_{n}(\frac{p − 1}{p}) ∫_{Ω} |v_{n}(x)|dx .

Using (2.22), we see that

(1 − θ)μr_{n}(\frac{p − 1}{p}) ∫_{Ω} |v_{n}(x)|dx < ϵ for all n ≥ n_{0}.

Then we have

r_{n} ∫_{Ω} |v_{n}(x)|dx → r_{0} ∫_{Ω} |v_{0}(x)|dx = 0

as n → +∞ and v_{0} = c_{0} = 0 which gives Ẽ(v_{0}) = 0. This contradicts M_{0} < 0.

Due to (2.20) and (2.21), there exists θ > 1 (i.e., θ^{p} = 1/ ∫_{Ω} |∇v_{0}(x)|^{p}dx > 1) such that v_{∗} = θv_{0} satisfies

∫_{Ω} |∇v_{∗}(x)|^{p}dx = 1

and

Ẽ(v_{∗}) = Ẽ(θv_{0}) = min_{r≥0} \{ − \frac{r^{p}}{p} + \frac{r^{q+1}}{q + 1}θ^{q+1} ∫_{Ω} |v_{0}|^{q+1}dx − μrθ ∫_{Ω} |v_{0}(x)|dx\}

= min_{ρ≥0} \{ − \frac{ρ^{p}}{pθ^{p}} + \frac{ρ^{q+1}}{q + 1} ∫_{Ω} |v_{0}|^{q+1}dx − μρ ∫_{Ω} |v_{0}(x)|dx\}

> min_{ρ≥0} \{ − \frac{ρ^{p}}{p} + \frac{ρ^{q+1}}{q + 1} ∫_{Ω} |v_{0}|^{q+1}dx − μρ ∫_{Ω} |v_{0}(x)|dx\}.

Thus,

Ẽ(v_{∗}) > Ẽ(v_{0}).

This inequality contradicts the definition of (2.8). Thus, we have obtained a solution to the variational problem. Step 3:

Ẽ(v_{0}) = sup {Ẽ(v); v ∈ W_{0}^{1,p}(Ω)/ ∫_{Ω} |∇v|^{p}dx = 1}

The fibering method implies r = r_{0} = r(v_{0}) where r_{0} > 0 and

− \frac{r_{0}^{p}}{p} + \frac{r_{0}^{q+1}}{q + 1} ∫_{Ω} |v_{0}|^{q+1}dx − μr_{0} ∫_{Ω} |v_{0}(x)| dx

= min_{r≥0} \{ − \frac{r^{p}}{p} + \frac{r^{q+1}}{q + 1} ∫_{Ω} |v|^{q+1}dx − μr ∫_{Ω} |v(x)|dx\}

To complete the proof, we must show that the equation (1.5) is verified. We can assume that v_{0} is nonnegative by replacing v_{n} by |v_{n}|. Moreover, there exists a Lagrange multiplier σ such that

Ẽ′(v_{0}).h = σ( ∫_{Ω} |∇(.)|^{p}dx)′(v_{0}).h ∀h ∈ W_{0}^{1,p}(Ω).  (2.23)

From the above equation, and by taking v_{0} as test function, we have

r_{0}{ ∫_{Ω}((r_{0}v_{0})^{q} − μ)v_{0}dx} = pσ ∫_{Ω} |∇(v_{0})|^{p}dx = pσ .

By (2.6) we obtain σ = \frac{r_{0}^{p}}{p} > 0. Then we can write

Ẽ′(v_{0}) = pσ(−Δ_{p}v_{0})

which is equivalent to

−Δ_{p}(r_{0}v_{0}) = (r_{0}v_{0})^{q} − μ .

Then if we set u = r_{0}v_{0} ≥ 0, we can see that u is a solution of problem (1.5). Step 4: For u ≥ 0, we have Ẽ(v_{0}) < 0, thus the solution u ≥ 0 is non trivial. Step 5: We have obtained the nonnegative nontrivial solution u to problem (1.5). A standard bootstrap argument (see Drabek [4]) shows that u ∈ L^{∞}(Ω). Then the asserted regularity of u ∈ C_{Loc}^{1,α}(Ω) follows by Tolksdorf [9]. Thus the theorem is proved.

References

[1] D. Arcoya and A. Zertiti, Existence and non-existence of radially symmetric non-negative solutions for a class of semi-positone problems in a annulus, Rendiconti di Matematica, Serie VII, Vol. 14, Roma (1994), 625-646.

[2] K. J. Brown, A. Castro and R. Shivaji, Nonexistence of radially symmetric solutions for a class of semipositone problems, Differential and Integral Eqns., 2(4) (1989) pp. 541-545.

[3] A. Castro and R. Shivaji, Nonnegative solutions for a class of nonpositone problems, Proc. Roy. Soc. Edin., 108(A)(1988), pp. 291-302.

[4] P. Drabek, Strongly nonlinear degenerate and singular elliptic problems, in Nonlinear Partial Differential Equations (Fes, 1994), Pitman Res. Notes Maths. Ser.343, Longman, Harlow, UK, 1996, pp. 112-146.

[5] S. I. Pohozaev, On eigenfunctions of quasilinear elliptic problems, Mat. Sb. 82, 192-212 (1970).

[6] S. I. Pohozaev, On one approach to nonlinear equations, Dokl. Akad. Nauk. 247, 1327-1331(in Russian)(1979).

[7] S. I. Pohozaev, On a constructive method in the calculus of variations, Dokl. AKad. Nauk. 288, 1330-1333 (in Russian)(1988).

[8] S. I. Pohozaev, On fibering method for the solution of nonlinear boundary value problems, Trudy Mat.Inst. Steklov 192, 140-163 (1997).

[9] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), pp. 126-150.

Naji Yebari Département de Mathématiques, Université Abdelmalek Essaadi, B. P. 2121, Tetouan, Maroc

   E-mail address: nyebari@hotmail.com, nyebari@fst.ac.ma

Abderrahim Zertiti Département de Mathématiques, Université Abdelmalek Essaadi, B. P. 2121, Tetouan, Maroc

   E-mail address: zertiti@fst.ac.ma