Positive Solutions Of Superlinear Kirchhoff-type And Asymptotically Linear Elliptic Equations

When studying the nonhomogeneous elliptic equations, many scholars assume that the nonlinearity f(x,t) satisfies the famous Ambrosetti-Rabinowitz condition, which is called (AR) condition for short.(AR) condition guarantees that the (PS) sequence of the studied elliptic equation is bounded. That is the important premise of variational methods. However, there are many functions in application which do not satisfy the (AR) condition. Then, our main work is applying a variant version of the Mountain Pass Theorem to study the existence of positive solutions of superlinear Kirchhoff-type and asymptotically linear elliptic equations without (AR) condition.(1) This paper discusses a superlinear Kirchhoff-type elliptic equation where Ω is a bounded domain in RN with smooth boundary (?)Ω. First, We prove that the energy functional I of the equation satisfies the Mountain Pass geometric properties and get the Cerami sequence of I. Second, We prove that the Cerami sequence of I is bounded. At last, We prove that the bounded Cerami sequence has a subsequence, which converges strongly to a positive solution of equation.(2) This paper discusses the asymptotically elliptic equationFirst, We apply Ekeland’s variational principle obtaining a bounded (PS) sequence of the energy functiona I and prove that the sequence converges to a positive solution of equation. Second, We prove that I satisfies the Mountain Pass geometric properties, then, we get the Cerami sequence of I and prove that it is bounded. Finally, We prove that the bounded Cerami sequence has a subsequence, which converges strongly to the other positive solution of equation.