In this paper, we consider the following semilinear elliptic equation:
. The exponent
with the real numbers b
By this definition, .
With respect to the functions V and K, we assume that
(A1) for every , and .
) There exist
A typical example for Eq. (1
.1) with V
) and (A2
) is the equation
When , the potentials are vanishing at infinity and when , the potentials are coercive.
Equation (1.1) arises in various applications, such as chemotaxis, population genetics, chemical reactor theory and the study of standing wave solutions of certain nonlinear Schrödinger equations. Therefore, they have received growing attention in recent years (one can see, e.g., [1–6] and [7–10] for reference).
Under the above assumptions, Eq. (1
.1) has a natural variational structure. For an open subset Ω in
be the collection of smooth functions with a compact support set in Ω. Let E
be the completion of
with respect to the inner product
From assumptions (A1
) and (A2
), we deduce that
are two equivalent norms in the space
Therefore, there exists
Moreover, assumptions (A1
) and (A2
) imply that there exists
Then, by the Hölder and Sobolev inequalities (see, e.g.
], Theorem 1.8]), we have, for every
is a constant independent of u
. It follows that there exists a constant
This implies that E
can be embedded continuously into the weighted
Then the functional
is well defined in E. And it is easy to check that Φ is a functional and the critical points of Φ are solutions of (1.1) in E.
In a recent paper , Alves and Souto proved that the space E can be embedded compactly into if and and Φ satisfies the Palais-Smale condition consequently. Then, by using the mountain pass theorem, they obtained a nontrivial solution for Eq. (1.1). Unfortunately, when , the embedding of E into is not compact and Φ no longer satisfies the Palais-Smale condition. Therefore, the ‘standard’ variational methods fail in this case. From this point of view, should be seen as a kind of critical exponent for Eq. (1.1). If the potentials V and K are restricted to the class of radially symmetric functions, ‘compactness’ of such a kind is regained and ‘standard’ variational approaches work (see  and ). However, this method does not seem to apply to the more general equation (1.1) where K and V are non-radially symmetric functions.
It is not easy to deal with Eq. (1.1) directly because there are no known approaches that can be used directly to overcome the difficulty brought by the loss of compactness. However, in this paper, through an interesting transformation, we find an equivalent equation for Eq. (1.1) (see Eq. (2.9) in Section 2). This equation has the advantages that its Palais-Smale sequence can be characterized precisely through the concentration-compactness principle (see Theorem 5.1), and it possesses partial compactness (see Corollary 5.8). By means of these advantages, a positive solution for this equivalent equation and then a corresponding positive solution for Eq. (1.1) are obtained.
Before stating our main result, we need to give some definitions.
be the Sobolev space endowed with the norm and the inner product
respectively, and let
be the function space consisting of the functions on
that are p
can be embedded continuously into
. Therefore, the infimum
We denote this infimum by .
Our main result reads as follows.
Theorem 1.1 Under assumptions
), if b
, s and p satisfy
then Eq. (1.1) has a positive solution .
We should emphasize that condition (1.10) can be satisfied in many situations. For
be the closure of
. Under assumptions (A1
) and (A2
), we have
Then, for any
, there exist
It follows from this inequality and
is small enough such that
This implies that (1.10) is satisfied if ϵ is chosen such that .
be a Banach space and
. We denote the Fréchet derivative of φ
. The Gateaux derivative of φ
is denoted by
. By → we denote the strong and by ⇀ the weak convergence. For a function u
denotes the functions
. The symbol
denotes the Kronecker symbol:
We use to mean as .