In this section, we shall give the proof of the existence of the positive eigenvalue for the problem (E), by applying the basic properties of the spaces and , which were given in the previous section.
Throughout this paper, let satisfy the log-Hölder continuity condition (2.2) and with the norm
which is equivalent to norm (2.1).
Definition 3.1 We say that is a weak solution of the problem (E) if
for all .
Denote
(we allow the case that one of these sets is empty). Then it is obvious that . We assume that:
(H1) , , and .
-
(HJ1)
satisfies the following conditions: is measurable on for all and is locally absolutely continuous on for almost all .
-
(HJ2)
There are a function and a nonnegative constant b such that
for almost all and for all .
-
(HJ3)
There exists a positive constant c such that the following conditions are satisfied for almost all :
(3.1)
for almost all . In case , assume that condition (3.1) holds for almost all , and in case , assume that for almost all instead
(3.2)
-
(HJ4)
For all and all , the estimate holds
where .
Let us put
and define the functional by
Then [5], and its Gateaux derivative is
(3.3)
Let be a real-valued function. We assume that the function f satisfies the Carathéodory condition in the sense that is measurable for all and is continuous for almost all . Denote
where q is given in (H1) and . We assume that
(F1) For all , , and there is a nonnegative measurable function m with such that
Denoting , it follows from (F1) that
Define the functional by
Then it is easy to check that , and its Gateaux derivative is
(3.4)
for any . Let us consider the following quantity:
(3.5)
For the case of and , where satisfies a suitable condition, Benouhiba [6] proved that . In this section, we shall generalize the conditions on f and ϕ to satisfy still.
The following lemma plays a key role in obtaining the main result in this section.
Lemma 3.2 Assume that assumptions (HJ3)-(HJ4), (H1), and (F1) hold and satisfy
then the functionals Φ and Ψ satisfy the following relations:
(3.6)
and
(3.7)
Proof Applying Lemmas 2.2, 2.4 and 2.6, we get
(3.8)
for some positive constant C. Let u in X with . Then it follows from (HJ3), (HJ4), (3.8) and Lemma 2.3(3) that
(3.9)
Since , we conclude that
Next, we show that relation (3.7) holds. From (H2), there exists a positive constant δ such that , and thus we have
(3.10)
Let be a measurable function such that
(3.11)
holds for almost all and
(3.12)
Then we have and . Let with . Then it follows from (F1′) and Lemma 2.2 that
Therefore, without loss of generality, we may suppose that . From the inequality above, by using Lemma 2.3, Lemma 2.2 and Lemma 2.4 in order, we have
where
By Young’s inequality, we get
Using (3.11), we get that
holds for almost all . Hence it follows from Lemma 2.6 that
(3.13)
for some positive constant C. Therefore, we obtain that
From (3.10), with the inequality above, we conclude that relation (3.7) holds. □
Lemma 3.3 Assume that (HJ1)-(HJ3) and (H1) hold. Then Φ is weakly lower semi-continuous, i.e., in X implies that .
Proof Suppose that in X as . Since (HJ3) implies that is strictly monotone on X, we have that Φ is convex, and so,
for any n. Then we get that
The proof is complete. □
Lemma 3.4 Assume that (H1) and (F1) hold. For any and all , the following estimate holds:
(3.14)
where is either or .
Proof Applying Lemmas 2.2, 2.4 and 2.6, we get
for some positive constant C. □
Lemma 3.5 Assume that (H1) and (F1) hold. For almost all and all , the following estimate holds:
(3.15)
Proof Since , estimate (3.15) is obtained from (F1′) and Young’s inequality. □
Lemma 3.6 Assume that (H1) and (F1) hold. Then Ψ is weakly-strongly continuous, i.e., in X implies that .
Proof Let be a sequence in X such that in X. Then is bounded in X. By Lemma 3.4, for each , there is a positive constant such that
(3.16)
holds for each . It follows from Lemma 3.5 that the Nemytskij operator
is continuous from into ; see Theorem 1.1 in [26]. This together with Lemma 2.5 yields that
(3.17)
Using (3.16) and (3.17), we deduce that as . The proof is complete. □
We are in a position to state the main result about the existence of the positive eigenvalue for the problem (E).
Theorem 3.7 Assume that (HJ1)-(HJ4), (H1), (H2), and (F1) hold. Then is a positive eigenvalue of the problem (E). Moreover, the problem (E) has a nontrivial weak solution for any .
Proof It is trivial by (3.5) that . Suppose to the contrary that . Let be a sequence in such that
As in (3.9), we have
for some positive constant C. Since , we obtain that as . Hence it follows from Lemma 3.2 that
which contradicts with the hypothesis. Hence we get . The analogous argument as that in the proof of Theorem 4.5 in [5] proves that is an eigenvalue of the problem (E); see also Theorem 3.1 in [6].
Finally, we show that the problem (E) has a nontrivial weak solution for any . Notice that u is a weak solution of (E) if and only if u is a critical point of . Assume that is fixed. Let with . With the help of (HJ3) and (HJ4), it follows from proceeding as in the proof of relation (3.13) in Lemma 3.2 that
Since , the inequality above implies that as for , that is, is coercive. Also since the functional is weakly lower semi-continuous by Lemmas 3.3 and 3.6, we deduce that there exists a global minimizer of in X. Since , we verify by definition (3.5) that there is an element ω in such that . Then . So we obtain that
Consequently, we conclude that . This completes the proof. □
Now, we consider an example to demonstrate our main result in this section.
Example 3.8 Let with satisfy the log-Hölder continuity condition (2.2). Suppose that , and there is a positive constant such that for almost all . Let us consider
In this case, put
for all . Denote the quantities
If conditions (H1)-(H2) hold, then we have
-
(i)
,
-
(ii)
is a positive eigenvalue of the problem (E0),
-
(iii)
the problem (E0) has a nontrivial weak solution for any ,
-
(iv)
λ is not an eigenvalue of (E0) for .
Proof It is clear that conditions (HJ1)-(HJ4) and (F1) hold. From the definitions of and , we know that
and thus . Also, from the same argument as that in Theorem 3.7, we have , and thus . Applying Theorem 3.7, the conclusions (ii) and (iii) hold. Let . Suppose that λ is an eigenvalue of the problem (E0). Then there is an element such that
By the definition of , we get that
a contradiction. □