We study the existence of an intermediate solution of nonlinear elliptic boundary value problems (BVP) of the form $$ (BVP) {Delta u = f(x,u,Delta u), in Omega {Bu(x) = phi(x), on partialOmega, $$ where $Omega$ is a smooth bounded domain in $R^n, n geq 1, and partialOmega in C^{2,alpha}, (0 < alpha < 1), Delta$ is the Laplacian operator, $ abla u = (D_1u, D_2u, cdots, D_nu)$ denotes the gradient of u and $$ Bu(x) = p(x)u(x) + q(x)frac{d u}{du} (x), $$ where $frac{d u}{du} denotes the outward normal derivative of u on $partialOmega$.