Compactness and existence results for the Laplace equation
Abstract
Given and two measurable functions and , , we define the weighted spaces
and study the compact embeddings of the radial subspace of into , and thus into () as a particular case. We consider exponents that can be greater or smaller than . Our results do not require any compatibility between how the potentials and behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately. We then apply these results to the investigation of existence and multiplicity of finite energy solutions to nonlinear Laplace equations of the form
where and with fixed may
be vanishing or unbounded at zero or at infinity. Both the cases of
super and sub linear in are studied and, in the sub linear case,
nonlinearities with are also
considered.
MSC (2010): Primary 35J92; Secondary 35J20, 46E35, 46E30
Keywords: Quasilinear elliptic equations with Laplacian, unbounded or decaying potentials, weighted Sobolev spaces, compact embeddings
1 Introduction
In this paper we pursue the work we made in papers [4, 3, 9], where we studied embedding and compactness results for weighted Sobolev spaces in order to get existence and multiplicity results for semilinear elliptic equations in , by variational methods.
In the present paper, we face nonlinear elliptic Laplace equations with radial potentials, whose prototype is
(1) 
(more general nonlinear terms will be actually considered in the following). Here , is a continuous nonlinearity satisfying and are given potentials, which may be vanishing or unbounded at the origin or at infinity.
To study this problem we introduce the weighted Sobolev space
equipped with the standard norm
(2) 
and we say that is a weak solution to (1) if
The natural approach in studying weak solutions to equation (1) is variational, since these solutions are (at least formally) critical points of the Euler functional
where . Then the problem of existence is easily solved if does not vanish at infinity and is bounded, because standard embeddings theorems of and its radial subspace into the weighted Lebesgue space
are available (for suitable ’s). As we let and to vanish, or to go to infinity, as or , the usual embeddings theorems for Sobolev spaces are not available anymore, and new embedding theorems need to be proved. This has been done in several papers: see e.g. the references in [4, 3, 9] for a bibliography concerning the usual Laplace equation, and [1, 13, 7, 14, 15, 17, 18, 5, 19, 6, 8, 10] for equations involving the laplacian.
The main novelty of our approach (in [4, 3] and in the present paper) is twofold. First, we look for embeddings of not into a single Lebesgue space but into a sum of Lebesgue spaces . This allows to study separately the behaviour of the potentials and at and , and to assume different set of hypotheses about these behaviours. Second, we assume hypotheses not on and separately but on their ratio, so allowing asymptotic behaviours of general kind for the two potentials.
Thanks to these novelties, our embedding results yield existence of solutions for (1) in cases which are not covered by the previous literature. Moreover, one can check that our embeddings are also new in some of the cases already treated in previous papers (see e.g. Example 3.5), thus giving existence results which improve some wellknown theorems in the literature.
The proofs of our embedding theorems for the space are generalizations of those presented in [4] for the Hilbertian case . The generalizations to the case are not difficult but boring and lengthy, because one needs to repeat a lot of detailed computations, the basic ideas remaining the same. In view of this, in the present paper we limit ourselves to state our embedding results and to present in detail some examples, leading to new existence results for equation (1). For all the proofs, with full details, we refer the reader to the specific document [2], which is essentially a longer version of Section 2 below.
This paper is organized as follows. In Section 2 we state our main results: a general result concerning the embedding properties of into (Theorem 2.1) and some explicit conditions ensuring that the embedding is compact (Theorems 2.2, 2.3, 2.5 and 2.7). In Section 3 we apply our compactness results to some examples, with a view to both illustrate how to use them in concrete cases and to compare them with the related literature. In Section 4 we present existence and multiplicity results for equations like (1), but with more general nonlinearities, whose proofs are given in Section 5.
Notations. We end this introductory section by collecting some notations used in the paper.
We denote , , , and for any .
and denote the norm and the dual space of a Banach space , in which and mean strong and weak convergence respectively.
denotes continuous embeddings.
is the space of the infinitely differentiable real functions with compact support in open.
For any measurable set , and are the usual real Lebesgue spaces and, if is a measurable function, is the real Lebesgue space with respect to the measure ( stands for the Lebesgue measure on ). In particular, if is measurable, we denote for any measurable set .
For , is the Sobolev critical exponent and is the usual Sobolev space, which identifies with the completion of with respect to the norm of the gradient. is the radial subspace of .
2 Compactness results
Assume and consider two functions such that:

is a measurable function such that for some

is a measurable function such that for some .
Define the function spaces
(3) 
and let be the standard norm (2) in (and ). Assumption implies that the spaces and are nontrivial, while hypothesis ensures that is compactly embedded into the weighted Lebesgue space for every and (see [2, Lemma 3.1]). In what follows, the summability assumptions in and will not play any other role than this.
Given and , we define the following functions of and :
(4)  
(5) 
Clearly is nondecreasing, is nonincreasing and both of them can be infinite at some .
Our first result concerns the embedding properties of into the sum space
We recall from [5] that such a space can be characterized as the set of measurable mappings for which there exists a measurable set such that . It is a Banach space with respect to the norm
and the continuous embedding holds for all . The assumptions of our result are quite general, sometimes also sharp (see claim (iii)), but not so easy to check, so that the next results will be devoted to provide more handy conditions ensuring such general assumptions.
Theorem 2.1.
Let , let , be as in , and let .

