Study On Solutions For Some Nonlinear Partial Differential Equations

The famous Bernstein Theorem led mathematician De Giorgi formulate in 1978 the celebrated conjecture. i.e., is the bounded and monotone solution for the equationΔu - u + u3 = 0 one dimension symmitric? From then on, many scholars tried to work out this problem. In 1998, Goussoub and Gui first validated the De Giorgi conjecture in dimension two. Namely, the bounded and monotone solution to the Allen-Cahn equation has one dimensional symmetry. Later, Berestycki, Caffarelli,Nirenberg proved it also. And then Ambrosio, Cabre proved it true for n = 3. In 2003, Savin proved it for 4≤n≤8 with some additional condition. In the subsequent years, the conjecture attracted more and more attention and was developed. For example, scolars tried to find the counter-example for De Giorgi conjecture and the saddle solution was considered as the condidate solution in high dimension. At the same time, the Allen-Cahn equation with constant coefficient was generalized to equations with variable coefficient, and to vector equations. Recently, it was extended to p-Laplacian equations with p> 1 and fractional Laplacian equations with 0< p< 1. The existence, symmetry and regularity of the solutions were studied extensively. In this paper, we study several calss of p-Laplacian equations and get some properties of the equation. The present papre is organized as follows:In Chapter one, the background and progress on De Giorgi conjecture are intro-duced. And we emphasize several important works and retell the shining points in them. Then introduce the trends to the problem briefly. In the last we list some preparations which will be usd in the following.In Chapter two, Since Cabre and Terra considered the saddle type solution as the counter-example for De Giorgi conjecture, and Danielli, Garofalo studied the existence of one dimension symmetric solution for the autonomous p-Laplace equation. Here we investigate a class of p-Laplacian equation with constant coefficient. By variational method, we get the existence of multi saddle solutions.In Chapter three, we study a class of p-Laplacian equation with variable coefficient. Where the coefficient a(x) which is periodic depends on one of the two variables, and u converges to±1 uniformly on y when x→±∞. Since the one dimension symmetric solution is certainly the solution, I wonder is there any other type solution. By exploring the distance of the set of the ODE solutions corresponding to the p-Laplacian equation, we get the existence of more than two solutions, where u(·,±∞) is the solution of the ODE. In Chapter four, Since Alessio, Jeanjean and Montecchiari studied a class of equa-tion-Δu+a(x)W'(u) = 0, (x,y)∈R2, and got some layered solution. Here, we deeply discussed the same equation, by constucting suitable admitted set, we get in-finitely many Multi-bump solutions. Namely, u(·,y) alternates between two sets when y→∞.In the last chapter, we consider the p-Laplacian equation-Δu+a(x, y)W'(u) = 0, (x,y)∈R2, where a(x,y) is periodic and symmetric with y. By the condition a(x, y)= a(x, -y) we know that u is periodic on y, and we get the entire solution by extending periodically. Since a(x, y) is not constant and depends on y, so the entire solution is not one dimension symmetric.