EXISTENCE OF BLOWUP SOLUTIONS FOR NONLINEAR PROBLEMS WITH A GRADIENT TERM

in a bounded domain. In the case f (u)= p(x)uγ, a > 0, and γ > max(1,a), Lair and Wood [7] dealt with the above equation in bounded domain and the whole space. They proved the existence of entire large solution under the condition ∫∞ 0 rmax|x|=r p(x)dr <∞ when the domain is RN . Recall that u is a large solution on a bounded domain Ω in RN , if u(x) → +∞ as dist(x,∂Ω) → 0, and u is called an entire large solution if u is defined on RN and lim|x|→+∞u(x)= +∞. Ghergu et al. [3] considered more general equation


Introduction and the main result
Semilinear elliptic problems involving gradient term with boundary blowup interested many authors.Namely, Bandle and Giarrusso [1] developed existence and asymptotic behaviour results for large solutions of Δu + ∇u(x) a = f (u) (1.1) in a bounded domain.
In the case f (u) = p(x)u γ , a > 0, and γ > max(1,a), Lair and Wood [7] dealt with the above equation in bounded domain and the whole space.They proved the existence of entire large solution under the condition ∞ 0 r max |x|=r p(x)dr < ∞ when the domain is R N .
Recall that u is a large solution on a bounded domain Ω in R N , if u(x) → +∞ as dist(x,∂Ω) → 0, and u is called an entire large solution if u is defined on R N and lim |x|→+∞ u(x) = +∞.
Ghergu et al. [3] considered more general equation where 0 ≤ a ≤ 2, p and q are Hölder continuous functions on (0, ∞).We note that the Keller-Osserman condition on f (see [6,8]) remains the key condition for the existence for their works.
In the present paper, we are interested in the study of nonlinear elliptic problems with boundary blowup, of the type where λ : [0,∞) → [0,∞) is a continuous function and ϕ satisfies the following hypotheses.
(H 1 ) ϕ : R N × [0,∞) → [0,∞) is measurable, continuous with respect to the second variable.(H 2 ) There exist nonnegative functions p, q, and f satisfying for each x ∈ R N and t ≥ 0, where f is required to satisfy.
In the sequel, we put where K(t) := t N−1 exp( t 0 λ(s)ds), for each t > 0, and we define the function F on [1, ∞) by From the hypotheses adopted on f , we note that the function Our main result is the following.
Motivation for the present contribution stems from the one of Ghergu and Rȃdulescu [4] who considered the following problem: where Ω is either a smooth bounded domain or the whole space and f is a nondecreasing function satisfying f ∈ Ꮿ 0,α loc (0,∞), f (0) = 0, f > 0 on (0,∞), and The authors studied the existence and nonexistence of explosive solutions under the assumption that where ∞ 0 r min |x|=r p(x)dr).More precisely, they showed in the case of Ω = R N that the above problem has positive solution if and only if We remark that the condition (1.4) adopted on f includes the sublinear case, sup x≥1 f (x)/ x < ∞, studied by Ghergu and Rȃdulescu [4].
The outline of the paper is as follows.In Section 2, we will give some auxiliary results.The comparison result obtained in Section 2, Theorem 2.6, is used in Section 3 to prove the main result of this work.

Auxiliary results
In this section, we suppose that (A, p) satisfies (H 5 ) A is a nonnegative continuous function on [0,∞), positive and differentiable on (0,∞), and p : (0,∞) → [0,∞) is continuous function satisfying For any given continuous function ψ on (0, ∞), we put We consider the following problem: ( We state the following. Theorem 2.1.Under the hypotheses (H 3 ) and (H 5 ), the problem (2.3) has a positive solution Proof.Let (u k ) k≥0 be the sequence of functions defined on [0,∞) by u 0 (r) = α and (2.5) Clearly, we have for each By induction, we prove that (u k ) k≥0 is a nondecreasing sequence.Since the function f is nondecreasing, we obtain by (2.5) that for each k ≥ 0, That is, (2.7) Faten Toumi 5 Then (2.8) It follows that for each r ≥ 0, Then the sequence (u k ) k≥0 converges and the function u = sup k∈N u k is finite and satisfies for each r ≥ 0, (2.11) Proof.Existence follows from Theorem 2.1.Now, let us prove the uniqueness.Let u and v be positive solutions of the problem (2.3).Then for each a > 0 and r ∈ [0,a], we have (2.14) Since u and v are continuous, it follows that there exists c > 0 such that u(r),v(r) ∈ [0,c] for each r ∈ [0,a].So, by hypothesis (H 6 ) and Fubini theorem, we obtain that (2.15) By Gronwall's lemma, we deduce that u(r) = v(r) on [0,a].This completes the proof.

Proof of the main result
Proof of Theorem 1.1.Recall that for each t > 0, K(t) := t N−1 exp( t 0 λ(s)ds).Necessity.We will proceed by contradiction.Suppose that (1.9) fails and let u be an entire large solution of problem (P).Let Define the spherical mean of v by where w N denotes the surface of the unit sphere in R N .Since u is a positive entire large solution of (P), it follows by (1.4) that v is positive and lim |x|→∞ v(x) = +∞.
By [2, Section 1, Proposition 6], we obtain By computation, we have on the ball (3.5) Using the fact that f ≥ 0, we obtain Sufficiency.Suppose that (1.9) holds.We will use the comparison result given by Theorem 2.6 for A(t) = B(t) = K(t) = t N−1 exp( t 0 λ(s)ds), p, q, and f satisfying, respectively, (H 4 ) and (H 3 ).