Nonselfadjoint resonance problems with nonlinearities of superlinear growth

Second Order Sturm-liouville Problems with Asymmetric, Superlinear Nonlinearities

We consider the nonlinear Sturm-Liouville problem −(p(x)u′(x))′ + q(x)u(x) = f(x, u(x), u′(x)), in (0, π), (1) c00u(0) + c01u ′(0) = 0, c10u(π) + c11u ′(π) = 0, (2) where p ∈ C1[0, π], q ∈ C0[0, π], with p(x) > 0, x ∈ [0, π], and ci0 + ci1 > 0, i = 0, 1. We suppose that f : [0, π] × R2 → R is continuous and there exist increasing functions ζl, ζu : [0,∞) → R, and a constant B, such that limt→∞ ...

Second Order, Sturm-liouville Problems with Asymmetric, Superlinear Nonlinearities Ii

We consider the nonlinear Sturm-Liouville problem −(p(x)u′(x))′ + q(x)u(x) = f(x, u(x)) + h(x), in (0, π), c00u(0) + c01u ′(0) = 0, c10u(π) + c11u ′(π) = 0, where: p ∈ C1[0, π], q ∈ C0[0, π], with p(x) > 0 for all x ∈ [0, π]; ci0 + ci1 > 0, i = 0, 1; h ∈ L2(0, π). We suppose that f : [0, π] × R → R is continuous and there exist increasing functions ζl, ζu : [0,∞)→ R, and positive constants A, B...

On the Number of Radially Symmetric Solutions to Dirichlet Problems with Jumping Nonlinearities of Superlinear Order

This paper is concerned with the multiplicity of radially symmetric solutions u(x) to the Dirichlet problem ∆u+ f(u) = h(x) + cφ(x) on the unit ball Ω ⊂ RN with boundary condition u = 0 on ∂Ω. Here φ(x) is a positive function and f(u) is a function that is superlinear (but of subcritical growth) for large positive u, while for large negative u we have that f ′(u) < μ, where μ is the smallest po...

Global Structure of Nodal Solutions for Second-Order m-Point Boundary Value Problems with Superlinear Nonlinearities

One-sided Resonance for Quasilinear Problems with Asymmetric Nonlinearities

for two consecutive variational eigenvalues, λl < λl+1 of −∆p on W 0 (Ω), and some ε > 0 (see Section 2 for the definition of the variational spectrum). The special case where α+(x) = α−(x) ≡ λl and q = 1 was recently studied by Arcoya and Orsina [1], Bouchala and Drábek [3], and Drábek and Robinson [8] (see also Cuesta et al. [6] and Dancer and Perera [7]). In the present paper, we prove a sin...