In this paper, a semilinear elliptic equation of fractional order is constructed by combining the semilinear elliptic equation with the fractional order equation. On this basis, the “standardized” solution, which is often sought by physicists, is studied. In order to overcome the problem of lack of boundness, we set up appropriate conditions to prove the existence of the solution by means of variational theorem and the mountain road theorem.
Cite this paper
Li, T. (2024). Semilinear Fractional Elliptic Equation of Normal Solution. Open Access Library Journal, 11, e1412. doi: http://dx.doi.org/10.4236/oalib.1111412.
Kulish, V.V. and Lage, J.L. (2002) Application of Fractional Calculus to Fluid Mechanics. Journal of Fluids Engineering, 124, 803-806.
https://doi.org/10.1115/1.1478062
Fefferman, C. and de la Llave, R. (1986) Relativistic Stability of Matter-I. Revista Matemática Iberoamericana, 2, 119-213. https://doi.org/10.4171/rmi/30
Gomez-Aguilar, J.F., Miranda-Hernandez, M., Lbpez-Lopez, M.G., Alvarado-Martinez, V.M. and Baleanu, D. (2016) Modeling and Simulation of the Fractional Space-Time Diffusion Equation. Communications in Nonlinear Science and Numerical Simulation, 30, 115-127. https://doi.org/10.1016/j.cnsns.2015.06.014
Bates, P. (2006) On Some Nonlocal Evolution Equations Arising in Materials Science. American Mathematical Society, Providence, 13-52.
https://doi.org/10.1090/fic/048/02
Gonzalez, M. and Monneau, R. (2012) Slow Motion of Particle Systems as a Limit of a Reaction-Diffusion Equation with Half-Laplacian in Dimension One. Discrete and Continuous Dynamical Systems, 32, 1255-1286.
https://doi.org/10.3934/dcds.2012.32.1255
Caffarelli, L., Mellet, A. and Sire, Y. (2012) Traveling Waves for a Boundary Reaction-Diffusion Equation. Advances in Mathematics, 230, 433-457.
https://doi.org/10.1016/j.aim.2012.01.020
Kurzke, M. (2006) A Nonlocal Singular Perturbation Problem with Periodic Well Potential. ESAIM: Control, Optimisation and Calculus of Variations, 12, 52-63.
https://doi.org/10.1051/cocv:2005037
Caffarelli, L. and Valdinoci, E. (2011) Uniform Estimates and Limiting Arguments for Nonlocal Minimal Surfaces. Calculus of Variations and Partial Differential Equations, 41, 203-240. https://doi.org/10.1007/s00526-010-0359-6
Thomas, B. and de Valeriola, S. (2013) Normalized Solutions of Nonlinear Schr？dinger Equations. Archiv der Mathematik, 100, 75-83.
https://doi.org/10.1007/s00013-012-0468-x
Jeanjean, L. (1997) Existence of Solutions with Prescribed Norm for Semilinear Elliptic Equations. Nonlinear Analysis: Theory, Methods & Applications, 28, 1633-1659.
https://doi.org/10.1016/S0362-546X(96)00021-1
Frank, R.L., Lenzmann, E. and Silvestre, L. (2016) Uniqueness of Radial Solutions for the Fractional Laplacian. Communications on Pure and Applied Mathematics, 69, 1671-1726. https://doi.org/10.1002/cpa.21591
Strauss, W.A. (1977) Existence of Solitary Waves in Higher Dimensions. Communications in Mathematical Physics, 55, 149-162. https://doi.org/10.1007/BF01626517