If
then is continuously embedded into .

If
then is compactly embedded into .

If and , then conditions and are also necessary to the above embeddings.
Observe that, of course, implies . Moreover, these assumptions can hold with and therefore Theorem 2.1 also concerns the embedding properties of into , .
We now look for explicit conditions on and implying for some and . More precisely, we will ensure through a more stringent condition involving the following functions of and :
(6)  
(7) 
Note that is nondecreasing, is nonincreasing and both can be infinite at some . Moreover, for every one has and , so that is a consequence of the following, stronger condition:
In Theorems 2.2 and 2.7 we will find ranges of exponents such that . In Theorems 2.3 and 2.5 we will do the same for exponents such that . Condition then follows by joining Theorem 2.2 or 2.7 with Theorem 2.3 or 2.5.
For and , define two functions and by setting
and
Note that and if and only if .
The first two Theorems 2.2 and 2.3 only rely on a power type estimate of the relative growth of the potentials and do not require any other separate assumption on and than and , including the case (see Remark 2.4.1).
Theorem 2.2.
Let and let , be as in , . Assume that there exists such that almost everywhere in and
(8) 
Then for every such that
(9) 
Theorem 2.3.
Let and let , be as in , . Assume that there exists such that for almost every and
(10) 
Then for every such that
(11) 
We observe explicitly that for every one has
Remark 2.4.

The inequality is equivalent to . Then, in (9), such inequality is automatically true and does not ask for further conditions on and .

The assumptions of Theorems 2.2 and 2.3 may hold for different pairs , . In this case, of course, one chooses them in order to get the ranges for as large as possible. For instance, if is essentially bounded in a neighbourhood of 0 and condition (8) holds true for a pair , then (8) also holds for all pairs such that and . Therefore, since is nondecreasing in and is increasing in and decreasing in , it is convenient to choose and the best interval where one can take is with (we mean ).
For any , and , define
(12) 
Of course and are undefined if and , respectively.
The next Theorems 2.5 and 2.7 improve the results of Theorems 2.2 and 2.3 by exploiting further informations on the growth of (see Remarks 2.6.2 and 2.8.3).
Theorem 2.5.
Let and let , be as in , . Assume that there exists such that for almost every and
(13) 
and
(14) 
Then for every such that
(15) 
where and
For future convenience, we define three functions , and by setting
(16) 
Then an explicit description of is the following: for every we have
(17) 
where for every and if .
Remark 2.6.

Denote for brevity. If , then one has
so that, under assumption (14), Theorem 2.5 improves Theorem 2.3. Otherwise, if , we have and Theorems 2.5 and 2.3 give the same result. This is not surprising, since, by Hardy inequality, the space coincides with if and thus, for , we cannot expect a better result than the one of Theorem 2.3, which covers the case of , i.e., of .

Description (17) shows that and are not relevant in inequality (15) if . On the other hand, if , both and turn out to be increasing in and hence it is convenient to apply Theorem 2.5 with the smallest for which (14) holds. This is consistent with the fact that, if (14) holds with , then it also holds with every such that .
In order to state our last result, we introduce, by the following definitions, an open region of the plane, depending on and . Recall the definitions (12) of the functions and . We set
(18) 
Notice that because . For more clarity, is sketched in the following five pictures, according to the five cases above. Recall the definitions (16) of the functions , and .





Theorem 2.7.
Let and let , be as in , . Assume that there exists such that almost everywhere in and
(19) 
and
(20) 
Then for every such that
(21) 
Remark 2.8.

Condition (21) also asks for a lower bound on , except for the case , as it is clear from Figures 15.
3 Examples
In this section we give some examples of application of our compactness results, which might clarify how to use them in concrete cases. We also compare them with the most recent and general related results [15, 14], which unify and extend the previous literature. Essentially, the spirit of the results of [15, 14] is the following: assuming that are continuous and satisfy power type estimates of the form:
(22) 
the authors find two limit exponents and such that the embedding is compact if . The case with is studied in [15], the one with in [14]. The exponent is always defined, while exists provided that suitable compatibility conditions between and occur. Moreover, the condition also asks for , which is a further assumption of compatibility between the behaviours of the potentials at zero and at infinity.
In the following it will be always understood that .
Example 3.1.
Consider the potentials
Since satisfies (14) with (cf. Remark 2.6.3 for the best choice of ), we apply Theorems 2.2 and 2.5, where we choose and . Note that implies and . Hence we get that holds for every exponents such that
(23) 
If , then one has and therefore Theorem 2.1 gives the compact embedding
(24) 
If , then and we get the compact embedding
Since and are power potentials, one can also apply the results of [15], which give two suitable limit exponents and such that the embedding is compact if . These exponents and are exactly exponents and of (23) respectively, so that one obtains (24) again provided that (which implies ). If , instead, one gets and no result is avaliable in [15]. The results of [14] do not apply to and , since the top and bottom exponents of [14] turn out to be equal to one another for every .