Multiple positive radial solutions to some Kirchhoff equations

In this paper, we discuss the existence and multiplicity of positive radial solutions to some Kirchhoff equations and elliptic equations. By using the Leggett–Williams three solutions theorem skillfully, we obtain two positive solutions to Kirchhoff equations under some appropriate conditions. As a direct corollary, we also obtain the same result to elliptic equations. In order to facilitate the proof of main results, we first establish a new simple three solutions theorems for the equation [a+bφ(x)]x=Ax[a+bφ(x)]x=Ax, where A   is a completely continuous operator, a>0a>0, b⩾0b⩾0 and φ(x)⩽k(‖x‖)φ(x)⩽k(‖x‖) for all x∈Px∈P, where k is a nondecreasing nonnegative continuous function, then apply it to prove the main results. In addition, we discover and prove a new inequality in the article. In the end of the paper, we present an example which makes the elliptic equation has infinite many positive radial solutions and give the approximate images of the nonlinear term.