某些非线性椭圆型方程的多重正解
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摘要
这篇论文主要研究三类非线性椭圆方程的多解性问题.本文由四章组成.
第一章,阐述论文的研究背景和我们所得到新的结果.
第二章,研究一类半线性Schr(o|¨)dinger方程多包解的存在性,其中N≥1,20是一个参数.设函数a满足下面的条件:a∈C(R~N),a(x)>0,(?)x∈R~N,当|x|→∞时,a(x)=o(1),ln(a(x))=o(|x|).对于任意正整数n,我们证出存在ε(n)>0使得对于0<ε<ε(n),方程存在一个正的n-包解.从而当ε→0时,方程有越来越多的正的多包解.
第三章,设ε>0是一个很小的参数.我们研究一类半线性Schr(o|¨)dinger方程多个正的多包解的存在性,其中N≥1,20,(?)x∈R~N,lim_(|x|→∞)a(x)=0.本章也研究如下指定纯量曲率方程多个证得多塔解的存在性,其中N≥3,K∈C([0,∞])满足K(r)>0,r>0,lim_(r→0)K(r)=0,lim_(r→∞)K(r)=0.
我们还考虑更一般的方程以及另一个相关的方程其中N≥3,q>1.我们要强调的是我们没有对q的上界做任务限制.
第四章,我们研究如下Caffarelli-Kohn-Nirenberg型临界椭圆方程多个正的多泡解的存在性.其中N≥3,a,b,p,v满足适当的条件,K∈C(R~N),K(x)>0,(?)x∈R~N,lim_(|x|→∞)K(x)=0,lim_(|x|→∞)K(x)=0,ε>0是一个小参数.
This paper mainly studies the existence of multiple solutions for three types of nonlinear elliptic equations. The thesis consists of four chapters.
In chapter one, we introduce some research background and our new results.
In chapter two, we study the existence of multi-bump solutions for the semilinear Schr(o|¨)dinger equationwhere N≥1,2 < p < 2~*,2~* is the critical Sobolev exponent defined by 2~* =(?) if N≥3 and 2~* =∞if N = 1 or N = 2, andε> 0 is a parameter. The function a is assumed to satisfy the following conditions: a∈C(R~N),a(x) > 0 in R~N,a(x) =o(1) and ln(a(x)) =o(|x|) as |x|→∞.For any positive integer n, we prove that there existsε(n) > 0 such that, for 0 <ε<ε(n), the equation has an n-bump positive solution. Therefore, the equation has more and more multi-bump positive solutions asε→0.
In chapter three, letε> 0 be a small parameter. We study existence of multiple multi-bump positive solutions for the semilinear Schr(o|¨)dinger equationwhere N≥1,2
第一章,阐述论文的研究背景和我们所得到新的结果.
第二章,研究一类半线性Schr(o|¨)dinger方程多包解的存在性,其中N≥1,2
第三章,设ε>0是一个很小的参数.我们研究一类半线性Schr(o|¨)dinger方程多个正的多包解的存在性,其中N≥1,2
我们还考虑更一般的方程以及另一个相关的方程其中N≥3,q>1.我们要强调的是我们没有对q的上界做任务限制.
第四章,我们研究如下Caffarelli-Kohn-Nirenberg型临界椭圆方程多个正的多泡解的存在性.其中N≥3,a,b,p,v满足适当的条件,K∈C(R~N),K(x)>0,(?)x∈R~N,lim_(|x|→∞)K(x)=0,lim_(|x|→∞)K(x)=0,ε>0是一个小参数.
This paper mainly studies the existence of multiple solutions for three types of nonlinear elliptic equations. The thesis consists of four chapters.
In chapter one, we introduce some research background and our new results.
In chapter two, we study the existence of multi-bump solutions for the semilinear Schr(o|¨)dinger equationwhere N≥1,2 < p < 2~*,2~* is the critical Sobolev exponent defined by 2~* =(?) if N≥3 and 2~* =∞if N = 1 or N = 2, andε> 0 is a parameter. The function a is assumed to satisfy the following conditions: a∈C(R~N),a(x) > 0 in R~N,a(x) =o(1) and ln(a(x)) =o(|x|) as |x|→∞.For any positive integer n, we prove that there existsε(n) > 0 such that, for 0 <ε<ε(n), the equation has an n-bump positive solution. Therefore, the equation has more and more multi-bump positive solutions asε→0.
