Monotonicity and one-dimensional symmetry for solutions of -? u = f( u) in half-spaces

Farina, Alberto; Montoro, Luigi; Sciunzi, Berardino
January 2012
Calculus of Variations & Partial Differential Equations;Jan2012, Vol. 43 Issue 1/2, p123
Academic Journal
We prove a weak comparison principle in narrow domains for sub-super solutions to -? u = f( u) in the case 1 < p = 2 and f locally Lipschitz continuous. We exploit it to get the monotonicity of positive solutions to -? u = f( u) in half spaces, in the case $${\frac{2N+2}{N+2} < p\leq 2}$$ and f positive. Also we use the monotonicity result to deduce some Liouville-type theorems. We then consider a class of sign-changing nonlinearities and prove a monotonicity and a one-dimensional symmetry result, via the same techniques and some general a-priori estimates.


Related Articles

  • Monotonicity and 1-Dimensional Symmetry for Solutions of an Elliptic System Arising in Bose-Einstein Condensation. Farina, Alberto; Soave, Nicola // Archive for Rational Mechanics & Analysis;Jul2014, Vol. 213 Issue 1, p287 

    We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system: for every dimension $${N \geqq 2}$$ . In particular, we prove a Gibbons-type conjecture proposed by Berestycki et al.

  • On 3-dimensional f-Kenmotsu manifolds and Ricci solitons. Yildiz, A.; De, U. C.; Turan, M. // Ukrainian Mathematical Journal;Oct2013, Vol. 65 Issue 5, p684 

    The aim of the present paper is to study 3-dimensional f-Kenmotsu manifolds and Ricci solitons. First, we give an example of a 3-dimensional f-Kenmotsu manifold. Then we consider a Riccisemisymmetric 3-dimensional f-Kenmotsu manifold and prove that a 3-dimensional f-Kenmotsu manifold is Ricci...

  • Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$. Poláčik, P. // Archive for Rational Mechanics & Analysis;Jan2011, Vol. 199 Issue 1, p69 

    This paper is devoted to a class of nonautonomous parabolic equations of the form u = Δu + f( t, u) on $${\mathbb{R}^N}$$ . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero,...

  • Local well-posedness of Chern-Simons-Higgs system in the Lorentz gauge. Yuan, Jianjun // Journal of Mathematical Physics;Oct2011, Vol. 52 Issue 10, p103706 

    In this paper, we investigate the local well-posedness of the 2+1-dimensional Chern-Simons-Higgs equations in the Lorentz gauge. By exploiting the null structure in the nonlinear terms of the equations, we reprove the low regularity result by Bournaveas ['Low regularity solutions of the...

  • New Korn-type inequalities and regularity of solutions to linear elliptic systems and anisotropic variational problems involving the trace-free part of the symmetric gradient. Schirra, Oliver // Calculus of Variations & Partial Differential Equations;Jan2012, Vol. 43 Issue 1/2, p147 

    The aim of this note is to investigate a regularity theory for minimizers of energies whose density depends on the trace-free part of the symmetric gradient, where integrands of anisotropic growth are considered. An adequate coercive inequality guarantees the existence of minimizers of such...

  • On relations between γ-operations. Min, Won // Acta Mathematica Hungarica;Jul2012, Vol. 136 Issue 1/2, p129 

    Let exp X be the power set of a non-empty set X. A function γ: exp X→exp X is said to be monotonic iff A⫅ B⫅ X implies γA⫅ γB. Császár [2] investigated relations between the monotonic functions. The purpose of the paper is to investigate some results...

  • On Some Properties for the Sequence of Brualdi-Li Matrices. Xiaogen Chen // Journal of Applied Mathematics;2013, p1 

    Let B2n denote the Brualdi-Li matrix, and let ▂n = Ï€(B2n) denote the Perron value of the Brualdi-Li matrix of order 2n. We prove that 2n(n-1/2-▂n) is monotonically decreasing for all n and ▂n< n-1/2-(e²-1)/4(e²+1)n, where e=2.718281828459045 . . . .

  • The Bernstein theorem for a class of fourth order equations. Zhou, Bin // Calculus of Variations & Partial Differential Equations;Jan2012, Vol. 43 Issue 1/2, p25 

    In this paper, we prove the 2-dimensional Bernstein theorem for a class of fourth order equations including Abreu's equation. The main ingredients of the paper are the a priori estimates and the proof of the strict convexity.

  • A parametrized version of the Borsuk–Ulam theorem. Schick, Thomas; Simon, Robert Samuel; Spież, Stanislaw; Toruńczyk, Henryk // Bulletin of the London Mathematical Society;Dec2011, Vol. 43 Issue 6, p1035 

    We show that for a ‘continuous’ family of Borsuk–Ulam situations, parametrized by points of a compact manifold W, its solution set also depends ‘continuously’ on the parameter space W. By such a family we understand a compact set Z⊂W×Sm×ℝm,...


Read the Article


Sign out of this library

Other Topics