Asymptotic behaviour of positive large solutions of quasilinear logistic problems

We are interested in the asymptotic analysis of singular solutions with blowup boundary for a class of quasilinear logistic equations with indefinite potential. Under natural assumptions, we study the competition between the growth of the variable weight and the behaviour of the nonlinear term, in order to establish the blow-up rate of the positive solution. The proofs combine the Karamata regular variation theory with a related comparison principle. The abstract result is illustrated with an application to the logistic problem with convection.

We first recall some notations used in this paper.For some α ∈ (0, 1), we denote by C 0,α loc (Ω) the Banach space of locally Hölder continuous functions, that is, real-valued functions defined on Ω which are uniformly Hölder continuous with exponent α on any compact subset of Ω.The local Hölder space C 1,α loc (Ω) consists of functions whose first order derivatives are locally Hölder continuous with exponent α in Ω.Similarly, for p > 1, we denote by W 1,p loc (Ω) the Banach space of locally L p -integrable functions with locally L p -integrable weak derivatives, that is, W 1,p loc (Ω) := u : Ω → R measurable; u| K ∈ W 1,p (K) for all compact set K ⊂ Ω .
B Corresponding author.Email: vicentiu.radulescu@imar.ro;vicentiu.radulescu@math.cnrs.fr 2 R. Alsaedi, H. Mâagli, V. D. Rȃdulescu and N. Zeddini In this paper, we study the existence and the boundary behaviour of solutions for the following quasilinear elliptic problem where p > 1, f : (0, ∞) → (0, ∞) is a C 1 function, a is a positive function, locally Hölder continuous in Ω and satisfies some conditions related to Karamata regular variation theory and δ(x) denotes the Euclidean distance from x to the boundary ∂Ω.
By a weak solution of (1.1), we mean a positive function u ∈ W 1,p loc (Ω) ∩ C 1,β loc (Ω) for some 0 < β < 1 which satisfies in the distributional sense A solution of (1.1) is called a large solution (or boundary blow-up or explosive solution).Problems such as (1.1) arise in the study of the subsonic motion of a gas [35], the electric potential in some bodies [23], and Riemann geometry [5].
When p = 2, problem (1.1) becomes The subject of large solutions to (1.2) has received much attention starting with the pioneering works of Bieberbach in 1916 with a(x) = 1, f (u) = e u , n = 2 and with a(x) = 1, f (u) = e u and n = 3 in Rademacher's work in 1943 (see [3] and [36]).In 1957, Keller [20] and Osserman [34] gave a necessary and sufficient condition for the existence of a solution to (1.2) when a(x) = 1 and Ω is bounded, namely ∞ 1/ F(s) ds < ∞, where F (s) = f (s) is an increasing nonlinearity.Later, many authors have considered questions such as existence, uniqueness and boundary behaviour of the solution and its normal derivative in different domains and for bounded positive weights a(x).Problem (1.2) arises from many branches of mathematics and applied mathematics, and has been discussed by many authors in many contexts.For p = 2, f = 0 and u ∈ C 2 one obtains the classical Laplace equation which was extensively studied in the literature (see, for example, [1,37,[39][40][41]).
In a significant development, Cîrstea and Rȃdulescu [7] use Karamata's regular variation theory to study the blow-up rate and uniqueness, near the boundary to problem (1.2), in the case where a(x) decays to zero on ∂Ω at a fixed rate along the entire boundary ∂Ω and f varies regularly at infinity.
In the general case (not necessarily p = 2), the problem (1.1) seems to have been first considered in [9] when a(x) = 1.The question of existence, uniqueness and boundary behaviour of solutions were dealt there.Since then, there have been some other papers which included similar results for different types of nonlinearities; we mention for instance [11,14,15,[30][31][32].We also point out the important contributions of Guo and Webb [17,18] in the understanding of the structure of boundary blow-up solutions for quasi-linear elliptic problems.
When the weight a(x) is bounded, problem (1.1) has been considered by several authors.But when the weight a(x) is not necessarily bounded very little is known about the global behaviour of the solution except for the case p = 2, see for example [11,30,32].
Our aim in this paper is to establish existence and asymptotic behaviour of solutions of (1.1) with more general nonlinearities f (u) and weights a(x).In particular, we give global estimates of solutions (1.1) in the case where a(x) may be unbounded and satisfies some hypotheses related to the Karamata class of regularly varying functions at zero.
In order to use the method of sub-and super-solutions for (1.1), we begin by giving an auxiliary result in the case f (u) = u α , α > p − 1.More precisely, we prove the existence and asymptotic behaviour of a positive solution for the following problem where b(x) satisfies the following hypothesis L belongs to the set of Karamata functions K defined on (0, η] by Under this hypothesis, we state our first main result. Theorem 1.1.Let p > 1, α > max{p − 1, 1} and assume that b satisfies (B 1 ).Then problem where C > 1 is a constant and θ L,λ,p,α is the function defined on (0, η] by In order to establish our main result for problem (1.1), we assume that the functions f and a satisfy the following conditions: (H 1 ) The function a is positive, belongs to C 0,γ loc (Ω), 0 < γ < 1 and there exist two γ-Hölder continuous functions a 1 and a 2 such that for each x ∈ Ω, We now give an example of weight a(x) that satisfies hypothesis (H 1 ).Consider the simplest case corresponding to Ω = B(0, 1) ⊂ R n and assume that p > 1, λ < p, and µ > p − 1.Then the function for some positive constant c > 0. We refer to [7] for more examples of functions belonging to the Karamata class.
We now state the second main result in this paper. where for i ∈ {1, 2} and C > 1.
Throughout this paper, we need the following notations.
For two nonnegative functions f and g defined on a set S, the notation f (x) ≈ g(x), x ∈ S, means that there exists c > 0 such that We denote by ϕ 1 the positive normalized (i.e., max x∈Ω ϕ 1 (x) = 1) eigenfunction corresponding to the first positive eigenvalue λ 1 of the p-Laplace operator (−∆ p ) in W 1,p 0 (Ω).By definition, ϕ 1 is the unique normalized function satisfying the following eigenvalue problem We recall that, from Moser iterations [33] and [24, Theorem 1], ϕ 1 ∈ C 1,β (Ω), for some 0 < β < 1, and from strong maximum principle for quasilinear operators (see [42,Theorem 10]), and (1.9) Our paper is organized as follows.In Section 2, we collect some useful properties of Karamata functions.Section 3 deals with the proof of our main results.The last section is reserved to some applications.

