临界点理论对非线性方程的应用
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摘要
本文利用变分方法研究全空间上一类拟线性椭圆方程的无穷多解的存在性以及二阶差分方程的多重解的存在性.
首先,我们应用喷泉定理研究了拟线性椭圆方程的无穷多解的存在性.其中Δ_pu=div(|(?)u|~(p-2)(?)u),1 其次,我们应用三临界点定理和Clark定理研究如下的非线性差分方程多重m-周期解的存在性.
这里m≥2为一固定的整数,δ>0,Δx_n=x_(n+1)-x_n,Δ~2x_n=Δ(Δx_n),{p_n}是个m-周期非负实序列,f是Z×R上的连续函数,且关于第二变元是m-周期的.我们约定(-1)~δ=-1.
In this thesis, the existence of infinitely many solutions for quasilinear elliptic equation on R~N, and multiple periodic solutions for nonlinear difference equations are obtained by variational methods. First, we use the Fountain theorem to obtain infinitely many solutions for quasilinear elliptic equationwhere△_pu = (?), 1 < p <∞. Here we consider the superlinear case: (?). Unlike the usual results, our f(x,u) does not satisfy the superlinear condition of Ambrosetti-Rabinowitz.
Next, using the three critical points theorem and the clark theorem, we consider the existence of multiple periodic solutions for nonlinear difference equations of the formwhere m≥2 is a fixed integer,δ> 0,△is the forward difference operator defined by△x_n = x_(n+1) - x_n, {p_n} is a nonnegative m-periodic real sequence; f is a continuous function on Z×R, which is m-periodic on the second variable. Throughout this thesis, the convention (-1)~δ= -1 is made.
首先,我们应用喷泉定理研究了拟线性椭圆方程的无穷多解的存在性.其中Δ_pu=div(|(?)u|~(p-2)(?)u),1
这里m≥2为一固定的整数,δ>0,Δx_n=x_(n+1)-x_n,Δ~2x_n=Δ(Δx_n),{p_n}是个m-周期非负实序列,f是Z×R上的连续函数,且关于第二变元是m-周期的.我们约定(-1)~δ=-1.
In this thesis, the existence of infinitely many solutions for quasilinear elliptic equation on R~N, and multiple periodic solutions for nonlinear difference equations are obtained by variational methods. First, we use the Fountain theorem to obtain infinitely many solutions for quasilinear elliptic equationwhere△_pu = (?), 1 < p <∞. Here we consider the superlinear case: (?). Unlike the usual results, our f(x,u) does not satisfy the superlinear condition of Ambrosetti-Rabinowitz.
Next, using the three critical points theorem and the clark theorem, we consider the existence of multiple periodic solutions for nonlinear difference equations of the formwhere m≥2 is a fixed integer,δ> 0,△is the forward difference operator defined by△x_n = x_(n+1) - x_n, {p_n} is a nonnegative m-periodic real sequence; f is a continuous function on Z×R, which is m-periodic on the second variable. Throughout this thesis, the convention (-1)~δ= -1 is made.
引文
[1] A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applications, J.Funct. Anal. 14, 349-381 ( 1973)
[2] P. Bartolo, V. Benci, D.Fortunato, Abstract critical point theorems and applications to some nonlinear problem with strong resonance, at infinity, Nonl.Anal.TMA,1983,7:981-1012.
[3] T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, . Nonlinear Analysis, TMA,20 (1993), 1205-1216.
[4] T. Bartsch, Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on R~N, Comm. PDE, 20 (1995), 1725-1741.
[5] T. Bartsch, M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation, J. Funct. Anal., 117 (1993), 447-460.
[6] T. Bartsch, and M. Willem: On an elliptic equation with concave and convex nonlinearities, Proc. Amer.Math.Soc.l23,3555-3561.(1995)
[7] H. Brezis, L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963.
[8] X. C. Cai, J. S. Yu, Z. M. Guo, Periodic solutions of a class of nonlinear difference equations via critical point method, Comput. Math. Appl., 52 (2006), 1630-1647.
[9] D. C. Clark, A variant of the Ljusternik-Schnirelmann theory,, Indiana Univ. Math. J., 22 (1972), 65-74.
[10] David.G. Costa, On a class of elliptic system in R~N , Electronic journal of differential equations, 1994, No.07,pp.1-14.
[11] Z.M. Guo and J.S. Yu, The existence of perodic and subharmonic solutions for second-order superlinear difference equations, Science in China (Series A)3, 2003, 226-235.
