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- 2018
分数阶椭圆方程近共振问题解的多重性
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Abstract:
考虑当线性项的参数从右边逼近非主特征值时分数阶椭圆方程的多解性.一方面，通过对泛函在不同特征子空间上的能量水平的估计可构造出一个具有鞍点结构的解；另一方面，当参数充分接近特征值时，结合鞍点定理、Galerkin逼近方法及对近共振对应的特征子空间上能量水平的仔细估算证明第二个解的存在性.
The present paper considers the multiplicity of the solution for fractional elliptic equations when the parameter of the linear term approximates to the non-principal eigenvalue from the right. On the one hand, we establish the existence of the first solution of saddle point geometry by calculating the energy level of the functional on different eigenspaces. On the other hand, we obtain the second solution by applying the saddle theorem and the Galerkin approximation method and by evaluating the energy level on the eigenspace when the linear part is near resonance
| [1] | CAFFARELLI L, VALDINOCI E. Uniform Estimates and Limiting Arguments for Nonlocal Minimal Surfaces[J]. Calc Var Partial Differential Equations, 2011, 41(1-2): 203-240. DOI:10.1007/s00526-010-0359-6 |
| [2] | SERVADEI R, VALDINOCI E. Mountain Pass Solutions for Non-Local Elliptic Operators[J]. J Math Anal Appl, 2012, 389(2): 887-898. DOI:10.1016/j.jmaa.2011.12.032 |
| [3] | 郭灵钟, 姚娟, 索洪敏, 等. 半线性分数阶椭圆型算子方程近共振问题的多重解[J]. 遵义师范学院学报, 2014, 16(1): 87-91. DOI:10.3969/j.issn.1009-3583.2014.01.021 |
| [4] | 梁志霞, 欧增奇. 拟线性椭圆方程近共振问题解的多重性[J]. 西南大学学报(自然科学版), 2018, 40(2): 64-69. |
| [5] | KE X F, TANG C L. Existence and Multiplicity of Solutions for Asymptotically Linear Noncooperative Elliptic Systems[J]. J Math Anal Appl, 2011, 375(2): 631-647. DOI:10.1016/j.jmaa.2010.09.041 |
| [6] | MASSA E, ROSSATO R A. Multiple Solutions for an Elliptic System Near Resonance[J]. J Math Anal Appl, 2014, 420(2): 1228-1250. DOI:10.1016/j.jmaa.2014.06.043 |
| [7] | SIRE Y, VALDINOCI E. Fractional Laplacian Phase Transitions and Boundary Reactions:a Geometric Inequality and a Symmetry Result[J]. J Funct Anal, 2009, 256(6): 1842-1864. DOI:10.1016/j.jfa.2009.01.020 |
| [8] | LASKIN N. Fractional Quantum Mechanics and Lévy Path Integrals[J]. Phys Lett A, 2000, 268(4-6): 298-305. DOI:10.1016/S0375-9601(00)00201-2 |
| [9] | ELEONORA D N, GIAMPIERO P, ENRICO V. Hitchhiker's Guide to the Fractional Sobolev Spaces[J]. Bull Sci Math, 2012, 136(5): 521-573. DOI:10.1016/j.bulsci.2011.12.004 |
| [10] | SERVADEI R, VALDINOCI E. Variational Methods for Non-Local Operators of Elliptic Type[J]. Discrete Contin Dyn Syst, 2013, 33(5): 2105-2137. |
| [11] | PAIVA F O, MASSA E. Semilinear Elliptic Problems Near Resonance with a Nonprincipal Eigenvalue[J]. J Math Anal Appl, 2008, 342(1): 638-650. DOI:10.1016/j.jmaa.2007.12.053 |
| [12] | SUO H M, TANG C L. Multiplicity Results for Some Elliptic Systems Near Resonance with a Nonprincipal Eigenvalue[J]. Nonlinear Anal, 2010, 73(7): 1909-1920. DOI:10.1016/j.na.2009.11.004 |
| [13] | SUO H M, TANG C L. Degenerate Semilinear Elliptic Problems Near Resonance with a Nonprincipal Eigenvalue[J]. Bull Korean Math Soc, 2012, 49(4): 669-684. DOI:10.4134/BKMS.2012.49.4.669 |
| [14] | KE X F, TANG C L. Multiple Solutions for Semilinear Elliptic Equations Near Resonance at Higher Eigenvalues[J]. Nonlinear Anal, 2011, 74(3): 805-813. DOI:10.1016/j.na.2010.09.031 |
| [15] | 陈清明, 李春, 欧增奇. 拟线性椭圆方程在近共振处解的多重性[J]. 西南大学学报(自然科学版), 2014, 36(8): 87-91. |