The Karamata class K
To make the paper self-contained, we begin this section by recapitulating some properties of Karamata regular variation theory established by Karamata in 1930.This theory has been applied to study the asymptotic behaviour of solutions to differential equations.We refer to [7,8,26,37,43,46] for more details.
We point out that the constants in asymptotic relation (2.5) depend on ε (the first one goes to zero but the second one goes to infinity as ε → 0 + ).

Proof of main results
First, we recall some classical results about the sub-and super-solution method.

Definition 3.1. A function
If the above inequality is reversed, v is called a weak super-solution of (1.1).
We point out that this definition agrees with the sub-and super-solutions used in the proofs of Theorem 1.1 and Theorem 1.2, since the corresponding relations in those proofs are viewed in the weak sense.
The following property is an adaptation of Lemma 2.1 in [13].In the statement of the next result, we can assume without loss of generality that the Hölder exponent is the same for all functions u, u, and u.Indeed, if the corresponding exponents are β 1 , β 2 and β 3 , it is enough to consider β = min{β 1 , β 2 , β 3 }.Lemma 3.2.Let a(x) be a locally γ-Hölder continuous function in Ω, 0 < γ < 1 and f be continuously differentiable on [0, ∞).Assume that there exist a weak sub-solution u and a weak super-solution u to the problem (1.1) such that u ≤ u.Then there exists at least one weak solution Proof.For n ∈ N, we set Consider the boundary value problem Since u is a sub-solution and u is a super-solution, this problem has at least one positive weak solution u n such that u ≤ u n ≤ u, see Rȃdulescu [37].This in particular gives bounds on any compact set K ⊂ Ω for the sequence u n which in turn leads to bounds in C 1,γ loc (Ω).Since the embedding of C 1,γ (Ω ) into C 1 (Ω ) is compact for every Ω ⊂ Ω, then for every k ∈ N, we can select a subsequence of u n which converges in C 1 (Ω k ).A diagonal procedure gives a subsequence (denoted again by u n ) which converges to a function u ∈ C 1 loc (Ω).Passing to the limit in (3.1) we see that u is a weak solution of the equation in (1.1), verifying u ≤ u ≤ u.In particular, we deduce that u = ∞ on ∂Ω.This proves the lemma.
Next, we give the proof of Theorem 1.1.
Proof of Theorem 1.1.Let ϕ 1 be the positive normalized eigenfunction associated to the first eigenvalue λ 1 of −∆ p in W 1,p 0 (Ω) and let 0 < ε < η.In order to construct a sub-solution u and a super-solution u of (1.1), we define the function Since L ∈ K and η 0 (L(t)) Hence, there exists ε > 0 such that for each x ∈ Ω, and This gives Therefore using these inequalities and (1.9) we obtain ∆ p v(x) ≈ (ϕ 1 (x)) −λ L(εϕ 1 (x))(v(x)) α .Now, using (1.8), Lemma 2.7 and hypothesis (B 1 ) we obtain This proves that for every λ ≤ p, there exists M > 0 such that for every x ∈ Ω, we have By putting c = M 1 α−p+1 , it follows from (3.2) that u = 1 c v and u = c v are respectively subsolution and super-solution of problem (1.3).Thus, we conclude by Lemma (3.2) that problem (1.3) has a positive solution u such that u ≤ u ≤ u.Applying Lemma 2.7, Remark 2.6, and Lemma 2.4, we deduce that The following proposition plays a key role in the proof of Theorem 1.2.Proposition 3.3.Let a 1 , a 2 be the functions defined in hypothesis (H 1 ) and let α 1 , α 2 be such that 0 < p − 1 < α 1 ≤ α 2 .Let u i be the solution, given in Theorem 1.1, of the following problem Then there exists a constant c 0 > 0 such that Proof.By Theorem 1.1, problem (3.3) has a solution u i and there exist two constants c 1 > 0, c 2 > 0 such that for each x ∈ Ω, we have, where for i ∈ {1, 2}, ψ L i ,λ i ,p,α i is the function defined on (0, η], by and θ L i ,λ i ,p,α i is given by (1.7).To prove Proposition (3.3), it is enough to show that ψ L 2 ,λ 2 ,p,α 2 ψ L 1 ,λ 1 ,p,α 1 is bounded in (0, η].Now, using Lemma 2.1 (i) and hypothesis (H 1 ), we deduce that λ 1 ≤ λ 2 ≤ p.On the other hand, since p − 1 < α 1 ≤ α 2 , then we deduce that .
Proof of Theorem 1.2.Let u i be a solution of the problem (3.3) and let c 0 be a positive constant such that u 2 ≤ c 0 u 1 .Since lim δ(x)→0 u 1 (x) = ∞, then inf x∈Ω u 1 (x) > 0. Let µ 1 , µ 2 be two positive constants chosen so that