Study On Solutions To Degenerate Elliptic Equations

In this thesis, we first study saddle solutions to a class of nonlinear elliptic equation involving the p-Laplacian in the whole space. We get the existence of saddle solutions to this class elliptic equation in spaces R2m(2m> p>2), by using sub-supersolutions method and monotone iteration method. We also consider the existence of maximal and minimal saddle solutions, and obtain that there exist maximal and minimal saddle solutions in any saddle solutions, namely, so-called quasi-maximal and quasi-minimal saddle solutions. Further more, we also prove that there exist strict maximal saddle solution among any bounded weak solutions (not necessary to be saddle solutions), which vanish in the Simons cone C and with the same sign as s—t, by applying monotone iteration method. We also obtain some convexity properties of saddle solutions to bistable diffusion equations, by applying a Maximum principle to the saddle solutions’derivatives of second order. Combining these convexity properties with some symmetry of the saddle solutions, we establish monotonicity and convexity properties of level sets of these saddle solutions in R2. Finally, we obtain existence and uniqueness of very weak solutions to a class of degenerate elliptic equations with a linear weighted Neumann boundary conditions.In chapter one, the background and main results are briefly presented. Several notations and some basic theorems used in the thesis, and the outline of this work are also given in this chapter.In chapter two, we study saddle solutions to a class of nonlinear elliptic equa-tion involving the p-Laplacian in the whole space△pu=f(u), x∈R2m. We obtain the existence of saddle solutions to this equation in more even-dimensional spaces R2m(2m≥p>2)(the existence of saddle solutions to this equation in spaces R2m(2m≥2p>4) had been obtained in [39].), by applying sub-supersolutions method. To this end, we need to introduce the notation of semi-stability of bound-ed weak solutions to the studied problem in domain Q(bounded or unbounded). Then we prove that there exist maximal and minimal saddles among any saddle solutions, namely, so-called quasi-maximal and quasi-minimal saddle solutions. It is import to establish that positive solutions is semi-stability in some domain and get appropriate supersolution and subsolution. Moreover, we need to show that these solutions are nontrival(not identically equal to constant). Duing the argu-ment we use the fact that the solution to the corresponding ODE is a supersolution of the previous problem in the Simons cone.In chapter three, we first define uR,0. Then we get the fact that the sequence solutions{uR:k} are monotone and bounded. Finally, we show that the limit of UR,k is actually the solution we are willing to established. To sum, we obtain the existence of strict maximal saddle solution among any bounded weak solutions (not necessary to be saddle solutions), which vanish in the Simons cone C and with the same sign as s—t.In chapter four, we consider the following bistable diffusion equations-△u(x)=f(u(x)), x∈R2m We obtain some convexity properties of saddle solutions u to these equations, by applying a maximum principle to the derivative of second order utt. More precisely, we obtain the results that the sign of utt in H={s> t} is negative and the sign of uss in{t> s} is positive. Combining the fat that the sign of utt in H={s> t} is negative with some symmetry of the saddle solutions, we establish monotonicity and convexity properties of level sets of these saddle solutions in R2.In chapter five, we obtain existence, uniqueness and an estimate of very weak solutions to the degenerate elliptic equation with a linear weighted Neumann boundary condition. where a∈(-1,1), and=-limI|1_>0+<^.