Existence Results for Quasilinear Elliptic Equations with Indefinite Weight

We provide the existence of a solution for quasilinear elliptic equation −div(∞()|∇|−2∇+̃(,|∇|)∇)=()||−2+(,)+ℎ() in Ω under the Neumann boundary condition. Here, we consider the condition that ̃(,)=(−2) as →+∞ and (,)=(||−1) as ||→∞. As a special case, our result implies that the following -Laplace equation has at least one solution: −Δ=()||−2+||−2+ℎ() in Ω,/=0 on Ω for every 1<<<∞, ∈ℝ, ≠0 and ,ℎ∈∞(Ω) with ∫Ω≠0. Moreover, in the nonresonant case, that is, is not an eigenvalue of the -Laplacian with weight , we present the existence of a solution of the above -Laplace equation for every 1<<<∞, ∈ℝ and ,ℎ∈∞(Ω)

Nonlinear Differential Equations on Bounded and Unbounded Domains

Differential equations represent one of the strongest connections between Mathematics and real life. This is due to the fact that almost all the physical phenomena, as well as many other in economy, biology or chemistry, are modelled by differential equations. This Thesis includes a detailed study of nonlinear differential equations, both on bounded and unbounded domains. In particular, we analyze the qualitative properties of the solutions of nonlinear differential equations, focusing on the study of constant sign solutions on the whole domain of definition or, at least, on some subset of it. The main technique is based on the construction of an abstract formulation included into functional analysis, in which the solutions of the differential equations coincide with the fixed points of certain operators