带有凹非线性项的平均曲率半正问题正解的确切个数.pdf

2023,43A(5):13411349http:/k5?-?K?)?(op?M(=?X=730070):T?Minkowski m-K?u01 u02?0=f(u),x (L,L),u(L)=0=u(L)?)?(9,0,L 0,f C2(0,),R)v f(0)0,u 0 u L,f00(u)K?u01 u02?0=f(u),x (L,L),u(L)=0=u(L)(1.1)?)?(,0,L 0,f C2(0,),R)v f(0)0,u 0 u L,f00(u)0.,K(1.1)Minkowski-Dirichlet Kdiv vp1|v|2!+f(|x|,v)=0,x B(R);v=0,x B(R)?/,?K3Adk-1,4,11,12.Cc5,rr13,Huang6,7,p?M3?0,?z 6=,(z)f(z)0,K(1.1)?)?(-.?(J,?f(u)uLogistic.,k?L 0,-NO?;?f(u)uf Allee effect.,uv?L 0,-.S-.?5?,zvk?5 f?,f(0)0,K(1.1)?)?(9-.d?K(1.1)?)?(9-,5 fv/(H1)f C2(0,),R),f(0)0,u 0 u L,f00(u)0?0 u 0?p u L,f0(u)0;(H2)f C2(0,),R),f(0)0,u 0 u 0,f00(u)0.b?(C1)3 0 a k r p q,?0 u r,f(u)uf0(u)0?r u 0,f(a)=f(q)=0,k F(u)=Ru0f(t)dt?1?:;(C2)3 k r L,?0 u r,f(u)uf0(u)0?r u 0,k F(u)=Ru0f(t)dt?1?:.?(Jn 1.1b?5 f v(H1)(C1),K3 0?(i)?0 ,K(1.1)Tk?);(iii)?0?(i)?0 ,K(1.1)Tk?);(iii)?0 0,T(s)=Zs01+F(s)F(u)p1+F(s)F(u)2 1du,F(u)=Ru0f(t)dt.w,K(1.1)?)u Aukuk=s T(s)=L.?B,A=A(s,u)=sf(s)uf(u),B=B(s,u)=F(s)F(u),C=C(s,u)=s2f0(s)u2f0(u),KT(s)=Zs01+B2B2+2Bdu=sZ101+B(s,st)p2B2(s,st)+2B(s,st)dt.5 2.1?y?0 u 0.?f v(H1)(C1),k s q L;?f v(H2)(C2),k s 0,limsk+T0(s)=.y?0 u k,k F(u)0,g(k)=limukF(u)u(u k)=1kf(k)0,L?0 u k,g(u)0.d3 M 0,?0 u k,g(u)M.?0 t f(0)kt B(k,kt)=F(kt)=kt(kt k)g(kt)k2Mt(1 t).(2.1)1344n?Vol.43 Al(2.1)?T(k)=kZ101+B(k,kt)p2B2(k,kt)+2B(k,kt)dt kZ101 f(0)kp2B(k,kt)dt k1 f(0)k2k2MZ101pt(1 t)dt=1 f(0)k2M?Z1201pt(1 t)dt+Z1121pt(1 t)dt?1 f(0)k2M?2Z1201tdt+2Z11211 tdt?=221 f(0)kM 0,T(k)3.N?2B(s,u)A(s,u)=(s)(u),(u)=2F(u)uf(u).L(C1)(C2),?0(u)=f(u)uf0(u)0,u (r,L),00(u)=uf00(u)0,(0)=0,(k)=kf(k)0.(u)”2.u q d r k 0()u 2(u)?”5?3 d,?(d)=0?0 t 1,(k)(kt)0.d(2.1),3 t(0,1),?B(k,kt)f(0)kt 1?0 t t 1,2B2(k,kt)+2B(k,kt)(2+2)B(k,kt)(2+2)f(0)kt.?Z10(k)(kt)2B2(k,kt)+2B(k,kt)32dt Zt0(k)(kt)2B2(k,kt)+2B(k,kt)32dt(k)(kt)(2+2)f(0)k32Zt01t32dtNo.5o?:k5?-?K?)?(1345=R10B32(k,kt)+3B12(k,kt)dt .d,limsk+T0(s)=Z103B3(k,kt)+32B2(k,kt)+(k)(kt)2B2(k,kt)+2B(k,kt)32dtZ103B3(k,kt)+32B2(k,kt)(2B(k,kt)32dt+Z10(k)(kt)2B2(k,kt)+2B(k,kt)32dt=12232Z10B32(k,kt)+3B12(k,kt)dt+Z10(k)(kt)2B2(k,kt)+2B(k,kt)32dt=.y?.n 2.5b?5 f v(H1),(C1)k 0.y 1+B 12(2+B),KT00(s)=1s2Zs0B(3A2 BC 2AB)3+(3A2 4AB 2BC)2(2B2+2B)52du=1s2Zs032A2(1+B)2B(2+B)(2A+C)(2B2+2B)52du12s2Zs02(2+B)(3A2 4AB 2BC)(2B2+2B)52du=12s2Zs02(2+B)(3(A 2B)A+2AB 2BC)(2B2+2B)52du.