Existence And Multiplicity Of Solutions For A Class Of P-Laplacian Equations

In this paper,firstly,we consider the following p-Laplacain equations with Dirichletboundary value condition:(1)whereâ–³_pu=div(|â–½u|^p-2â–½u),is the p-Laplacian operator with 1N (N≥1)with smooth boundary (?)Ω,λ>0 is a real parameter f:(?)×R→'R is a continuous function.We will proof the existence of positive solution and ground statesolution of (1)by using the Mountain Pass Theorem.The main result is the following theoremTheorem 1 Suppose that the function f satisfies the following assumptions(f₁)f∈C((?)×R,R),f(x,t≥0 for all t∈R,x∈(?)ï¼›(f₂)f(x,t)=o(t^p-1 as t→0⁺ uniformly for a.e.x∈(?);(f₃)there exists q>0 if N≤p,pp such that(?)f(x,t)/t^q=0,uniformly for a.e x∈(?);(f₄)there existθ>p,r>0 such that0<θF(x,t)≤tf(x,t),(?)(x,t)∈(?)×R⁺,t≥r,where F(x,t)=∫₀^tf(x,s)ds.Then problem (1)has a positive solution for everyλ>0.Theorem 2 Assume that f satisfies the following conditions (f₁^*)f∈C((?)×R,R),f(x,t)t≥0 for all t∈R,x∈(?);(f₂^*)f(x,t)o(|t|^p-2t)as|t|→0 uniformly for a.e.x∈(?);(f₃^*)there exists q>0 if N≤p,p*-1,where p^*=pN/N-p,if N>p such that(?)f(x,t)/|t|^q=0,uniformly for a.e x∈(?);(f₄^*)there existθ>p,r>0 such that 0<θF(x,t)≤tf(x,t),(?)(x,t)∈(?)×R,|t|≥r.Then problem (1)has at least two nontrivial solutions for everyλ>0 in which one ispositive and the other is negative.Next we consider the existence of ground state solution for the following p-Laplacianequation in R^N.-â–³_pu+|u|^p-2u=f(u),u∈W^1,p(R^N) (2)where 15)f(t)=o(|t|^p-2t)as|t|→0;(f₆)there exists p*=p^*(?)p^N/(N-p)such that(?)f(t)/|t|^q-2t=0;(f₇)the function t→f(t)/|t|^p-1 is increasing in t∈R\{0};(f₈)(?)F(t)/|t|^p=+∞;(f₉)there existμ>N/p(q-p)and a constant a>0 such that f(t)t-pF(t)≥a|t|^μ>0 for all t∈R\{0}.Then problem (2)has a ground state solution.At last,we consider the following p-Laplacian equation in R^N-â–³_pu+V(x)|u|^p-2u=f(u),u∈W^1,p(R^N) (3)where 1N,R).We get positive solution,multiple solution and ground state solution of (3)under conditions that around 0and at∞are required and using the Mountain Pass Theorem without (PS)condition.The main results are the following:Theorem 4 Assume that f and V satisfy the following conditions (f₁₀)f(t)≥0 for all t>0 and f(t)≡0 for all t≤0;(f₁₁) there is q<∞if N=p,q*-1 where p^*=Np/N-p if N>p such that(?)f(t)/t^q=0(f₁₂)(?)f(t)/t^p-1=a,where a≥0;(f₁₃)(?)f(t)/t(p-1)=+∞;(v₁)a<α₀,whereα₀=(?)∫R^N(|â–½u|^p+V(x)|u|^p)dx/∫R^N|u|^pdx;(v₂)0＜(?)V(x)≤V(x)≤(?)V(x)=V(∞)＜+∞;(v₃) there exists a function (?)∈L²(R^N)∩W^1,∞(R^N) such that|x||â–½V(x)|≤(?)(x)²,(?)_x∈R^N.Then the problem (3) has a non trivial positive solution.Theorem 5 Under the assumption of Theorem 4, the problem (3) has a ground statesolution. Namely, there exists a solution w∈W^1,p(R^N) such that I(w)=m, where m= inf{I(u)|u≠0,I′(u)=0}.