On positive solutions of a class of nonlinear elliptic equations

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Abstract

We investigate the existence of positive solutions vanishing as |x|+ to the semilinear elliptic equation Δu+f(x,u)+g(|x|)xu=0, |x|>A>0, in Rn, n3, under mild assumptions on the functions f, g. Conditions for these solutions to be radial are also given.

Introduction

The semilinear elliptic equation Δu+f(x,u)+g(|x|)xu=0,|x|>A>0, constitutes the subject of numerous investigations in the last few years. Constantin [2] studied the existence of positive solutions assuming that f(x,u)0. A further work [3] discusses the presence of positive, vanishing as |x| goes to +, solutions to (1) under some flexible hypotheses. Similar investigations were performed in [14], [15], [16], [5]. Throughout these papers a variety of approaches are developed: the Banach contraction principle and exponentially rescaled metrics, subsolutions and supersolutions, variational techniques. Recent contributions to the field are [6], [7].

For our hypotheses, consider GA={xRn:|x|>A}, where n3. We assume in the following that f:GA×RR is locally Hölder continuous and g:[A,+)R is continuously differentiable. We suppose also that 0f(x,t)a(|x|)w(t),t[0,+),xGA, where a:[A,+)[0,+), w:[0,+)[0,+) are continuous. In some cases we shall assume further that there exist ε, M>0 and α1 such that w(t)Mtα,t[0,ε].

To establish the existence of positive solutions u to (1) in GB, where BA is large enough, with u(x)0 as |x|+ (we shall refer to such u’s as asymptotically vanishing solutions), the recent research literature relies on the condition A+ra(r)dr<+.

The aim of this note is threefold. Firstly, we show that such solutions exist even if condition (4) does not hold. Secondly, we give results in the spirit of [6], [7] under some mild hypotheses. Thirdly, we provide conditions for the positive, asymptotically vanishing solutions of (1) to be radial. The results will be established using the subsolution and supersolution approach, certain techniques from the qualitative theory of ordinary differential equations, and the strong maximum principle.

By a subsolution of (1) we understand a function wC2(GB)C(G¯B) such that Δw+f(x,w)+g(|x|)xw0 for |x|>B. As for the supersolution, the sign of the inequality should be reversed.

Let us close this section by recalling [4], where a phase-plane analysis has been employed to undertake similar investigations in the case n=2. It seems that there is no known way of translating such techniques to the n3 case. This is why we emphasize that our results are specifically designed for the latter case.

Section snippets

The asymptotically vanishing solution

The following lemma will be needed in our investigation.

Lemma 1

Cf. [13]

If, for some B>A , there exist a nonnegative subsolution w and a positive supersolution v to(1)in GB , such that w(x)v(x) for xG¯B , then(1)has a solution u in GB such that wuvthroughout G¯B . In particular, u=v on |x|=B .

In the first result dealing with positive and asymptotically vanishing solutions to (1), we shall employ the Wronskian technique [12].

Theorem 1

  • (i)

    Assume that(2)holds,A+rg(r)dr>,where g(r)=min(g(r),0) for all rA , and

Radial solutions

Define the range of u:GBR to be R(u)={u(x):xGB}. We have the following lemma.

Lemma 2

Cf. [6]

Suppose that u is a solution to(1)with f(x,u)=f(|x|,u) nonincreasing in u in GB×R(u) , such thatu(x)0as |x|+andu(x)=uBwhenever |x|=B.Then u(x)=u(|x|) is a radial function, and it is the unique solution to(1)with range within R(u) satisfying these criteria.

Lemma 2 includes two assumptions generally not satisfied by the solutions found in Theorem 1, Theorem 2. These both involve f(x,u). There exist solutions to

Acknowledgements

The second author (O.G.M.) was partially financed during this research by the National Science Foundation of Sweden, Grant VR 621-2003-5287. Both M.E. and O.G.M. wish to express their gratitude to an anonymous referee for very valuable comments.

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