Nonlinear elliptic equations with unbounded coefficient and singular lower order term
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Journal of Fixed Point Theory and Applications
Nonlinear elliptic equations with unbounded coefficient and singular lower order term Amine Marah, Hicham Redwane and Khaled Zaki Abstract. In this paper, we are interested in the existence result of solutions for nonlinear and singular Dirichlet problem whose model is ⎧ 2 ⎨ − div b(u)∇u + μ(x) |∇u| sign(u) = f in Ω, |u|θ ⎩ u = 0 on ∂Ω, where Ω is a bounded open subset of RN (N ≥ 2), b(s) is a positive continuous function which blows up for a finite value of the unknown, μ(x) is positive, bounded and measurable, 0 < θ < 1, and the source f belongs to L1 (Ω). Mathematics Subject Classification. 35J60, 35J75. Keywords. Nonlinear elliptic equations, blowing-up coefficients, renormalized solutions, existence, singular gradient term, changing sign data.
1. Introduction This paper is concerned with the study of a class of nonlinear elliptic problems having a singular lower order term. More precisely, we are going to study the existence of the renormalized solutions for the problem whose prototype is ⎧ 2 ⎨ − div b(u)∇u + μ(x) |∇u| sign(u) = f in Ω, |u|θ (1.1) ⎩ u = 0 on ∂Ω, where Ω is a bounded open subset of RN (N ≥ 2), b is a continuous function of C 0 ((−∞, m), R+ ), where m is a positive m real number, and satisfies b(s) ≥ α, for α > 0, lims→m− b(s) = +∞ and 0 b(s) ds < +∞. The function μ(x) is positive and measurable such that 0 < ξ ≤ μ(x) ≤ η, 0 < θ < 1 and the data f belongs to L1 (Ω). The main difficulties in studying (1.1) are due to the presence of the lower order term which is singular at u = 0 and has quadratic growth with 0123456789().: V,-vol
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respect the gradient and to the fact that the principal part of the operator blows up for a finite value m of u. If μ ≡ 0, the problem of existence and uniqueness (sometimes partial uniqueness) results of (1.1) have been investigated in different contexts by several authors (see, for instance [6,7,16,17,23]). In all these papers, the existence and uniqueness of a solution have been obtained using the framework of renormalized solution which was introduced in [21]. We emphazise that, in the papers we quoted before, the idea of weak solutions is not suitable for this kind of problems, because m the flux b(u)∇u is not defined on the subset {u = m}. However, the case 0 b(s) ds = +∞ (see [14,22]) allows solutions to avoid the value m and to prove existence of weak solutions in the case of p-Laplacian operator and the data f belongs to some Lebesgue space. Now, if μ(x) ≡ 0 and b does not blow up for a finite value of the unknown, problems like (1.1) have been extensively studied in [1–4,9,12,15] under different assumptions on the singular lower order term and on the data f . More precisely, the existence and regularity of the solutions have been proved in the case of nonnegative data f belonging to a suitable Lebesgue space. If the data do not have a constant sign, the solution u vanishes inside Ω so that the measure of the set {u = 0} is positive. In [19,20], the authors deal with the sign-ch
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