Existence and multiplicity results for a class of p-Laplacian problems with Neumann–Robin boundary conditions

In this paper, we study the following Neumann–Robin boundary value problem whereα R, λ > 0 are parameters and p > 1, and is the conjugate exponent of p and p(x): = x p−2x for all x R where ( p(u′))′ is the one dimensional p-Laplacian and f C2[0, ∞) such that f(0) < 0, or f(0) > 0, and also f is increasing and concave up. We shall investigate the existence and multiplicity of nonnegative solutions. Note that in this paper, we shall establish our existence results by using the quadrature method.