Existence of a positive solution for quasilinear elliptic equations with nonlinearity including the gradient
© Tanaka; licensee Springer 2013
Received: 15 May 2013
Accepted: 10 July 2013
Published: 24 July 2013
We provide the existence of a positive solution for the quasilinear elliptic equation
in Ω under the Dirichlet boundary condition. As a special case (), our equation coincides with the usual p-Laplace equation. The solution is established as the limit of a sequence of positive solutions of approximate equations. The positivity of our solution follows from the behavior of as t is small. In this paper, we do not impose the sign condition to the nonlinear term f.
where is a bounded domain with boundary ∂ Ω. Here, is a map which is strictly monotone in the second variable and satisfies certain regularity conditions (see the following assumption (A)). Equation (P) contains the corresponding p-Laplacian problem as a special case. However, in general, we do not suppose that this operator is -homogeneous in the second variable.
, where for all , and there exist positive constants , , , , and such that
for every , and ;
for every , and ;
for every , ;
for every , with .
- (f)f is a continuous function on satisfying for every and the following growth condition: there exist , and a continuous function on such that(1)
for every .
for all .
A similar hypothesis to (A) is considered in the study of quasilinear elliptic problems (see [, Example 2.2.], [2–5] and also refer to [6, 7] for the generalized p-Laplace operators). From now on, we assume that , which is without any loss of generality as can be seen from assumptions (A)(ii), (iii).
In particular, for , that is, stands for the usual p-Laplacian , we can take in (A). Conversely, in the case where holds in (A), by the inequalities in Remark 3(ii) and (iii), we see that whence . Hence, our equation contains the p-Laplace equation as a special case.
In the case where f does not depend on the gradient of u, there are many existence results because our equation has the variational structure (cf. [1, 4, 8]). Although there are a few results for our equation (P) with f including ∇u, we can refer to [7, 9] and  for the existence of a positive solution in the case of the -Laplacian or m-Laplacian (). In particular, in  and , the nonlinear term f is imposed to be nonnegative. The results in  and  are applied to the m-Laplace equation with an -superlinear term f w.r.t. u. Here, we mention the result in  for the p-Laplacian. Faria, Miyagaki and Motreanu considered the case where f is -sublinear w.r.t. u and ∇u, and they supposed that for some and . The purpose of this paper is to remove the sign condition and to admit the condition like for large as . Concerning the condition for f as , Zou in  imposed that there exists an satisfying as for the m-Laplace problem. Hence, we cannot apply the result of  and  to the case of as for and (admitting sign changes), but we can do our result if is large.
In , the positivity of a solution is proved by the comparison principle. However, since we are not able to do it for our operator in general, after we provide a non-negative and non-trivial solution as a limit of positive approximate solutions (in Section 2), we obtain the positivity of it due to the strong maximum principle for our operator.
which is equal to 1 in the case of (i.e., the case of the p-Laplacian) because we can choose . Then, we introduce the hypothesis (f1) to the function in (f) as t is small.
and ν denotes the outward unit normal vector on ∂ Ω.
Next, we consider the case where A is asymptotically -homogeneous near zero in the following sense:
Under (AH0), we can replace the hypothesis (f1) with the following (f2):
Theorem 2 Assume (AH0) and (f2). Then equation (P) has a positive solution .
Throughout this paper, we may assume that for every , and because we consider the existence of a positive solution only. In what follows, the norm on is given by , where denotes the usual norm of for (). Moreover, we denote .
1.2 Properties of the map A
for all , is maximal monotone and strictly monotone in y;
for every ;
for every ,
where and are the positive constants in (A).
Proposition 4 ([, Proposition 1])
for . Then A is maximal monotone, strictly monotone and has property, that is, any sequence weakly convergent to u with strongly converges to u.
2 Constructing approximate solutions
In , the case in the above equation is considered.
Lemma 5 Suppose (f1) or (f2). Then there exists such that for every , and .
holds, where is a positive constant independent of . Therefore, for , we easily see that for every , and holds. □
Proposition 6 If is a non-negative solution of () for , then . Moreover, for any , there exists a positive constant such that holds for every .
Proof Set if , and in the case of , is an arbitrarily fixed constant. Let be a non-negative solution of () with (some ). For , choose a smooth increasing function such that if , if and if for some . Define for .
provided by (8) and (9), where , comes from the continuous embedding of into and is a positive constant independent of , ε and r. Consequently, Moser’s iteration process implies our conclusion. In fact, we define a sequence by and . Then, we see that holds if by applying Fatou’s lemma to (10) and letting . Here, we also see as . Therefore, by the same argument as in Theorem C in , we can obtain and for some positive constant D independent of and ε. □
Lemma 7 Suppose (f1) or (f2). If is a solution of () for , then .
because of if and by Remark 3(iii). Hence, follows. Because Proposition 6 guarantees that , we have (for some ) by the regularity result in . Note that because of and . In addition, Lemma 5 implies the existence of such that in the distribution sense. Therefore, according to Theorem A and Theorem B in , in Ω and on ∂ Ω, namely, . □
The following result can be shown by the same argument as in [, Theorem 3.1].
Proposition 8 Suppose (f1) or (f2). Then, for every , () has a positive solution .
where is the identity map on , and denotes the degree on for a continuous map (cf. ). Hence, this yields the existence of such that , and so the desired is obtained by setting since is injective.
as , where we use the boundedness of , the growth condition of f and in . In addition, since is bounded, by the boundedness of , we see that as , whence as holds. As a result, it follows from the property of A that in as .
Since l is arbitrary, (12) holds for every . Moreover, the density of in guarantees that (12) holds for every . This means that is a solution of (). Consequently, our conclusion follows from Lemma 7. □
3 Proof of theorems
holds, where is the positive constant defined by (2).
in Ω by (ii) and (iii) in Remark 3 and Young’s inequality. □
The conclusion follows from (15) and (16). □
Under (f1) or (f2), we denote a solution of () for each obtained by Proposition 8.
Lemma 11 Assume (f1) or (f2). Let . Then is bounded in .
by Remark 3(iii), the growth condition of f, Hölder’s inequality and the continuity of the embedding of into , where (<p) and is a positive constant independent of . Because of , this yields the boundedness of (). □
Lemma 12 Assume (f1) or (f2). Then and hold for every , where denotes the Lebesgue measure of Ω, and where and are positive constants as in (A) and Lemma 5, respectively.
Therefore, (17) and (18) imply the inequality for every . As a result, by letting , our conclusion is shown. □
holds, where is a continuous function as in (AH0).
because of by Lemma 12. Since is arbitrary, our conclusion is shown. □
3.1 Proof of main results
Proof of Theorems
Let . Due to Proposition 6 and Lemma 11, we have for some independent of . Hence, there exist and such that and for every by the regularity result in . Because the embedding of into is compact and by , there exists a sequence and such that and in as . If occurs, then by the same reason as in Lemma 7, and hence our conclusion is proved. Now, we shall prove by contradiction for each theorem. So, we suppose that , whence in as .
This is a contradiction. □
As a result, by taking a limit superior with respect to n in (19), we have . This is a contradiction. □
The author would like to express her sincere thanks to Professor Shizuo Miyajima for helpful comments and encouragement. The author thanks Professor Dumitru Motreanu for giving her the opportunity of this work. The author thanks referees for their helpful comments.
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