The Existence Of Solutions To Semilinear Elliptic And Kirchhoff-type Equations

When studying the nonlinear differential equations, many scholars assume that the nonlinearity f(x,t) satisfies the famous Ambrosetti-Rabinowitz condition, which iscalled (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 Mountain Pass Lemma without (PS) condition and a variant version of the Mountain Pass Theorem to study the existence of solutions of Semilinear elliptic and Kirchhoff-type equations.(1) This paper discusses Semilinear elliptic equation Where Q is a bounded domain in RN with smooth boundary (?)?. First, We prove that the energy functional I of the equation satisfies the Mountain Pass geometric properties and get the (PS) sequence of I. Second, We prove that the (PS) sequence of I is bounded. Third, We prove that the bounded (PS) sequence has a subsequence, then we obtain the existence theorem of the equation. The Last, by using the parity of f(x,t), we will discuss the number of the solutions.(2) This paper discusses the Kirchhoff-type equation First, We improve that if a sequence satisfies the (Ce) condition, then it is bounded. Then we improve the (Ce) sequence has a strong convergence sequence, it converges to a point. Then by using the variational Mountain Pass theorem We prove that the energy functional I of the equation satisfies the Mountain Pass geometric properties and get the (Ce) sequence of I. Finally, according to the conclusions previously obtained, we can know that the point which the (Ce) sequence has converged is a solution of the discussed equation.