Existence Of Infinitely Many Solutions To (2,p)-Laplacian Equations

As an important branch of mathematics,nonlinear partial differential equation has a wide range of applications,and always promotes the development of mathematics.(2,p)-Laplacian equation is one of the important equations,which has a wide range of applications in many mathematical physical models,such as in elasticity theory,reaction diffusion system and quantum physics,etc.In recent years,the existence and multiplicity of solutions of(2,p)-Laplacian equation have been widely concerned by scholars.The thesis consists of three sections,multiplicity of solutions of(2,p)-Laplacian equation is discussed by means of variational method,the generalized Clark theorem,truncation technique and so on.Chapter 1 is the preface,which mainly describes the latest research progress of the(2,p)-Laplacian equation,and introduces the research content,research ideas and main conclusions of this paper.Chapter 2 is one of the main contents,we consider the following(2,p)-Laplacian equa-tions where Ω(?)RN is a bounded domain with a smooth boundary,Δpu=div(|▽u|p-2▽u)is the p-Laplacian operator,2<p<N,f∈C(R,R).the nonlinear term.f also needs to meet the following condition:(f0)f is an odd function near zero and there exist q∈(1,2)such that Using the generalized Clark theorem,the truncation technique and the L∞ estimation of the solution,the main result we obtain is that if the assumption(f0)is holds,then there are infinitely many solutions to the above equation.Chapter 3 is also one of the main contents,we consider the following(2,p)-Laplacian equations with non-odd perturbation termswhere Ω(?)RN is a bounded domain with a smooth boundary,1<q<2<p<N and p+N(p-q)/q<p*:=Np/(N-p),a∈C(Ω)can change sign,and f need not be odd in u.Suppose a and f meet the following conditions(A)a∈C(Ω)and there exists x0∈Ω such that a(x0)>0(F)f∈C(Ω×(-δ,δ)),where δ>0,and there exists r∈(p+N(p-q)/q,p*)such that Using variational method,we obtain a sequence of solutions {un} to the above equation,satisfying ‖un‖→0,|un|∞→0,n→∞;Besides,J(un)<0 and J(un)→0,n→∞,wherewith F(x,t)=∫0 t f(x,s)ds,(x,t)∈Ω×R.