Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary

Abstract

In this paper we perform an extensive study of the existence, uniqueness (or multiplicity) and stability of nonnegative solutions to the semilinear elliptic equation −∆u = λu − u p in Ω, with the nonlinear boundary condition ∂u/∂ν = u r on ∂Ω. Here Ω is a smooth bounded domain of IR d with outward unit normal ν, λ is a real parameter and p, r > 0. We also give the precise behavior of solutions for large |λ| in the cases where they exist. The proofs are mainly based on bifurcation techniques, sub-supersolutions and variational methods.

The effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation

“…Under the higher regularity condition (1.10), it is well known [14,Theorem 4.6.3] that if b − ≡ 0 then, for every λ > 0, (P λ ) has at most one classical positive solution u λ ∈ C 2 (Ω), meaning that u λ > 0 in Ω. Indeed, a problem similar to (P λ ) has been studied by García-Melián, MoralesRodrigo, Rossi, and Suárez [8,Theorem 1.2] in the case where the coefficients are positive constants, which allows the authors to obtain existence and uniqueness results for positive solutions by means of the sub and super-solution method. In this way, the multiplicity issue for positive solutions naturally arises if b − ≡ 0.…”

Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions

“…For general f, h and an arbitrary very weak solution u we still have (29) with C 1 = C 1 (u) so that Theorem 8 guarantees u ∈ L ∞ ( ) and we can use known a priori estimates for strong solutions to get a uniform a priori bound for very weak solutions (see e.g. [5,6,11] and the references therein).…”

“…The same exponents appear as critical exponents for Liouville-type theorems for the corresponding problems in the halfspace and the whole of R N , respectively. For reference in the case of (7) see [6,7,11,14].…”

Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions

“…3. For the global nature of bifurcation components of semilinear elliptic problems with nonlinear boundary conditions arising from population dynamics, we refer to Morales-Rodrigo and Suárez [15], García-Melián, Morales-Rodrigo, Rossi, and Suárez [11], and Cantrell, Cosner, and Martinez [7], where linear terms with constant coefficients are considered. The rest of this paper is organized as follows.…”