-g1(u)=uf(u)2F(u),K?u (0,s),A 2B=g1(s)g1(u).d(C1)(C2),?u (0,r),g01(u)=uf0(u)f(u)0,L3(A 2B)A+2AB 2BC 6B(A 2B)+2AB 2BC=2B(4A 6B C).2-g2(u)=4uf(u)6F(u)u2f0(u),K?u (0,s),4A6BC=g2(s)g2(u).d(H1),(C1)(H2),(C2),?u (0,s),g02(u)=2uf0(u)2f(u)u2f00(u)0.d,T00(s)0.y?.n 2.6b?5 f v(H1),(C1)(H2),(C2),K?s (r,d),T00(s)+2sT0(s)0(?s (r,L),T00(s)+2sT0(s)0).y-(u)=u0(u)(u),?u 0,0(u)=u00(u)=u2f00(u)0,(0)=0 (s)(u)=2A 2B C.LO,T00(s)+2sT0(s)=1s2Zs032A2(1+B)+2B(2+B)2B(1+B)(2+B)4A C(2B2+2B)52du.1+B 12(2+B),32A2(1+B)+2B(2+B)2B(1+B)(2+B)4A C1346n?Vol.43 A122(2+B)3A2 8AB 2BC+8B2=122(2+B)3(s)(u)2+2B(s)(u).d,T00(s)+2sT0(s)12s2Zs03(s)(u)2+2B(s)(u)12B52(2+B)32du1s2Zs0(s)(u)12B32(2+B)32du.du?u (0,s),0(u)0.?s (r,d),T00(s)+2sT0(s)0.A?y,?s (r,L),T00(s)+2sT0(s)0.y?.n 2.7b?5 f v(H1)(C1),K?s (d,q),T0(s)0.y?s (d,q),k 2B(s,u)A(s,u)=(s)(u)0.?T0(s)=1sZs03B3+32B2+(2B A)(2B2+2B)32du=1sZs03B3(s,u)+32B2(s,u)+(s)(u)(2B2(s,u)+2B(s,u)32du 0.y?.n 2.8b?5 f v(H1)(C1),K?0,limsqT0(s)=.yd f(a)=f(q)=0,3 g3(u)C0,q),?f(u)=(u a)(q u)g3(u).Nwg3(a)=limuaf(u)(u a)(q u)=f0(a)q a 0,g3(q)=limuqf(u)(u a)(q u)=f0(q)q a 0,L?0 u q,g3(u)0.3 M1 0,?0 u q,0 g3(u)M1.?0 t 1,?B(q,qt)=Zqqtf(s)ds=qZ1tf(qv)dv=qZ1tq(qv a)(1 v)g3(qv)dv q2(q a)M1Z1t(1 v)dv=12q2(q a)M1(1 t)2.d,3 t(0,1),?t t 1,B(q,qt)0,K(1.1)?)?(?du T(s)3m k,q)?/G.Ln 2.3,lim0+T(s)=,limT(s)=s.LO,?s0 0,lim+T(s0)=s0,L3 s0 k,q),?lim+T(s0)0,?(0,),K(1.1);?(,),K(1.1)Tk);?(,K(1.1)Tk?).=L k 0 L()sTL qk=0L s s ()sT (a)(b)3(a)n 1.1 T(s)?”;(b)n 1.2 T(s)?”y?.eyn 1.2.ymN”?,?0,K(1.1)?)?(?du T(s)3m k,L)?/G.Ln 2.3,lim0+T(s)=,limT(s)=s.LO,?s0 0,lim+T(s0)=s0,L3 s0 k,L),?lim+T(s0)0,?(0,),K(1.1);?(,),K(1.1)Tk);?(,K(1.1)Tk?).y?.3nf 3.1f(u)=u2+4u 1.LO,f(0)=1,f0(u)=2u+4,f00(u)=2 0,?(0,),TK);?(,),TKTk);?(,TKTk?).3.2f(u)=u2+8u 9.LO,f(0)=9,f0(u)=2u+8,f00(u)=2 0,?(0,),TK);?(,),TKTk);?(,TKTk?).3.3f(u)=u 1.LO,f(0)=1,f0(u)=12u,f00(u)=14u32 0,?(0,),TK);?(,),TKTk);?(,TKTk?).z1 Bratu G.Sur les equation integrals non linWarires.Bull Soc Math France,1914,42:1131422 Corsato C.Mathematical Analysis of Some Differential Models Involving the Euclidean or the MinkowskiMean Curvature Operator.Trieste:University of Trieste,20153 Gao H L,Xu J.Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems withthe Minkowski-curvature equation.Boundary Value Problems,2021,Article number 814 Gelfand I M.Some problems in the theory of quasilinear equations.American Mathematical Society Trans-lations,1963,29(2):2953815 Huang S Y.Global bifurcation diagrams for Liouville-Bratu-Gelfand problem with Minkowski-curvatureoperator.Journal of Dynamics and Differential Equations,2021,34:215721726 Huang S Y.Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application.Communications on Pure and Applied Analysis,2018,17(3):127112947 Huang S Y.Classifification and evolution of bifurcation curves for the one-dimensional Minkowski-curvatureproblem and its applications.Journal of Differential Equations,2018,264(9):597760118 Huang S Y.Global bifurcation and exact multiplicity of positive solutions for the one-dimensionalMinkowski-curvature problem with sign-changing nonlinearity.Communications on Pure and Applied Anal-ysis,2019,18(6):326732849 Huang S Y.Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problemand its applications.Discrete and Continuous Dynamical Systems-A,2019,39(6):3443346210 Huang S Y,Hwang M S.Bifurcation curves of positive solutions for the Minkowski-curvature problem withcubic nonlinearity.Electronic Journal of Qualitative Theory of Differential Equations,2021,Article number4111 Hutten E H.Relativistic(non-linear)Oscillator.Nature,1965,205(4974):89212 Walter G.Classicai Mechanics-point Particles and Relativity.New York:Springer,2004:1485No.5o?:k5?-?K?)?(134913 Zhang X M,Feng M Q.Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensionalprescribed mean curvature equation in Minkowski space.Communications in Contemporary Mathematics,2019,21(3):111137Exact Multiplicity of Positive Solutions for a Semipositone MeanCurvature Problem with Concave NonlinearityLi XiaodongGao HongliangXu Jing(Department of Mathematics,Lanzhou Jiaotong University,Lanzhou 730070)Abstract:In this paper,we study the exact multiplicity and bifurcation diagrams of positive solutionsfor the prescribed mean curvature problem in one-dimensional Minkowski space in the form of?u01 u02?0=f(u),x (L,L),u(L)=0=u(L),where 0 is a bifurcation parameter and L 0 is an evolution parameters,f C2(0,),R)satisfiesf(0)0 and f is concave for 0 u L.In two different cases,we obtain that the above problem haszero,exactly one,or exactly two positive solutions according to different ranges of.The argumentsare based upon a detailed analysis of the time map.Key words:Minkowski space;Semipositone;Positive solution;Time map;Exact multiplicity.MR(2010)Subject Classification:34B15;34B18