Dirichlet problems with singular and gradient quadratic lower order terms

Abstract

Abstract. We present a revisited form of a result proved in [Boccardo, Murat and Puel, Portugaliae Math. 41 (1982) 507-534] and then we adapt the new proof in order to show the existence for solutions of quasilinear elliptic problems also if the lower order term has quadratic dependence on the gradient and singular dependence on the solution.Mathematics Subject Classification. 35J20, 35J25, 35J65.

Bifurcation for Quasilinear Elliptic Singular BVP

“…Since g is integrable in a neighborhood of zero, this problem has a unique continuous [8] for existence and [6] for uniqueness). Using that v ∈ C(Ω) and v > 0 in Ω, if ω ⊂⊂ Ω we infer the existence of…”

“…From the pioneering works by Boccardo, Murat and Puel [12,13] this kind of quasilinear operators with g a continuous function in [0, +∞) has been extensively studied, especially if λ = 0 in the right hand side. More recently and also for λ = 0, the case of a function g with a singularity at zero has been studied in [5,6,8].…”

“…Indeed, in the case that g is singular at zero with 0 f 0 ∈ L 2N/(N +2) (Ω) the existence is due to [8] and the uniqueness to [6]. Moreover, if g is continuous at zero, the existence ( [12]) and uniqueness ( [6]) results remain also true in the case f 0 ≡ 0.…”

A Contribution to the Theory of Quasilinear Elliptic Equations and Application to the Minimization of Integral Functionals

“…(1.1) Note that the lower order term g(x, v, ∇v) = v|v| r−1 |∇v| 2 depends quadratically on the gradient and satisfies v g(x, v, ∇v) ≥ 0. (1.2) We recall here only the elliptic case: the papers [11], [8], [1], [9], [4], [6] where the the lower order term satisfies (1.2) and we recall the papers [12], [13], [14], [15], [7] where the the lower order term does not satisfy (1.2). In particular, [8] is the first paper where the regularizing effect of the lower order term is used (so that there exist finite energy solutions even if the data are L 1 functions).…”

Uniqueness of solutions for some elliptic equations with a quadratic gradient term

“…Recently, in [15] this result is improved by replacing condition (1.3) by the weaker condition that 0 ≤ f ∈ L (2 * /r) (Ω) with f ≡ 0 and λ < 1/2 in the case r = 1. On the other hand, a similar equation is studied in [24].…”