Nodal solutions for some semipositone problemsvia bifurcation theory

Abstract

We show the existence of nodal solutions of the second-order nonlinear boundary value problem

$$\begin{array}{l}-{u}^{^{\prime\prime} }\left(x\right)=\lambda \left(g\left(u\left(x\right)\right)+p\left(x,u\left(x\right),{u}^{\mathrm{^{\prime}}}\left(x\right)\right)\right),x\in \left(\mathrm{0,1}\right),\\ u\left(0\right)=u\left(1\right)=0,\end{array} ({\text{P}})$$

where λ > 0 is a parameter, p : [0, 1]×ℝ² → ℝ and g : ℝ →ℝ are continuous functions, and g(0) = 0. For a nonnegative integer k, we say that a solution is nodal if it has only simple zeros in (0, 1) and has exactly k-1 such zeros. Under some suitable conditions, we obtain that there exists λ_∗ > 0 (or λ^∗ > 0) such that for fixed k ∈ {1, 2,…}, problem (P) has at least one nodal solution for λ ∈ (k²π²/g_∞, λ^∗) (or λ ∈ (λ^∗, k²π²/g_∞)), where g_∞ = lim_|s|→∞ g(s)/s. The proof of our main results relies on the bifurcation technique.