Nonlinear Analysis: Theory, Methods & Applications
On positive solutions of a class of nonlinear elliptic equations
Introduction
The semilinear elliptic equation constitutes the subject of numerous investigations in the last few years. Constantin [2] studied the existence of positive solutions assuming that . A further work [3] discusses the presence of positive, vanishing as 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 , where . We assume in the following that is locally Hölder continuous and is continuously differentiable. We suppose also that where , are continuous. In some cases we shall assume further that there exist , and such that
To establish the existence of positive solutions to (1) in , where is large enough, with as (we shall refer to such ’s as asymptotically vanishing solutions), the recent research literature relies on the condition
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 such that for . 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 . It seems that there is no known way of translating such techniques to the 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 If, for some , there exist a nonnegative subsolution and a positive supersolution to(1)in , such that for , then(1)has a solution in such that throughout . In particular, on .Cf. [13]
In the first result dealing with positive and asymptotically vanishing solutions to (1), we shall employ the Wronskian technique [12]. Theorem 1 Assume that(2)holds,where for all , and
Radial solutions
Define the range of to be . We have the following lemma. Lemma 2 Suppose that is a solution to(1)with nonincreasing in in , such thatThen is a radial function, and it is the unique solution to(1)with range within satisfying these criteria.Cf. [6]
Lemma 2 includes two assumptions generally not satisfied by the solutions found in Theorem 1, Theorem 2. These both involve . 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.
References (16)
- et al.
Existence and asymptotic behavior of solutions of a boundary value problem on an infinite interval
Math. Comput. Modelling
(2005) Positive solutions of quasilinear elliptic equations
J. Math. Anal. Appl.
(1997)- et al.
Positive solutions of quasilinear elliptic equations in two-dimensional exterior domains
Nonlinear Anal. TMA
(2000) On radial solutions of certain semi-linear elliptic equations
Nonlinear Anal. TMA
(2006)Positive solutions for second-order nonlinear differential equations
Nonlinear Anal. TMA
(2006)- et al.
Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations
Nonlinear Anal. TMA
(2002) - et al.
Positive solutions of quasilinear elliptic equations in exterior domains
J. Math. Anal. Appl.
(1980) On the existence of positive radial solutions for a certain class of elliptic BVPs
J. Math. Anal. Appl.
(2004)
Cited by (25)
Uncountable sets of finite energy solutions for semilinear elliptic problems in exterior domains
2019, Journal of Mathematical Analysis and ApplicationsIncreasing sequences of positive evanescent solutions of nonlinear elliptic equations
2015, Journal of Differential EquationsPositive solutions to a class of second-order semilinear elliptic equations in an exterior domain
2013, Nonlinear Analysis, Theory, Methods and ApplicationsCitation Excerpt :We present a criterion without conditions (1.3) and (1.4). We use the supersolution-subsolution method and the Schauder–Tikhonov fixed point theory, which are based on a work of Mats Ehrnström (see [6]). We now conclude this introduction by outlining the rest of this paper.
A class of function spaces and its application in solving second-order nonlinear differential equations
2011, Nonlinear Analysis, Theory, Methods and ApplicationsThe continuous dependence on parameters of solutions for a class of elliptic problems on exterior domains
2010, Nonlinear Analysis, Theory, Methods and ApplicationsExistence of global positive solutions of semilinear elliptic equations
2009, Nonlinear Analysis, Theory, Methods and Applications