Let and be the unit open ball in . This paper is concerned with the behavior of nonnegative solutions of
where α and β are real numbers satisfying
We say that u is a nonnegative solution of (1.1) if is nonnegative and satisfies (1.1) pointwise. In addition, we say that a nonnegative solution u of (1.1) is singular if u is unbounded in any punctured ball , with .
The case in (1.1) is by now well understood; in their pioneering work , Gidas and Spruck established a series of results that completely characterize the asymptotic behavior of local solutions of (1.1) (with ). The main goal of this paper is to obtain similar results for (1.1) when the exponents α and β are in the range given by (1.2).
Our main result is the following.
either u has a removable singularity at the origin,
or u is a singular solution and satisfies
For , we recover the result in [4, Theorem 1.3]. Let us note that in the case , the approach in  relies to a large extend on the properties of the scaling function (). Thus, if u is a solution of (1.1) (with ), then so is . A similar scaling is not available to us in case due to the presence of the logaritmic term in (1.1). In turn, we shall take advantage of the result in [1, Theorem 1.1] which allows us to derive that singular solutions of (1.1) are asymptotically radial. The exact asymptotic behavior (1.3) is further deduced by looking at the corresponding ODE of the scaled function in polar coordinates.
The asymptotic behavior of nonnegative singular solutions has been studied in various settings. In addition to the classical results  and , Korevaar et al.  derived the improved asymptotic behavior of the nonnegative singular solutions of by a more geometric approach. Meanwhile, C. Li  extended the result on the asymptotic radial symmetry of singular solutions of for a more general considered in . Recently, the asymptotic radial symmetry has been achieved for other operators, such as conformally invariant fully nonlinear equations [5, 8], fractional equations , and fractional p-laplacian equations .
This paper extends the classical argument in  and  to a log-type nonlinearity. One of the key observations is that from the asymptotic radial symmetry achieved in  for nonnegative solutions of , one can obtain an optimal asymptotic upper bound for . Hence, we are left with preserving the optimality by transforming to u under a suitable inverse mapping.
This observation indeed allows us to consider a more general class of equations of the type
where f is a slowly varying function at infinity, under some additional assumptions. A typical example is
where are positive integers, are real numbers and for with . However, we shall not specify the additional assumptions for the nonlinearity f as they turn out to involve technical and cumbersome computations. Hence, we present the argument only with in order to simplify the presentation.
Throughout the paper, we shall write if uniformly in x, where depends at most on n, α and β. We shall also use the notation as to denote that as .
2 Asymptotic behavior around a non-removable singularity
Let denote the spherical average of u on the ball of radius r, that is,
The following result is a slight modification of [1, Theorem 1.1].
Let u be a nonnegative solution of
with an isolated singularity at the origin. Suppose that is a locally Lipschitz function, which in a neighborhood of infinity satisfies the conditions below:
is nondecreasing in t,
for some and .
The original result in [1, Theorem 1.1] requires condition (i) above to be satisfied for all , but a careful analysis of its proof shows that this condition is enough to hold in a neighborhood of infinity.
The next lemma provides an asymptotic upper bound for .
Throughout this proof, depends at most on n, α and β, and may differ from one line to another. As mentioned earlier, we have , and thus from the divergence theorem and (1.1), we deduce that
Taking r small enough, and using (2.1) and the fact that is increasing for large s, we deduce that
Hence, from the assumption as and the fact , it follows that
Note that for any sufficiently large s satisfying , we have
whence we may proceed from the integral above as
for sufficiently small . Thus, we arrive at
However, since for sufficiently large s, we arrive at (2.2). ∎
Let us next define
with and .
for large and , where
We take small enough such that in , and set . In what follows, we take and , unless stated otherwise. For notational convenience, let us write
so that . Since and , we have
where the left and right side are evaluated in and, respectively, in , and by and , we denoted and, respectively, . Setting
we observe that and , and therefore
where we used the fact that and in deriving the second identity.
On the other hand, we know from (2.6) that
from which we may also deduce that
One may also notice from (2.6) that
Let us define
Averaging (2.7) over , we obtain
for large t.
For notational convenience, let denote . Also let us denote by the annulus . From (2.1), we have
Therefore, it follows from the interior gradient estimates that
for any and . Due to (2.18), we have
Using the above estimate together with Lemma 2.4, we have
An integration over in the above estimate, will lead us to (2.22). ∎
We have either
with A given by (1.4).
Now we multiply (2.17) by and integrate it over , which leads us to
On the other hand, since , a further integration by parts produces
where the second equality can be deduced analogously to the derivation of (2.24). Similarly, we also observe that
Although we do not know yet if converges as , we still know from (2.20) that it converges along a subsequence. Denoting by a limit value of along a subsequence, say , after passing to the limit in (2.17), with , we obtain, from (2.22), (2.27) and (2.33), that
Thus, in view of (1.4), we have
Now the continuity of implies that converges as (without extracting any subsequence) either to 0 or A. If there are two distinct sequences and such that and , then by the intermediate value theorem, there must exist some other such that , which violates (2.34). Thus, the proof is completed. ∎
We are now in a position to prove Theorem 1.1.
Proof of Theorem 1.1.
Henceforth, let us suppose that
The rest of the argument follows closely that of the proof in [1, Theorem 1.3].
Thus, the decay of is determined by the negative root of
Since , the root λ is
Therefore, we have
In view of (2.15), we obtain
Now if , then we deduce from (2.35) that as , from which, combined with (2.1), it follows that as . Hence, the origin is a removable singularity. Similarly, if , then (2.35) implies that , and thus the origin is again a removable singularity.
Hence, we are only left with the case . Since , (2.35) implies that
for each , for some constant depending on n, α, β and q. Therefore, for any , and in particular for . This implies that for , so for , proving again that the origin is a removable singularity. Thus, the proof of Theorem 1.1 (i) is completed. ∎
This work was initiated in June 2017 when M. Ghergu was visiting the Royal Institute of Technology (KTH) in Stocholm. The invitation and hospitality of the Department of Mathematics in KTH in greatly acknowledged.
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About the article
Published Online: 2018-06-14
Funding Source: National Research Foundation of Korea
Award identifier / Grant number: NRF-2014-Fostering Core Leaders of the Future Basi
Award identifier / Grant number: CH2015-6380
S. Kim has been supported by National Research Foundation of Korea (NRF) grant funded by the Korean government (NRF-2014-Fostering Core Leaders of the Future Basic Science Program). H. Shahgholian has been supported in part by Swedish Research Council.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 995–1003, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0261.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0