Nội dung được dịch bởi AI, chỉ mang tính chất tham khảo
Phương trình Schrödinger–Kirchhoff bậc 4 không đồng nhất quan trọng
Springer Science and Business Media LLC - Trang 1-18 - 2023
Tóm tắt
Trong bài báo này, chúng tôi nghiên cứu lớp phương trình Schrödinger–Kirchhoff bậc 4 tĩnh sau: $$\begin{aligned} \Delta ^{2} u-M\left( \Vert \nabla u\Vert ^2_2 \right) \Delta u+V(x)u=h(x)|u|^{q-2}u+|u|^{2_*-2}u+ g(x)|u|^{\tau -2}u, ~~x \in \mathbb {R}^{N}, \end{aligned}$$
trong đó $$N\ge 8,$$
và $$2_*=\frac{2N}{N-4}$$ là số mũ Sobolev quan trọng. Dưới một số giả định về hàm Kirchhoff M, tiềm năng V(x) và g(x), bằng cách sử dụng Nguyên lý Biến thiên Ekeland và Định lý Điểm núi, chúng tôi thu được sự tồn tại của nhiều nghiệm cho vấn đề trên. Những kết quả này là mới ngay cả đối với trường hợp cục bộ, tương ứng với các phương trình Schrödinger bậc 4 phi tuyến.
Từ khóa
#phương trình Schrödinger-Kirchhoff #phương trình bậc 4 #số mũ Sobolev #nghiệm phi tuyến #Nguyên lý Biến thiên EkelandTài liệu tham khảo
Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348, 305–330 (1996)
Bernis, F., Garcia Azorero, J., Peral, I.: Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Adv. Differ. Equ. 1, 219–240 (1996)
Bernstein, S.: Sur une classe d’équations fonctionnelles aux dérivées partielles. Bull. Acad. Sci. URSS Sér. Math. (Izv. Akad. Nauk SSSR) 4, 17–26 (1940)
Cabada, A., Figueiredo, G.M.: A generalization of an extensible beam equation with critical growth in \(\mathbb{R} ^N\). Nonlinear Anal. Real World Appl. 20, 134–142 (2014)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation. Adv. Differ. Equ. 6, 701–730 (2001)
Chabrowski, J.: On multiple solutions for the nonhomogeneous \(p\)-Laplacian with a critical Sobolev exponent. Differ. Integr. Equ. 8, 705–716 (1995)
Chabrowski, J., Yang, J.: Nonnegative solutions for semilinear biharmonic equations in \(\mathbb{R} ^N\). Analysis 17, 35–59 (1997)
Chabrowski, J., Marcos do Ó, J.: On some fourth-order semilinear problems in \(\mathbb{R} ^{N}\). Nonlinear Anal. 49, 861–884 (2002)
Edmunds, D.E., Fortunato, D., Jannelli, E.: Critical exponent, critical dimensions and the biharmonic operator. Arch. Rat. Mech. Anal. 112, 269–289 (1990)
Ferrero, A., Warnault, G.: On solutions of second and fourth order elliptic equations with power-type nonlinearities. Nonlinear Anal. 70, 2889–2902 (2009)
Kirchhoff, G.: Vorlesungen uber Mechanik. Teubner, Leipzig (1883)
Lazer, A., McKenna, P.: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 32, 537–578 (1990)
Li, C., Tang, C.L.: Three solutions for a Navier boundary value problem involving the p-biharmonic. Nonlinear Anal. 72, 1339–1347 (2010)
Lions, J.-L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of the International Symposium of Inst. Mat., Univ. Fed. Rio de Janeiro, 1997, North-Holland Mathematical Studies, 30, pp. 284–346. North-Holland, Amsterdam (1978)
Lions, P.-L.: The concentration-compactness principle in the calculus of variation. The limit case—part 1. Revista Matematica Iberoamericana 1, 145–201 (1985)
Myers, T.G.: Thin films with high surface tension. SIAM Rev. 40, 441–462 (1998)
Noussair, E.S., Swanson, C.A., Yang, J.: Critical semilinear biharmonic equations in \(\mathbb{R} ^N\). Proc. R. Soc. Ed. 121A, 139–148 (1992)
Noussair, E.S., Swanson, C.A., Yang, J.: Transcritical biharmonic equations in \(\mathbb{R} ^N\). Funkcialaj Ekvacioj 35, 533–543 (1992)
Pimenta, M.T.O., Soares, S.H.M.: Existence and concentration of solutions for a class of biharmonic equations. J. Math. Anal. Appl. 390, 274–289 (2012)
Pohozaev, S.: On a class of quasilinear hyperbolic equations. Math. Sborniek 96, 152–166 (1975)
Pucci, P., Serrin, J.: Critical exponent and critical dimensions for polyharmonic operators. J. Math. Pures Appi. 69, 55–83 (1990)
Rabinowitz, P.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Providence, RI (1986)
Song, H., Chen, C.: Infinitely many solutions for Schrödinger–Kirchhoff-type fourth-order elliptic equations. Proc. Edinb. Math. Soc. 4, 1–18 (2017)
Tarantello, G.: On nonhomogeneous elliptic involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 9, 281–304 (1992)
Xiang, M., Radulescu, V.D., Zhang, B.: Existence results for singular fractional p-Kirchhoff problems. Acta Math. Sci. 42, 1209–1224 (2022)
Wang, F., Hu, D., Xiang, M.: Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems. Adv. Nonlinear Anal. 10, 636–658 (2021)
Wang, W., Zhao, P.: Nonuniformly nonlinear elliptic equations of p-biharmonic type. J. Math. Anal. Appl. 348, 730–738 (2008)
Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in \({{ R}}^N\). Nonlinear Anal. Real World Appl. 12(2), 1278–1287 (2011)