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A class of quasilinear differential inequalities whose solutions are ultimately constant

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Abstract

Let Ω be a domain in R n with compact complement and let T be a quasilinear elliptic or degenerate elliptic operator associated with functions u ε C 2(Ω). This paper is a study of solutions of (sgn u) Tu≧f(¦u¦, ¦ grad u¦) where f belongs to a class of functions here termed bifurcation functions. The main condition on f is that uniqueness fails for the ordinary differential equation y′'=f(y, y′) with the initial condition y(0)=y′(0)=0. The conclusion is that u is constant for large ¦x¦ and hence, under mild supplementary hypotheses, u has compact support. Examples show that the results fail if the assumptions on f are only slightly weakened, so that the class of f is, essentially, the largest class for which the results can be stated truly.

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  1. Department of Mathematics, University of California, Los Angeles

    Ray Redheffer

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  1. Ray Redheffer
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To James Serrin, in celebration of his 60th birthday

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Redheffer, R. A class of quasilinear differential inequalities whose solutions are ultimately constant. Arch. Rational Mech. Anal. 99, 165–187 (1987). https://doi.org/10.1007/BF00275876

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  • DOI: https://doi.org/10.1007/BF00275876

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