Existence Of Infinite Solutions And Ground State Solution For A Class Of P-laplacian Equation With A Diffusion Term

The p-Laplacian equation is a very important nonlinear equation,which plays an impor-tant role in many branches of mathematical physics.Many nonlinear physical phenomena,such as nonlinear diffusion and filtration,non-Newtonian nows,elastic-plastic torsional creep and nows in porous media,can be described by p-Laplacian problems,thus,studying in which certainly has practical significance and application value.In this paper,we prove the exis-tence of infinite solutions and ground state solution of a class of p-Laplacian equation with a diffusion term by using variational methods,such as change of variables,the mountain pass theorem,cutoff techniques and Moser iteration.The thesis consists of three sections.In Chapter 1,we mainly introduces the research background and research status at home and abroad,besides,we give the research contents of this paper.In Chapter 2,we consider the following p-Laplacian equation with a diffusion termwhere △pu=div(|▽u|p-2▽u),N≥3,α∈[2,N),p∈[α,N),Ω(?)RN is a smooth bounded domain.The nonlinear term ∫∈C(Ω×R)satisfy the following hypotheses:(f1)there exists δ1>0 such that f(x,-t)=-f(x,t)for all(x,t)∈Ω×[-δ1,δ1];(f2)there exist δ2>0,r ∈(1,2)such that |f(x,t)|≤|t|r-1 for all(x,t)∈×[-δ2,δ2];(f3)limt→0 f(x,t)/|t|p-2t=∞ uniformly for x∈Ω.In this chapter,we prove that(0.3)has infinitely many solutions via truncation tech-nique and Moser iteration theory.Firstly,according to variable substitution and truncation technique,we convert the work space to Wo1,p(Ω)Secondly,we use Moser iteration theory to prove that vn ∈ L∞(Ω)and |vn|∞→0.Fnically,we prove(0.3)possesses infinitely many solutions.The conclusion is as follows:Theorem 2.1.1.Under the hypotheses(f1)-(f3).Then(0.3)admits a weak solution sequence {un} such that un→0,J(un)≤0.In Chapter 3,we consider the following p-Laplacian equation with a diffusion term where N≥3,β∈[2,N),p ∈[α,N),V ∈C(RN),f ∈ C(R).We assume that the potential function V and the nonlinear term f satisfy the following conditions:(V)V ∈C(RN)1-periodic in each of x1,x2,...,and there exists a constant V0>0 such that V(x)≥VO for all x ∈RN;(f1)there exist constants CO>0 and r ∈(αp,αp*)such that|f(t)|≤C0(1+|t|r-1),t ∈R,where p*=Np/(N-p);(f2)limt>0 f(t)/|t|p-2t=0;(f3)limt-∞ F(t)/|t|βp=∞,where F(t)=f0t f(s)ds,t ∈R;(f4)t→f(t)/|t|αp-2t is positive for t≠0,nonincreasing on(-∞,0)and non decreasing on(0,∞).In this chapter,our purpose is to prove the existence of ground state solutions on Nehari manifold for(0.4).However,we do not know whether the Nehari manifold is of class Cl under our assumptions.Therefore the problem becomes relatively difficult.In order to overcome difficulties,we explain the existence of Cerami sequence by using the mountain pass lemma,and then we prove that the minimum on Nehari manifold of energy functional is achieved by proving c=c1 and c1 is achieved.The conclusion is as follows:Theorem 3.1.1.Assume that(V)and(f1)-(f4)are satisfied.Then(0.4)has at least a ground state solution w ∈ M such that I(w)=infMI.