Existence and uniqueness of solutions of nonlinear Neumann problems

Pairs of Nodal Solutions for a Class of Nonlinear Problems with One-sided Growth Conditions

“…With this respect, see [6] where, for the case (3.3) with f(x)x > 0 for x 0, some conditions on the weight function q(t) are found for the existence or nonexistence of positive periodic solutions. Analogous results can be derived for the Neumann problems (compare also with [2]). …”

“…Hence we can pass to polar coordinates and define the rotation number associated to the solution ζ(t) = ζ(t; t 0 , z) on a time interval [t 0 , t 0 + τ], as Rot (5.1) (z; [t 0 , t 0 + τ]) := 1 2π τ 0 ∂ ∂y H(t, ζ(t))ζ 2 (t) + ∂ ∂x H(t, ζ(t))ζ 1 (t) ζ 1 (t) 2 + ζ 2 (t) 2 dt.…”

“…has been widely investigated for its significance in many applications and also for the interest in understanding the role of the weight function q(s) with respect to existence or nonexistence results (see, for instance, [2]). …”

An indefinite concave-convex equation under a Neumann boundary condition I

“…On the other hand, it is known that if b − ≡ 0 then dead core solutions may arise [4], which makes delicate the study of the non-negative solutions set of (P λ ), as shown in [1]. For instance, when b changes sign the existence of a minimal non-negative solution for λ > 0 small is still unknown.…”

Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem

“…The investigation of the case in which the nonlinear term g(s) has superlinear growth at infinity (namely, g(s) ∼ |s| p−1 s, with p > 1) led to multiplicity results of oscillatory solutions for various boundary value problems associated to (1.1) (see [13,30,34] and the references therein). The search of positive solutions has been addressed both to the case of ODEs and to nonlinear elliptic PDEs of the form 4) under different conditions for u| ∂Ω (see [1,2,3,4,5,6] for some classical results in this direction). In particular, multiplicity results for (1.4), with Dirichlet boundary conditions on a bounded domain Ω, have been obtained in [7,17,19,21,28] in the superlinear case g(s) ∼ s p , with p > 1.…”