The Existence Of Nontrivial Solutions To Quasilinear Elliptic Equations

In this paper, we studied the existence of nontrivial solutions to quasilinearelliptic equations by the variational methods and critical theory. By making d-iferent hypotheses to the nonlinear term f(x, u), we discussed respectively theexistence of nontrivial solutions to the elliptic equation under the superlinear con-dition and the asymptotically linear condition, and we also obtained the existenceof positive and negative solutions and infinitely many solutions to the equations.In detail, this paper has been divided into the following three chapters:Chapter1We mainly introduced the writing background, and some pre-liminary knowledge and symbols which will be used in the process of the proofs.Chapter2Under the superlinear condition, we studied the existence ofthe solution to the elliptic equation. By using the symmetric mountain passtheorem, we obtained infinitely many solutions to the equation.Chapter3Under the asymptotically linear condition, we studied the ex-istence of the solution to the elliptic equation, and obtained the positive andnegative solutions by the truncation method.