Multiple existence and structure of solutions for semilinear elliptic equations.

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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.