Existence Of Multiple Solutions To (p,q)-Laplacian Equations With Nonlocal Terms

Nonlinear partial differential equations are important branch of modern mathematics.No matter in theoretical or practical application,they both have great significance and value,and they get extensive attention.Reaction diffusion equation is an important portion of the nonlinear partial differential equation,which has extensive applications in related sciences such as physics,chemical and biophysics.As the stationary version of a general reaction-diffusion equation.（p,q）-Laplacian equation received a large number research about the existence and multiplicity of its solution in recent years.In this paper,we use variational method,quantitative deformation lemma and topological degree theory to discuss the exis-tence of nonnegative solution,sign-changing solution and ground state solution for a kind of（p,q）-Laplacian equation under different conditions.In this paper,we consider the following（p,q）-Laplacian equation with nonlocal terms:（?）where 2 ≤ q<p<q*,N<2p,△m = div（｜▽u｜m-2▽u）is the m-Laplacian operator,m*= ∞for N≤m and m*= Nm/（N—m）for N>m.a,b are positive constants,c,d ≥ 0.h,g are continuous,coercive and positive functions.In the second chapter of the paper,in order to obtain the existence of nonnegative solution,we assume that f satisfies the following hypotheses:（f1）f is a Caratheodory function such that:f（x,0）= and F（x,t）>0 if t>0,where（?）（f2）存在s ∈[q.p*),k∈L+∞（RN）∩Ls/（s-1）（RN）and C>0 such that（?）（f3）lim｜t｜→∞F（x,t）/｜t｜2p = ∞,uniformly in x ∈ RN;（f4）there exists β ∈ L+（RN）such that（?）for all 0<t≤s或s≤t≤0,where σ（x,t）= f（x,t）t-2pF（x,t）;（f5）there exists l∈ L+∞（RN）such that（?）uniformly in x∈RN.Applying variational method,we get the main result as follow:Theorem 2.3.1.If f satisfies the conditions（f1）-（f5）,then the equation above has atleast one nonnegative solution.In the third chapter,in order to obtain the existence of sign-changing solution and ground state solution,we assume that f（x,u）= f（u）∈C1（R,R）satisfies the following hypotheses:（（?）1）lims→0f（s）/|s|q-1 = 0;（（?）2）for some constant r∈（2p,p*）,lim|s|→∞（?）（s）/|s|r-1=0;（（?）3）lim｜s｜→∞（?）（s）/｜s｜2p=∞,where F（s）=∫₀^s （?）（t）dt for all s∈R;（（?）4）f（s）/|s|2p-1 is increasing on（-∞,0）and（0,∞）respectively.Applying constraint variational method,quantitative deformation lemma and topolog-ical degree theory,we get three main theorems as follows:Theorem 3.3.1.Suppose the assumptions（f1）-（f4）hold.Then the equation above has one least energy sign-changing solution.Theorem 3.3.2.Suppose the assumptions（f1）-（f4）hold.For any sequence {（c_n,d_n）}with c_n,d_n≥0,as（c_n,d_n）→0,there exists a subsequence,still denoted by {（c_n,d_n）},such that uc_n,d_n→u0,where uc_n,d_n is a least energy sign-changing solution of the equation above with c_n,d_n replace c,d,and u0:=u0,0.Theorem 3.4.1.Suppose the assumptions（f1）-（f4）hold.（ⅰ）There exists a ground state solution v of the equation above.（ⅱ）m>2m,where m and m represent the energy of the sign-changing solution and the ground state solution obtained above respectively.In particular,the ground state solution must maintain the sign unchanged.