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-(ϕp(u′(x)))′=λf(u(x)),x∈(0,1),u′(0)=0,u′(1)+αu(1)=0,whereα ∈ R, λ > 0 are parameters and p > 1, and p′=pp-1 is the conjugate exponent of p and ϕp(x): = ∣x∣p−2x for all x ∈ R where (ϕp(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.