Existence Of Solutions For Nonlinear Elliptic Equation With Hardy Potential And Critical Exponents

As the nonlinear differential equation is playing an important role in many practical applications, it has been studied more and more recently. And at the beginning times, people usually use Ambrosetti-Rabinowitz condition, in short AR condition, to prove Mountain Pass Lemma, whose aim is to solve the existence of the solutions for the nonlinear differential equations. As time passes, people try to weaken, or even to remove AR condition to get the solutions for the nonlinear differential equations, which is the main work of this articale either. The first step is to transforme the solution of Equation to find the critical point. Finally, we use Mountian Pass Lemma to get the existence of solutions for the nonlinear differential equations.The main work of the first chapter outlines the status of such researches.The main work of the second chapter is to discuss the solutions of the p-Laplace equation That is to transforme the solution of the p-Laplace equation to find the critical point, using the given conditions to prove PS condition and Mountain Pass Lemma in order to get the solutions for the equation above.The main work of the third chapter is to discuss the solutions of the elliptic equation That is to transforme the solution of the elliptic equation to find the critical point, using the given conditions to prove PS condition and Mountain Pass Lemma and Ekeland variational principle and minmax principle to get the solutions for the equation above.