Multiple nontrivial solutions for nonlinear eigenvalue problems

Multiple Nontrivial Solutions for Nonlinear Eigenvalue Problems

In this paper we study a nonlinear eigenvalue problem driven by the p-Laplacian. Assuming for the right-hand side nonlinearity only unilateral and sign conditions near zero, we prove the existence of three nontrivial solutions, two of which have constant sign (one is strictly positive and the other is strictly negative), while the third one belongs to the order interval formed by the two opposi...

Nontrivial Solutions for Singular Nonlinear Three-Point Boundary Value Problems

The singular nonlinear three-point boundary value problems { −(Lu)(t) = h(t)f (u(t)), 0 < t < 1, βu(0)− γ u′(0) = 0, u(1) = αu(η) are discussed under some conditions concerning the first eigenvalue corresponding to the relevant linear operator, where (Lu)(t) = (p(t)u′(t))′+q(t)u(t), 0 < η < 1, h(t) is allowed to be singular at both t = 0 and t = 1, and f need not be nonnegative. The associated ...

Nontrivial Solutions for Boundary Value Problems of Nonlinear Differential Equation

The nonlinear three-point boundary value problem { −u′′(t) = f(t, u(t)), t ∈ I, βu(0)− γu′(0) = 0, u(1) = αu(η), is discussed under some conditions concerning the first eigenvalue corresponding to a special linear operator, where I = [0, 1], η ∈ (0, 1), α, β, γ ∈ [0,∞) with β + γ 6= 0, f : [0, 1] × (−∞,+∞) → (−∞,+∞) is sign–changing continuous function and may be unbounded from below. By applyi...

Nontrivial Solutions for Nonlinear Problems with One Sided Resonance

We find nontrivial smooth solutions for nonlinear elliptic Dirichlet problems driven by the p-Laplacian (1 < p < ∞), when one sided resonance occurs at the principal spectral interval.

Existence of Positive Solutions for Nonlinear Eigenvalue Problems