PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION*

doi:10.1007/s10473-023-0310-5

Abstract

Abstract: In the present paper, we consider the problem
$\begin{equation} \left\{\begin{array}{ll}\label{0001} -\Delta u=u^{\beta_1}|\nabla u|^{\beta_2}, &\ \ { in} \ \Omega,\\ u=0,&\ \ { on} \ \partial{\Omega},\\ u>0,&\ \ { in} \ {\Omega},\\ \end{array}\right. \end{equation}$ $ \ \ \ \ \ $ (0.1)
where $\beta_1,\beta_2>0$ and $\beta_1+\beta_2<1$, and $\Omega $ is a convex domain in $ \mathbb{R}^{n} $. The existence, uniqueness, regularity and $\frac{2-\beta_{2}}{1-\beta_1-\beta_2}$-concavity of the positive solutions of the problem (0.1) are proven.

Key words: Harmonic-Mappings type equation, positive solution, existence, regularity, $\alpha$-concavity

Bo Chen, Zhengmao Chen, Junhui Xie. PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION^*[J].Acta mathematica scientia,Series B, 2023, 43(3): 1161-1174.

Trendmd

References

[1] Alvarez O, Lasry J M, Lions P L.Convexity viscosity solutions and state constraints. J Math Pures Appl, 1997, 76(3): 265-288
[2] Amann H, Crandall M G.On some existence theorems for semi-linear elliptic equations. Indiana Univ Math J, 1978,27: 779-790
[3] Brezis H, Turner R.On a class of superlinear elliptic problems. Comm Partial Differential Equations, 1977, 2(6): 601-614
[4] Buffa A, Costabel M, Dauge M.Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron. C R Acad Sci Paris Ser I, 2003, 336(1): 565-570
[5] Caffarelli L, Friedman A.Convexity of solutions of some semilinear elliptic equations. Duke Math J, 1985, 52(2): 431-456
[6] Chaira A, Touhami S. Riesz bases for L²(∂Ω) andregularity for the Laplace equation in Lipschitz domains.2018, arXiv:1803.07550
[7] Chen C Q, Hu B W.A microscopic convexity principle for spacetime convex solutions of fully nonlinear parabolic equations. Acta Math Sin, 2013, 29(4): 651-674
[8] Coffman C V.On the positive solutions of boundary-value problem for a class of nonlinear differential equation. J Differential Equations, 1967, 3(1): 92-111
[9] Colesanti A.Brunn-Minkowski inequalities for variational functionals and related problems. Adv Math, 2005, 194(1): 105-140
[10] Colesanti A, Salani P.The Brunn-Minkowski inequality for p-capacity of convex bodies. Math Ann, 2003, 327: 459-479
[11] Dai Q Y, Gu Y G.Positive solutions for non-homogeneous semilinear elliptic equations with data that changes sigh. Proc Roy Soc Edinburgh Sect A, 2003, 133(2): 297-306
[12] Damascelli L, Grossi M, Pacella F.Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle. Ann Inst H Poincaré Anal Non Linéaire, 1999, 16(5): 631-652
[13] Dong R, Li D S.Uniform Hüolder estimates for a type of nonlinear elliptic equations with rapidly oscillatory coefficients. Acta Math Sci, 2017, 37(6): 1841-1860
[14] Figueiredo D G, Girardi M, Matzeu M.Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques. Differential Integral Equations, 2004, 17(1/2): 119-126
[15] Gidas B, Spruck J.A priori bounds for positive solution of nonlinear elliptic equations. Comm Partial Differential Equations, 1981, 6(8): 883-901
[16] Greco D.Nuove formole integrali di maggiorazione per le soluzioni di un'equazione lieare di tipo ellittico ed applicazioni alla teoria del potenzile. Ricerche di Mat, 1956, 5: 126-149
[17] Guo C Y, Xiang C L.Regularity of p-harmonic mappings into NPC spaces. Acta Math Sci, 2021, 41B(2): 633-645
[18] Han Q, Lin F H.Elliptic Partial Differential Equations. Providence, RI: American Mathematical Society, 2011
[19] Kawohl B, Payne L.A remark on N. Korevaar's concavity maximum principle and on the asymptotic uniqueness of solutions to the plasma problem. Math Methods Appl Sci, 1986, 8(1): 93-101
[20] Kennington A.Power comcavity and boundary value problems. Indiana Univ Math J, 1985, 34(3): 687-704
[21] Koshelev A I.On the boundedness in L^p of derivatives of solutions of elliptic differential equations. Mat Sb (NS), 1956, 38(80): 359-372
[22] Korevaar N J.Capillary surface convexity above convex domains. Indiana Univ Math J, 1983, 32(1): 73-81
[23] Korevaar N J, Lewis J.Convex solutions of certain elliptic equations have constant rank Hessians. Arch Ration Mech Anal, 1987, 97(1): 19-32
[24] Leray J, Schauder J.Topologie et équations fonctionelles. Ann Sci école Norm Sup, 1934, 51(3): 45-78
[25] Li Y Y.Existence of many positive solutions of semilinear elliptic equations on annulus. J Differential Equations, 1990, 83(2): 348-367
[26] Lin F H, Wang C Y.The Analysis of Harmonic Maps and Their Heat Flows. Singapore: World Scientific Publishing, 2008
[27] Lin C S.Uniqueness of least energy solutions to a semilinear elliptic equation in $\mathbb{R}$². Manuscripta Math, 1994, 84(1): 13-20
[28] Salani P.A Brunn-Minkowski inequality for the Monge-Ampère eigenvalue. Adv Math, 2005, 194(1): 67-86
[29] Werner P.Regularity properties of the Laplace operator with respect to electric and magnetic boundary conditions. J Math Anal Appl, 1982, 87(2): 560-602
[30] Xiang C L.Gradient estimates for solutions to quasilinear elliptic equations with critical Sobolev growth and Hardy potential. Acta Math Sci, 2017, 37(1): 58-68
[31] Ye Y H.Power convexity of a class of elliptic equations involving the Hessian operator in a 3-dimensional bounded convex domain. Nonlinear Anal, 2013, 84: 29-38

PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION^*

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Zhenhai Liu, Nikolaos S. Papageorgiou. SINGULAR DOUBLE PHASE EQUATIONS^* [J]. Acta mathematica scientia,Series B, 2023, 43(3): 1031-1044.
[2]	Qing Guo, Leiga Zhao. POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHRÖDINGER EQUATIONS^* [J]. Acta mathematica scientia,Series B, 2023, 43(3): 1116-1130.
[3]	Leilei CUI, Jiaxing GUO, Gongbao LI. THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS^* [J]. Acta mathematica scientia,Series B, 2023, 43(3): 1131-1160.
[4]	Xuemei Zhang, Shikun Kan. SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS^* [J]. Acta mathematica scientia,Series B, 2023, 43(3): 1175-1194.
[5]	Hongxia Lin, Sen Liu, Heng Zhang, Ru Bai. GLOBAL REGULARITY OF 2D INCOMPRESSIBLE MAGNETO-MICROPOLAR FLUID EQUATIONS WITH PARTIAL VISCOSITY^* [J]. Acta mathematica scientia,Series B, 2023, 43(3): 1275-1300.
[6]	Mingxuan Zhu, Zaihong Jiang. THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION^* [J]. Acta mathematica scientia,Series B, 2023, 43(2): 736-750.
[7]	Boling Guo, Yamin Xiao. THE EXISTENCE OF WEAK SOLUTIONS AND PROPAGATION REGULARITY FOR A GENERALIZED KDV SYSTEM^* [J]. Acta mathematica scientia,Series B, 2023, 43(2): 942-958.
[8]	Hakho Hong. THE EXISTENCE OF GLOBAL SOLUTIONS FOR THE FULL NAVIER-STOKES-KORTEWEG SYSTEM OF VAN DER WAALS GAS^∗ [J]. Acta mathematica scientia,Series B, 2023, 43(2): 469-491.
[9]	Quansen Jiu, Lin Ma. GLOBAL WEAK SOLUTIONS OF COMPRESSIBLE NAVIER-STOKES-LANDAU-LIFSHITZ-MAXWELL EQUATIONS FOR QUANTUM FLUIDS IN DIMENSION THREE^* [J]. Acta mathematica scientia,Series B, 2023, 43(1): 25-42.
[10]	Narimane AISSAOUI, Wei LONG. POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS [J]. Acta mathematica scientia,Series B, 2022, 42(5): 1817-1830.
[11]	Wenjun WANG, Zhen CHENG. LARGE TIME BEHAVIOR OF GLOBAL STRONG SOLUTIONS TO A TWO-PHASE MODEL WITH A MAGNETIC FIELD [J]. Acta mathematica scientia,Series B, 2022, 42(5): 1921-1946.
[12]	Myong-Hwan RI, Reinhard FARWIG. MAXIMAL $L^1$-REGULARITY OF GENERATORS FOR BOUNDED ANALYTIC SEMIGROUPS IN BANACH SPACES [J]. Acta mathematica scientia,Series B, 2022, 42(4): 1261-1272.
[13]	Jianghao HAO, Yajing ZHANG. ESTIMATES FOR EXTREMAL VALUES FOR A CRITICAL FRACTIONAL EQUATION WITH CONCAVE-CONVEX NONLINEARITIES [J]. Acta mathematica scientia,Series B, 2022, 42(3): 903-918.
[14]	Ruiying WEI, Yin LI, Zheng-an YAO. THE GLOBAL EXISTENCE AND A DECAY ESTIMATE OF SOLUTIONS TO THE PHAN-THEIN-TANNER MODEL [J]. Acta mathematica scientia,Series B, 2022, 42(3): 1058-1080.
[15]	Salvatore LEONARDI, Nikolaos S. PAPAGEORGIOU. ARBITRARILY SMALL NODAL SOLUTIONS FOR PARAMETRIC ROBIN (p,q)-EQUATIONS PLUS AN INDEFINITE POTENTIAL [J]. Acta mathematica scientia,Series B, 2022, 42(2): 561-574.