Existence and uniqueness of positive solution of a logistic equation with nonlinear gradient term

Abstract

Abstract. The main goal of this work is to study the existence and uniqueness of positive solution of a logistic equation including a nonlinear gradient term. In particular, we use local and global bifurcation together with some a-priori estimates. To prove uniqueness, the sweeping method of Serrin is employed.

Existence and non-existence of positive solutions for nonlinear elliptic singular equations with natural growth

“…Moreover, multiplicity of solution is shown for λ > λ * . Finally if g ≡ 1, the case f (λ, s) = λs − s p , for some p > 1, was analyzed in [15] showing the existence and uniqueness of conti nuously differentiable positive solution for λ > λ 1 .…”

Bifurcation for Quasilinear Elliptic Singular BVP

“…To our knowledge, the case in which the (non-variational) differential operator (with g a continuous function in [0, +∞)) is faced up to a true nonlinear right hand side has been less studied [17,19]. See the works by Arcoya and Boccardo [4], Canino [15] and the references therein for the case of variational differential operators.…”

“…Compare this approach with this one in the work by Ruiz and Suárez [19], for g ≡ 1 and a logistic nonlinearity, where the authors cleverly combine regularity in C 1 (Ω) with the properties of the inverse (−∆) −1 of the Laplacian operator in C(Ω) in order to use bifurcation techniques. Unfortunately, this idea does not work in the case g singular at zero by the lack of regularity.…”

Quasilinear elliptic system in exterior domains with dependence on the gradient

“…Their argument relied essentially on the maximum principle with a sub‐super solution method. In regards to the scalar case, the subcritical and critical case, elliptic problems involving gradient terms and/or models in an exterior domain we would like to cite , , , , , , , , , , , , , and . Some of the above mentioned papers involve more general operator such as the p‐Laplacian operator.…”