[12] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on R~N,Proc. Roy.Soc. Edinburgh, 1999,129 A: 787-809.
[13] G.B. Li, H.S. Zhou, Asymptotically linear Dirichlet problem for the p-Laplacian , Nonl.Anal.TMA,2001,43:1043-1055.
[14] S.J. Li and J.Q. Liu, Some existence theorem on mutiple critical points and their applications,Kexue Tongbao,17( 1984), 1025-1027.
[15] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations , Cbms, American Mathematical Society, vol. 65, 1986.
[16] W. Stauss, Existence of solitary waves in higher dimensions,Comm. Math. Phys., 55 (1977), 149-162.
[17] C. Stuart, H. S. Zhou, A variational problem related to self-trapping of an electromagnetic field , Math.Methods in the Applied Sciences, 1996, 19: 1397-1407.
[18] M. Willem, Minimax Theorems.Boston: Birkhauser, 1996.
[19] H. Yang , X. L. Fan, Existence of infinitely mang solutions of the p-Laplace equation with p- concave and convex nonlinearities , J.Lanzhou Univ.Nat.Sci., 1992,35(2):12-16(in Chinese).
[20] J. F. Zhao,Structure theory for Banach space, Wuhan Univ. Press, 1991 (in Chinese).
[21] W. M. Zou, Variant fountain theorems and their applications, manuscripta math. 104 (2001), 343-358.
[22] 刘轼波，李树杰 一类超线性椭圆方程的无穷多解，数学学报，第46卷，第四期(2003) 625-630．
[2] P. Bartolo, V. Benci, D.Fortunato, Abstract critical point theorems and applications to some nonlinear problem with strong resonance, at infinity, Nonl.Anal.TMA,1983,7:981-1012.
[3] T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, . Nonlinear Analysis, TMA,20 (1993), 1205-1216.
[4] T. Bartsch, Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on R~N, Comm. PDE, 20 (1995), 1725-1741.
[5] T. Bartsch, M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation, J. Funct. Anal., 117 (1993), 447-460.
[6] T. Bartsch, and M. Willem: On an elliptic equation with concave and convex nonlinearities, Proc. Amer.Math.Soc.l23,3555-3561.(1995)
[7] H. Brezis, L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963.
[8] X. C. Cai, J. S. Yu, Z. M. Guo, Periodic solutions of a class of nonlinear difference equations via critical point method, Comput. Math. Appl., 52 (2006), 1630-1647.
[9] D. C. Clark, A variant of the Ljusternik-Schnirelmann theory,, Indiana Univ. Math. J., 22 (1972), 65-74.
[10] David.G. Costa, On a class of elliptic system in R~N , Electronic journal of differential equations, 1994, No.07,pp.1-14.
[11] Z.M. Guo and J.S. Yu, The existence of perodic and subharmonic solutions for second-order superlinear difference equations, Science in China (Series A)3, 2003, 226-235.
[12] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on R~N,Proc. Roy.Soc. Edinburgh, 1999,129 A: 787-809.
[13] G.B. Li, H.S. Zhou, Asymptotically linear Dirichlet problem for the p-Laplacian , Nonl.Anal.TMA,2001,43:1043-1055.
[14] S.J. Li and J.Q. Liu, Some existence theorem on mutiple critical points and their applications,Kexue Tongbao,17( 1984), 1025-1027.
[15] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations , Cbms, American Mathematical Society, vol. 65, 1986.
[16] W. Stauss, Existence of solitary waves in higher dimensions,Comm. Math. Phys., 55 (1977), 149-162.
[17] C. Stuart, H. S. Zhou, A variational problem related to self-trapping of an electromagnetic field , Math.Methods in the Applied Sciences, 1996, 19: 1397-1407.
[18] M. Willem, Minimax Theorems.Boston: Birkhauser, 1996.
[19] H. Yang , X. L. Fan, Existence of infinitely mang solutions of the p-Laplace equation with p- concave and convex nonlinearities , J.Lanzhou Univ.Nat.Sci., 1992,35(2):12-16(in Chinese).
[20] J. F. Zhao,Structure theory for Banach space, Wuhan Univ. Press, 1991 (in Chinese).
[21] W. M. Zou, Variant fountain theorems and their applications, manuscripta math. 104 (2001), 343-358.
[22] 刘轼波，李树杰 一类超线性椭圆方程的无穷多解，数学学报，第46卷，第四期(2003) 625-630．