In chapter three, letε> 0 be a small parameter. We study existence of multiple multi-bump positive solutions for the semilinear Schr(o|¨)dinger equationwhere N≥1,2
0 for x∈R~N,and lim_(|x|→∞) a(x) = 0.We also study existence of multiple multi-tower positive solutions for the prescribed scalar curvature equationwhere N≥3, K∈C([0,∞)), K(r) > 0 for r > 0, lim_(r→0) K(r) = 0, and lim_(r→∞) K(r) = 0.
We also consider the more general equationand another related equation iswhere N≥3 and q > 1.We would like to emphasize that we do not impose any upper bound on q.
In chapter four, the goal of this paper is to establish the existence of multiplicity of multi-bubble positive solutions for the following critical elliptic equations of Caffarelli-Kohn-Nirenberg type Where N≥3, a,b,p,v satisfy some conditions, K∈C(R~N),K(x) > 0 for x∈R~N, lim_(|x|→∞) K(x) = 0,lim_(|x|→∞)K(x) = 0 andε> 0 is a small parameter.
引文
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[3] N. Ackermann and T. Weth, Multibump solutions of nonlinear periodic schrodinger equations in a degenerate setting, Comm. Contemp. Math., 7 (2005), 269-298.
[4] S. Alama and Y. Y. Li, On "multibump" bound states for certain semilinear elliptic equations,Indiana Univ. Math. J., 41(1992), 983-1026.
[5] H. Amann, Saddle points and multiple solutions of differential equations, Math. Z. 169 (1979),122-166.
[6] H. Amann and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems,Manuscripta Math., 32 (1980), 149-189.
[7] A. Ambrosetti and M. Badiale, Homoclinics: Poincar(?)-Melnikov type results via a variational approach, Ann.Inst. H. Poincar(?) Anal. Non Lineaire, 15 (1998), 233-252.
[8] A. Ambrosetti and M. Badiale, Variational perturbative methods and bifurcation of bound states from the essential spectrum, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1131-1161.
[9] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical States of Nonlinear Schrodinger Equations, Arch. Rational Mech. Anal., 140 (1997), 285-300.
[10] A. Ambrosetti, J. Garcia Azorero and I. Peral, Perturbation of Δu+u~((?)) = 0 the scalar curvature problem in R~N,and related topics, J. Funct. Anal., 165 (1999),117-149.
[11] A. Ambrosetti, J. Garcia Azorero and I. Peral, Remarks on a class of semilinear elliptic equations on R~N via perturbation methods, Adv. Nonlinear Stud.,1(2001),1-13.
[12] A. Ambrosetti and A. Malchiodi, On the symmetric scalar curvature problem on S~n, J.Differential Equations, 170 (2001), 228-245.
[13] A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problem on R~N,Progress in Mathematics, 240, Birkhauser, Boston, 2005.
[14] A. Ambrosetti, A. Malchiodi and W. -M. Ni, Singularly perturbed elliptic equations with symmetry:existence of solutions concentrating on spheres, Part Ⅰ, Commun. Math. Phys., 235 (2003), 427-466.
[15] A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrodinger equations with potentials, Arch. Rational Mech. Anal., 159 (2001), 253-271.
[16] T. Aubin, Problemes isoperimetriques espaces de Sobolev, J. Diff. Geom.,11(1976), 573-598.
[17] A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations on unbounded domains, Ann. Inst. H. Poincar(?) Anal. Non Lin(?)aire,14(1997), 365-413.
[18] H. Berestycki and P. L. Lions, Nonlinear scalar field equations:Ⅰ,Ⅱ,Arch. Rational Mech. Anal.,82(1983), 313-375.
