带深井位势双调和方程的解
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摘要
本文研究下述双调和方程极小能量解的存在性:?~2u+[λV (x)-δ]u=|u|_(p-2)u, x∈R~N,(0.1)其中N≥5,λ> 0. p是次临界或临界的Sobolev指标,即2 0充分大时,(0.1)存在一个在V~(-1)(0)附近的极小能量解.
In this paper,we are concerned with the existence of least energy solutions for the following biharmonic equations:?~2u + [λV(x)-δ]u = |u|~(p-2u), x ∈ R~N,(0.1)where N≥5, 2 < p≤2N/N-4, λ > 0 is a parameter, V(x) is a nonnegative potential function with nonempty interior part of the zero set int V~(-1)(0), 0 < δ < μ0 and μ0 is the principal eigenvalue of ?~2in the zero set int V~(-1)(0) of V(x). We prove that the equation(0.1) admits a least energy solution which is trapped near the zero set V~(-1)(0)for λ > 0 large enough.
In this paper,we are concerned with the existence of least energy solutions for the following biharmonic equations:?~2u + [λV(x)-δ]u = |u|~(p-2u), x ∈ R~N,(0.1)where N≥5, 2 < p≤2N/N-4, λ > 0 is a parameter, V(x) is a nonnegative potential function with nonempty interior part of the zero set int V~(-1)(0), 0 < δ < μ0 and μ0 is the principal eigenvalue of ?~2in the zero set int V~(-1)(0) of V(x). We prove that the equation(0.1) admits a least energy solution which is trapped near the zero set V~(-1)(0)for λ > 0 large enough.
引文
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2 McKenna P J,Walter W.Travelling waves in a suspension bridge.SIAM J Appl Math,1990,50:703-715
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7 Salvatore A,Squassina M.Deformation from symmetry for Schr?dinger equations of higher order on unbounded domains.Electron J Differential Equations,2003,2003:1-15
8 Pimenta M T O,Soares S H M.Existence and concentration of solutions for a class of biharmonic equations.J Math Anal Appl,2012,390:274-289
9 Figueiredo G M,Pimenta M T.Multiplicity of solutions for a biharmonic equation with subcritical or critical growth.Bull Belg Math Soc Simon Stevin,2013,20:519-534
10 Zhang W,Tang X,Zhang J.Existence and concentration of solutions for sublinear fourth-order elliptic equations.Electron J Differential Equations,2015,2015:1-9
11 Alves C O,Nobrega A B.Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator.ArXiv:1602.03112,2016
12 Bartsch T,Wang Z Q.Multiple positive solutions for a nonlinear Schr?dinger equation.Z Angew Math Phys,2000,51:366-384
13 Deng Y,Shuai W.Non-trivial solutions for a semilinear biharmonic problem with critical growth and potential vanishing at infinity.Proc Roy Soc Edinburgh Sect A,2015,145:281-299
14 Carri?ao P,Demarque R,Miyagaki O H.Nonlinear biharmonic problems with singular potentials.Commun Pure Appl Anal,2014,13:2141-2154
15 Hu S,Wang L.Existence of nontrivial solutions for fourth-order asymptotically linear elliptic equations.Nonlinear Anal,2014,94:120-132
16 Gazzola F,Grunau H C.Radial entire solutions for supercritical biharmonic equations.Math Ann,2006,334:905-936
17 Guo Z,Huang X,Zhou F.Radial symmetry of entire solutions of a bi-harmonic equation with exponential nonlinearity.J Funct Anal,2015,268:1972-2004
18 Wang Y,Shen Y.Multiple and sign-changing solutions for a class of semilinear biharmonic equation.J Differential Equations,2009,246:3109-3125
19 Davila J,Dupaigne L,Wang K,et al.A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem.Adv Math,2014,258:240-285
20 Wang F,An Y.Existence and multiplicity of solutions for a fourth-order elliptic equation.Bound Value Probl,2012,2012:6
21 Gazzola F,Grunau H C,Sweers G.Polyharmonic Boundary Value Problems.Heidelberg:Springer,2010,52-59
22 Bartsch T,Pankov A,Wang Z Q.Nonlinear Schr?dinger equations with steep potential well.Commun Contemp Math,2001,3:549-569
23 Reed M,Simon B.Methods of Modern Mathmatical Physics,Vol.IV.London:Academic Press,1978,75-79;106-120
24 Lions P L.The concentration-compactness principle in the calculus of variations:The limit case,part 1.Rev Mat Iberoam,1985,1:145-201
25 Lin C S.A classification of solutions of a conformally invariant fourth order equation in RN.Comment Math Helv,1998,73:206-231
26 Wei J,Xu X.Classification of solutions of higher order conformally invariant equations.Math Ann,1999,313:207-228
27 Brezis H,Nirenberg L.Positive solutions of nonlinear elliptic equations involving critical sobolev exponents.Comm Pure Appl Math,1983,36:437-477
28 Brezis H,Lieb E.A relation between pointwise convergence of functions and convergence of functionals.Proc Amer Math Soc,1983,88:486-486
