Các trạng thái ổn định dương của một phương trình vô định với điều kiện biên phi tuyến: sự tồn tại, số lượng, tính ổn định và các cấu hình tiệm cận

Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Math. 12, 623–727 (1959) Alama, S., Tarantello, G.: On semilinear elliptic equations with indefinite nonlinearities. Calc. Var. Partial. Differ. Equ. 1, 439–475 (1993) Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122, 519–543 (1994) Bandle, C., Pozio, M.A., Tesei, A.: Existence and uniqueness of solutions of nonlinear Neumann problems. Math. Z. 199, 257–278 (1988) Berestycki, H., Capuzzo-Dolcetta, I., Nirenberg, L.: Superlinear indefinite elliptic problems and nonlinear Liouville theorems. Topol. Methods Nonlinear Anal. 4, 59–78 (1994) Berestycki, H., Capuzzo-Dolcetta, I., Nirenberg, L.: Variational methods for indefinite superlinear homogeneous elliptic problems. NoDEA Nonlinear Differ. Equ. Appl. 2, 553–572 (1995) Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011) Brown, K.J.: Local and global bifurcation results for a semilinear boundary value problem. J. Differ. Equ. 239, 296–310 (2007) Brown, K.J., Hess, P.: Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem. Differ. Integral Equ. 3, 201–207 (1990) Brown, K.J., Lin, S.S.: On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function. J. Math. Anal. Appl. 75, 112–120 (1980) Brown, K.J., Zhang, Y.: The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J. Differ. Equ. 193, 481–499 (2003) Cantrell, R.S., Cosner, C.: Spatial ecology via reaction-diffusion equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons Ltd, Chichester (2003) Chipot, M., Fila, M., Quittner, P.: Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions. Acta. Math. Univ. Comen. 60, 35–103 (1991) Fleming, W.H.: A selection-migration model in population genetics. J. Math. Biol. 2, 219–233 (1975) García-Melián, J., Morales-Rodrigo, C., Rossi, J.D., Suárez, A.: Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary. Ann. Mat. Pura Appl. (4) 187, 459–486 (2008) García-Melián, J., Rossi, J.D., Sabina de Lis, J.C.: Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions. Commun. Contemp. Math. 11, 585–613 (2009) Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983) Gómez-Reñasco, R., López-Gómez, J.: The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations. J. Differ. Equ. 167, 36–72 (2000) López-Gómez, J.: On the existence of positive solutions for some indefinite superlinear elliptic problems. Commun. Partial Differ. Equ. 22, 1787–1804 (1997) López-Gómez, J., Márquez, V., Wolanski, N.: Dynamic behavior of positive solutions to reaction-diffusion problems with nonlinear absorption through the boundary. Rev. Un. Mat. Argentina 38, 196–209 (1993) Morales-Rodrigo, C., Suárez, A.: Some Elliptic Problems with Nonlinear Boundary Conditions, Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology. World Scientific Publishing, Hackensack, pp. 175–199 (2005) Ouyang, T.: On the positive solutions of semilinear equations \(\Delta u+\lambda u+hu^p=0\) on compact manifolds. II. Indiana Univ. Math. J. 40, 1083–1141 (1991) Ramos Quoirin, H., Umezu, K.: The effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation. J. Differ. Equ. 257, 3935–3977 (2014) Ramos Quoirin, H., Umezu, K.: Bifurcation for a logistic elliptic equation with nonlinear boundary conditions: a limiting case. J. Math. Anal. Appl. 428, 1265–1285 (2015) Rossi, J.D.: Elliptic Problems with Nonlinear Boundary Conditions and The Sobolev Trace Theorem, Stationary Partial Differential Equations, vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, pp. 311–406 (2005) Stampacchia, G.: Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie hölderiane. Ann. Mat. Pura Appl. (4) 51, 1–37 (1960) Stavrakakis, N.M.: Global bifurcation results for semilinear elliptic equations on \({\mathbf{R}}^N\): the Fredholm case. J. Differ. Equ. 142, 97–122 (1998) Umezu, K.: Bifurcation approach to a logistic elliptic equation with a homogeneous incoming flux boundary condition. J. Differ. Equ. 252, 1146–1168 (2012) Umezu, K.: Global structure of supercritical bifurcation with turning points for the logistic elliptic equation with nonlinear boundary conditions. Nonlinear Anal. 89, 250–266 (2013) Vázquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984) Wu, T.-F.: A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential. Electron. J. Differ. Equ. 131, 15 (2006)