[19] I. M. Besieris, Solitons in randomly inhomogeous medea, Nonlinear Electromagnetics, 87-116,Academic Press, New York,1980.
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[21] J. Byeon and Y. Oshita, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrodinger equations, Comm. Partial Differential Equations, 29 (2004), 1877-1904.
[22] J. Byeon and Z. -Q. Wang, Standing waves with a critical frequency for nonlinear Schrodinger equations, Arch. Rational Mech. Anal., 165 (2002), 295-316.
[23] J. Byeon and Z. -Q. Wang, Standing waves with a critical frequency for nonlinear Schrodinger equations,Ⅱ,Calc.Var. PDEs, 18 (2003), 207-219.
[24] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
[25] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.
[26] D. M. Cao and E. S. Noussair, Multi-bump standing waves with a critical frequency for nonlinear Schrodinger equations, J. Differential Equations, 203 (2004), 292-312.
[27] D. M. Cao, E. S. Noussair and S. Yan, On the scalar curvature equation -Δu-(1+εK)u~((?)) in R~N,Calc. Var. PDEs, 15 (2002), 403-419.
[28] D. M. Cao and S. J. Peng, Multi-bump bound states of Schrodinger equations with a critical frequency, Math. Ann., 336 (2006), 925-948.
[29] F. Catrina and Z. -Q. Wang, On the Caffarelli-Nirenberg inequalities: Sharp constants, existence (and nonexistence) and symmetry of extremal functions, Comm. Pure Appl. Math., LIV (2001),229-258.
[30] F. Catrina and Z. -Q. Wang, Positive bound states having prescribed symmetry for a class of nonlinear elliptic equations in R~N,Ann.Inst. H. Poincar(?) Anal. Non Lin(?)aire, 18 (2001), 157-178.
[31] G. Cerami, G. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. PDEs, 23 (2005), 139-168.
[32] G. Cerami and D. Passaseo, Existence and multiplicity results for semilinear elliptic Dirichlet problems in exterior domains, Nonlinear Analysis TMA, 24 (1995), 1533-1547.
[33] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkh(a|¨)user(1993).
[34] K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J.London Math. Soc., 48 (1993), 137-151.
[35] V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer.Math. Soc., 4 (1991), 693-727.
[36] V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on R~N,Comm.Pure Appl.Math., XLV (1992), 1217-1269.
[37] T. D'Aprile and J. C. Wei, Standing waves in the Maxwell-Schr(o|¨)dinger equation and an optimal configuration problem, Calc.Van PDEs,25 (2005), 105-137.
[38] M. del Pino and P. Felmer,Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. PDEs, 4 (1996), 121-137.
[39] M. del Pino and P. Felmer, Semi-classical states for Nonlinear Schrodinger equations, J. Funct.Anal., 149(1997), 245-265.
[40] M. del Pino and P. Felmer, Multi-peak bound states of nonlinear Schrodinger equations, Ann.Inst.H. Poincar(?) Anal. Non Lineaire, 15 (1998), 127-149.
[41] M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrodinger equations: A variational reduction method, Math. Ann., 324 (2002),1-32.
[42] M. del Pino, M. Kowalczyk and J. C. Wei, Concentration on curves for nonlinear Schrodinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.
[43] V. Felli and A. Pistoia, Existence of blowing-up solutions for a nonlinear elliptic equation with hardy potential and critical growth, Comm. Partial Differential Equations, 31 (2006), 21-56.
[44] V. Felli and M. Schneider, A note on regularity of solutions to degenerate elliptic equations of Caffarelli-Kohn-Nirenberg type, Adv. Nonlinear Stud., 3 (2003), 431-443.
[45] V. Felli and M. Schneider, Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type, J. Differential Equations, 191 (2003), 121-142.
[46] V. Felli and M. Schneider, Compact and existence results for degenerate critical elliptic equations,Comm. Contemp.Math., 7 (2005), 37-73.
[47] V. Felli and S. Terracini, Fountain-like solutions for nonlinear elliptic equations with critical growth and Hardy potential, Comm. Contemp. Math., 7 (2005),867-904.
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