Liouville type theorems for nonlinear elliptic equations on the whole space ℝ^{ℕ}

Mounir, Hsini; Wahid, Sayeb

doi:10.1090/S0002-9939-2011-11112-2

Liouville type theorems for nonlinear elliptic equations on the whole space $\mathbb {R}^N$
HTML articles powered by AMS MathViewer

by Hsini Mounir and Sayeb Wahid PDF

Proc. Amer. Math. Soc. 140 (2012), 2731-2738 Request permission

Abstract:

The aim of this paper is to study the properties of the solutions of $\Delta _{p}u+f_{1}(u)-f_{2}(u)=0$ in all $\mathbb {R}^{N}.$ We obtain Liouville type boundedness for the solutions. We show that $|u|\leq (\frac {\alpha }{\beta })^{\frac {1}{m-q+1}}$ on $\mathbb {R}^{N},$ under the assumptions $f_{1}(u)\leq \alpha u^{p-1}$ and $f_{2}(u)\geq \beta u^{m},$ for some $0<\alpha \leq \beta$ and $m>q-1\geq p-1>0.$ If $u$ does not change sign, we prove that $u$ is constant.

References

E. Acerbi and N. Fusco, Regularity for minimizers of nonquadratic functionals: the case $1<p<2$, J. Math. Anal. Appl. 140 (1989), no. 1, 115–135. MR 997847, DOI 10.1016/0022-247X(89)90098-X
Azeddine Baalal and Nedra BelHaj Rhouma, Dirichlet problem for quasi-linear elliptic equations, Electron. J. Differential Equations (2002), No. 82, 18. MR 1927896
H. Berestycki, F. Hamel, and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J. 103 (2000), no. 3, 375–396. MR 1763653, DOI 10.1215/S0012-7094-00-10331-6
Haïm Brezis, Frank Merle, and Tristan Rivière, Quantization effects for $-\Delta u=u(1-|u|^2)$ in $\textbf {R}^2$, Arch. Rational Mech. Anal. 126 (1994), no. 1, 35–58. MR 1268048, DOI 10.1007/BF00375695
Luis Caffarelli, Nicola Garofalo, and Fausto Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math. 47 (1994), no. 11, 1457–1473. MR 1296785, DOI 10.1002/cpa.3160471103
Lucio Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré C Anal. Non Linéaire 15 (1998), no. 4, 493–516 (English, with English and French summaries). MR 1632933, DOI 10.1016/S0294-1449(98)80032-2
J. I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries. Vol. 1. Elliptic Equations. Pitman Res. Notes in Math., Vol. 106, Boston, 1985.
Y. Du and Z. Guo, Liouville type results and eventual flatness of positive solutions for $p$-Laplacian equations, Adv. Differential Equations 7 (2002), no. 12, 1479–1512. MR 1920542
Yihong Du and Zongming Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. Anal. Math. 89 (2003), 277–302. MR 1981921, DOI 10.1007/BF02893084
Yihong Du and Li Ma, Logistic type equations on $\Bbb R^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc. (2) 64 (2001), no. 1, 107–124. MR 1840774, DOI 10.1017/S0024610701002289
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
R. M. Hervé and M. Hervé, Quelques propriétés des solutions de l’équation de Ginzburg-Landau sur un ouvert de $\textbf {R}^2$, Potential Anal. 5 (1996), no. 6, 591–609 (French, with English summary). MR 1437586, DOI 10.1007/BF00275796
L. K. Martinson and K. B. Pavlov, The effect of magnetic plasticity in non-Newtonian fluids. Magnit. Gidrodinamika, 2 (1969), 69-75.
L. K. Martinson and K. B. Pavlov, Unsteady shear flows of a conducting fluid with a rheological power flow. Magnit. Gidrodinamika, 3 (1970), 5869-5875.
V. M. Mikljukov, Asymptotic properties of subsolutions of quasilinear equations of elliptic type and mappings with bounded distortion, Mat. Sb. (N.S.) 111(153) (1980), no. 1, 42–66, 159 (Russian). MR 560463
Yu. G. Reshetnyak, Index boundedness condition for mappings with bounded distortion. Siberian Math. J. 9 (1968), 281-285.
James Serrin and Henghui Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 (2002), no. 1, 79–142. MR 1946918, DOI 10.1007/BF02392645
K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), no. 3-4, 219–240. MR 474389, DOI 10.1007/BF02392316
A. Zhao, Qualitative properties of solutions of quasilinear equations. Electronic J. Differential Equations, No. 99 (2003), 1-18.