29 Willem M.Minimax Theorems.Progress in Nonlinear Differential Equations and Their Applications,vol.24.Boston:Birkh?user,1996,20-23
30 Lions P L.The concentration-compactness principle in the calculus of variations:The locally compact case,part 1.Ann Inst H Poincar′e Anal Non Lin′eaire,1984,1:109-145
2 McKenna P J,Walter W.Travelling waves in a suspension bridge.SIAM J Appl Math,1990,50:703-715
3 Abrahams I D,Davis A M.Deflection of a partially clamped elastic plate.In:IUTAM Symposium on Diffraction and Scattering in Fluid Mechanics and Elasticity.Fluid Mechanics and Its Applications,vol.68.Dordrecht:Springer,2002,303-312
4 Hebey E,Robert F.Compactness and global estimates for the geometric paneitz equation in high dimensions.Electron Res Announc Amer Math Soc,2004,10:135-142
5 Humbert E,Raulot S.Positive mass theorem for the Paneitz-Branson operator.Calc Var Partial Differential Equations,2009,36:525-531
6 Alves C O,doO J M,Miyagaki O H.Nontrivial solutions for a class of semilinear biharmonic problems involving critical exponents.Nonlinear Anal,2001,46:121-133
7 Salvatore A,Squassina M.Deformation from symmetry for Schr?dinger equations of higher order on unbounded domains.Electron J Differential Equations,2003,2003:1-15
8 Pimenta M T O,Soares S H M.Existence and concentration of solutions for a class of biharmonic equations.J Math Anal Appl,2012,390:274-289
9 Figueiredo G M,Pimenta M T.Multiplicity of solutions for a biharmonic equation with subcritical or critical growth.Bull Belg Math Soc Simon Stevin,2013,20:519-534
10 Zhang W,Tang X,Zhang J.Existence and concentration of solutions for sublinear fourth-order elliptic equations.Electron J Differential Equations,2015,2015:1-9
11 Alves C O,Nobrega A B.Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator.ArXiv:1602.03112,2016
12 Bartsch T,Wang Z Q.Multiple positive solutions for a nonlinear Schr?dinger equation.Z Angew Math Phys,2000,51:366-384
13 Deng Y,Shuai W.Non-trivial solutions for a semilinear biharmonic problem with critical growth and potential vanishing at infinity.Proc Roy Soc Edinburgh Sect A,2015,145:281-299
14 Carri?ao P,Demarque R,Miyagaki O H.Nonlinear biharmonic problems with singular potentials.Commun Pure Appl Anal,2014,13:2141-2154
15 Hu S,Wang L.Existence of nontrivial solutions for fourth-order asymptotically linear elliptic equations.Nonlinear Anal,2014,94:120-132
16 Gazzola F,Grunau H C.Radial entire solutions for supercritical biharmonic equations.Math Ann,2006,334:905-936
17 Guo Z,Huang X,Zhou F.Radial symmetry of entire solutions of a bi-harmonic equation with exponential nonlinearity.J Funct Anal,2015,268:1972-2004
18 Wang Y,Shen Y.Multiple and sign-changing solutions for a class of semilinear biharmonic equation.J Differential Equations,2009,246:3109-3125
19 Davila J,Dupaigne L,Wang K,et al.A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem.Adv Math,2014,258:240-285
20 Wang F,An Y.Existence and multiplicity of solutions for a fourth-order elliptic equation.Bound Value Probl,2012,2012:6
21 Gazzola F,Grunau H C,Sweers G.Polyharmonic Boundary Value Problems.Heidelberg:Springer,2010,52-59
22 Bartsch T,Pankov A,Wang Z Q.Nonlinear Schr?dinger equations with steep potential well.Commun Contemp Math,2001,3:549-569
23 Reed M,Simon B.Methods of Modern Mathmatical Physics,Vol.IV.London:Academic Press,1978,75-79;106-120
24 Lions P L.The concentration-compactness principle in the calculus of variations:The limit case,part 1.Rev Mat Iberoam,1985,1:145-201
25 Lin C S.A classification of solutions of a conformally invariant fourth order equation in RN.Comment Math Helv,1998,73:206-231
26 Wei J,Xu X.Classification of solutions of higher order conformally invariant equations.Math Ann,1999,313:207-228
27 Brezis H,Nirenberg L.Positive solutions of nonlinear elliptic equations involving critical sobolev exponents.Comm Pure Appl Math,1983,36:437-477
28 Brezis H,Lieb E.A relation between pointwise convergence of functions and convergence of functionals.Proc Amer Math Soc,1983,88:486-486
29 Willem M.Minimax Theorems.Progress in Nonlinear Differential Equations and Their Applications,vol.24.Boston:Birkh?user,1996,20-23
30 Lions P L.The concentration-compactness principle in the calculus of variations:The locally compact case,part 1.Ann Inst H Poincar′e Anal Non Lin′eaire,1984,1:109-145
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