Multiple existence and structure of solutions for semilinear elliptic equations.

我们发现了一个与对称山路引理有关的新的临界点定理。我们的定理断言Banach空间上的偶泛函有一列收敛到零的临界点。将其应用于一个次线性椭圆型方程，在一个很弱的条件下证明了无穷多个解的存在性，当非线性项不为奇时，证明了一个次线性椭圆型方程有无穷多个解.通过将椭圆型方程的拉格朗日泛函看作是偶泛函的扰动，利用泛函的对称性证明了解的存在性。研究了一类超线性椭圆型方程多解的存在性。然而，除了我们的结果外，关于次线性椭圆型方程的多解问题还知之甚少。我们证明了具有奇异系数的一维p-Laplace方程解的正则性。利用它，我们给出了特征值存在的一个充分必要条件。然后我们研究了一维p-Laplace方程解的分支结构。利用解的零点个数研究了分歧分支的方向，证明了分歧曲线的整体存在性。

We discover a new critical point theorem related to the symmetric mountain pass lemma. Our theorem asserts that an even functional on a Banach space has a sequence of critical points converging to zero. By applying it to a sublinear elliptic equation, we prove the existence of infinitely many solutions under a very weak condition.When the, nonlinear term is not odd, we prove that a sublinear elliptic equation has infinitely many solutions. By considering the Lagrangian functional associated with the elliptic equation as a perturbation from an even functional, we use the symmetry of the functional to prove the existence of solutions. The existence of multiple solutions has been studied for the superlinear elliptic equations. However, little is known about the multiple solutions of the sublinear elliptic equations except for our results.We prove the regularity of solutions for one-dimensional p-Laplace equations with singular coefficients. By using it, we give a necessary and sufficient condition for the existence of eigenvalues. Then we investigate the structure of the bifurcation of solutions for the one-dimensional p-Laplace equations. Moreover, by using the number of zeros of solutions, we study the direction of the bifurcation branch and show the global existence of the bifurcation curve.