The Existence Of Multiplicity Solutions For P-Laplacian Function

In this paper we present a result of existence of infinitely many arbitrarily small positive solutions to the following Dirichlet problem: In (0.3),whereΩ∈RN is a bounded open set with sufficiently smooth bounded (?)Ω, p, q> 1,λ> 0, and F:Ω×R×R→R is a measurable function fromΩ×R×R to R which is Lipschitz continuous with respect to the second and third variable. F:Ω×R×R→is the partial derivative with respect to u, Fv is the partial derivative with respect to v.There exist a>0 and t>0, such that for every (u,v)∈[0,t]×[0.t] and for a.e. x∈Ω,one has |F(x,u,v)|≤a(|u|+|v|) and F(x,0,0)= F(x,u,0)= F(x,0,v)= Fu(x,0,0)=Fu(x,0,0)= 0 for every u, v∈R.Precisely, our result ensure the existence of a sequence of non-zero and non-negative weak solutions to the above problem..In (0.4),whereΩ∈RN is a bounded open set with sufficiently smooth bounded (?)Ω, p> n,λ>0,and f:Ω×R→R is a Caratheodory function satisfying the following condition:There exists t>0 such that Precisely,our result ensure the existence of at least three weak solutions to the above problem.