Definition 3.1 A weak solution of problem (P) is any such that
for all . We define the corresponding energy functional of problem (P) as
Then is a functional and the critical points of it are weak solutions of problem (P).
Proof of Theorem 1.1 Let Φ, J, Ψ as above. So, for each , one has
Therefore, the weak solutions of problem (P) are exactly the solutions of the equation
In view of Proposition 2.4 (or  for details), certainly, Φ is a continuously Gâteaux differentiable and sequentially weakly lower semi-continuous functional whose Gâteaux derivative admits a continuous inverse on . Moreover, J and Ψ are continuously Gâteaux differentiable functionals whose Gâteaux derivatives are compact. Obviously, Φ is bounded on each bounded subset of X under our assumptions.
By Proposition 2.3, set just as before, then for ,
Actually, for , set , then we can obtain
Hence we have
So, there exists a constant such that
holds for any .
holds for any , where constants , . Here, by using conditions (j3) and (ii) of Proposition 2.1, combining the two inequalities above, we can obtain
Due to , , we get
Then assumption (2.1) of Proposition 2.5 is fulfilled.
Next, we derive that assumption (2.2) is also fulfilled. It is easy to verify the conditions of Proposition 2.6. Let , we can easily have
Then there exist and such that and (2.3) is satisfied.
There is a point since it is a nonempty bounded open set. Let , put
where is the open ball in of radius r centered at x,
Let , then by (j1) we can derive that
From (j2), , such that
By (j3), there are nine positive real numbers () according to , larger or smaller than η and 1. For example, when , some
Set , then
Hence, fix γ such that . And for , by the Sobolev embedding theorem ( is continuous), there exist suitable positive constants and such that
Since , , we have
We choose as above such that . Fix such that . Then we divide the proof into two cases.
For , by (3.2) we have
By (3.3), we obtain
For , from (3.1) we get
From (3.3), we have
For any , we can obtain , i.e.,
Then we can have
This inequality implies
Therefore we have
So, we can get that
Hence we can find , and satisfying (2.3). Also, we can find ρ satisfying
Put , moreover, , fulfil the assumption of Proposition 2.6. So, applying Proposition 2.6, we can easily get that (2.2) is fulfilled.
Thus, Φ, J and Ψ fulfil all the assumptions of Proposition 2.5, and our conclusion follows from Proposition 2.5. □
Remark Applying Theorem 2.1 in  to the proof of Theorem 1.1, an upper bound of the interval of parameters λ, for which (P) has at least three weak solutions, is obtained. To be precise, in the conclusion of Theorem 1.1, one has
for each and as in the proof of Theorem 1.1 (namely, ).