$$p-$$ Harmonic Functions in the Upper Half-space
Springer Science and Business Media LLC - Trang 1-24 - 2023
Tóm tắt
This paper investigates the existence, nonexistence, and qualitative properties of p-harmonic functions in the upper half-space
$$\mathbb {R}^N_+$$
$$(N\ge 3)$$
satisfying nonlinear boundary conditions for
$$1
Tài liệu tham khảo
Abreu, E., do Ó, J.M., Medeiros, E.: Properties of positive harmonic functions on the half-space with a nonlinear boundary condition. J. Differential Equations 248, 617–637 (2010)
Adams, R., Fournier, J.: Sobolev spaces. Second edition. Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam (2003)
Aleksandrov, A.: Uniqueness theorems for surfaces in the large. I. Amer. Math. Soc. Transl. 21, 341–354 (1962)
Alexandrov, A.: A characteristic property of spheres. Ann. Mat. Pura Appl. 58, 303–315 (1962)
Allegretto, W., Huang, Y.: A Picone’s identity for the \(p\)-Laplacian and applications. Nonlinear Anal. 32, 819–830 (1998)
Bonder, J., Rossi, J.: Existence results for the \(p\)-Laplacian with nonlinear boundary conditions. J. Math. Anal. Appl. 263, 195–223 (2001)
Chipot, M., Chlebík, M., Fila, M., Shafrir, I.: Existence of positive solutions of a semilinear elliptic equation in \(\mathbb{R} ^n_{+}\) with a nonlinear boundary condition. J. Math. Anal. Appl. 223, 429–471 (1998)
Cuesta, M., Takáč, P.: A strong comparison principle for positive solutions of degenerate elliptic equations. Differential Integral Equations 13, 721–746 (2000)
Damascelli, L., Pacella, F.: Monotonicity and symmetry of solutions of \(p\)-Laplace equations, \(1<p<2\), via the moving plane method. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26, 689–707 (1998)
Damascelli, L., Sciunzi, B.: Regularity, monotonicity and symmetry of positive solutions of \(m\)-Laplace equations. J. Differential Equations 206, 483–515 (2004)
Degiovanni, M., Musesti, A., Squassina, M.: On the regularity of solutions in the Pucci-Serrin identity. Calc. Var. Partial Differential Equations 18, 317–334 (2003)
do Ó, J.M., Medeiros, E.: Remarks on least energy solutions for quasilinear elliptic problems in \(R^N\). Electron. J. Differential Equations (83), 14 pp
Escobar, J.: Sharp constant in a Sobolev trace inequality. Indiana Univ. Math. J. 37, 687–698 (1988)
Farina, A., Montoro, L., Sciunzi, B.: Monotonicity and one-dimensional symmetry for solutions of \(-\Delta _pu=f(u)\) in half-spaces. Calc. Var. Partial Differential Equations 43, 123–145 (2012)
Farina, A., Montoro, L., Sciunzi, B.: Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces. Math. Ann. 357, 855–893 (2013)
Symmetry and related properties via the maximum principle: Gidas, B., Ni, w-M., Nirenberg, L. Comm. Math. Phys. 68, 209–243 (1979)
Guo, Y., Liu, X.: A multiple critical points theorem and applications to quasilinear boundary value problems in \(\mathbb{R} ^N_+\). Nonlinear Anal. 75, 3787–3808 (2012)
J. Harada, J.:Positive solutions to the Laplace equation with nonlinear boundary conditions on the half space. Calc. Var. Partial Differential Equations 50, 399–435 (2014)
Hu, B.: Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition. Differential Integral Equations 7, 301–313 (1994)
Il’yasov, Y., Takáč, P.: Optimal \(W^{2,2}_{\rm loc}\)-regularity, Pohozhaev’s identity, and nonexistence of weak solutions to some quasilinear elliptic equations. J. Differential Equations 252, 2792–2822 (2012)
F. Isaia, F.: Superposition operators between Sobolev spaces and a non-existence result of higher-order regular solutions for the \(p\)-Laplacian. Nonlinear Anal. 117, 87–98 (2015)
Li, Y., Zhu, M.: Uniqueness theorems through the method of moving spheres. Duke Math. J. 80, 383–417 (1995)
Lieberman, G.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)
Lou, H.: On singular sets of local solutions to \(p\)-Laplace equations. Chin. Ann. Math. Ser. B 29, 521–530 (2000)
Nazaret, B.: Best constant in Sobolev trace inequalities on the half-space. Nonlinear Anal. 65, 1977–1985 (2006)
Pohožaev, S.: On the eigenfunctions of the equation \(\Delta u+\lambda f(u)=0\). Dokl. Akad. Nauk SSSR 165, 36–39 (1965)
Pucci, P., Serrin, J.: A general variational identity. Indiana Univ. Math. J. 35, 681–703 (1986)
Pucci, P., Servadei, R.: Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations. Indiana Univ. Math. J. 57, 3329–3363 (2008)
Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304–318 (1971)
Simon, J.: Régularité de la solution d’un problème aux limites non linéaires. Ann. Fac. Sci. Toulouse Math. 5(3), 247–274 (1982)
Terracini, S.: Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions. Differential Integral Equations 8, 1911–1922 (1995)
Vázquez, J.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984)
Willem, M.: Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, 24 Birkhäuser Boston, Inc., Boston, MA, (1996)