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.
MSC: 35J92, 35P30.
Keywords:nonhomogeneous elliptic operator; positive solution; the first eigenvalue with weight; approximation
In this paper, we consider the existence of a positive solution for the following quasilinear elliptic equation:
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.
Throughout this paper, we assume that the mapAand the nonlinear termfsatisfy the following assumptions (A) and (f), respectively.
(A) , where for all , and there exist positive constants , , , , and such that
(ii) for every , and ;
(iii) for every , and ;
(iv) for every , ;
(v) 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
for every .
In this paper, we say that is a (weak) solution of (P) if
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.
To state our first result, we define a positive constant by
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.
(f1) There exist and such that the Lebesgue measure of is positive and
where is the continuous function in (f) and is the first positive eigenvalue of the p-Laplacian with the weight function m obtained by
Theorem 1Assume (f1). Then equation (P) has a positive solution , where
andνdenotes the outward unit normal vector on∂Ω.
Next, we consider the case where A is asymptotically -homogeneous near zero in the following sense:
(AH0) There exist a positive function and such that
Under (AH0), we can replace the hypothesis (f1) with the following (f2):
(f2) There exist and such that (3) and the Lebesgue measure of is positive, where is the first positive eigenvalue of with a weight function m obtained by
Theorem 2Assume (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
Remark 3 The following assertions hold under condition (A):
(i) for all , is maximal monotone and strictly monotone in y;
(ii) for every ;
(iii) for every ,
where and are the positive constants in (A).
Proposition 4 ([, Proposition 1])
Let be a map defined by
for . ThenAis maximal monotone, strictly monotone and has property, that is, any sequence weakly convergent touwith strongly converges tou.
2 Constructing approximate solutions
Choose a function . In this section, for such ψ and , we consider the following elliptic equation:
In , the case in the above equation is considered.
Lemma 5Suppose (f1) or (f2). Then there exists such that for every , and .
Proof From the growth condition of and (3), it follows that
holds, where is a positive constant independent of . Therefore, for , we easily see that for every , and holds. □
Proposition 6If 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 .
If , then by taking as a test function (note that is bounded), we have
due to Remark 3(iii) and . Putting , we see that provided (note ). Similarly, if , then , and if , then (note ). Thus, according to Young’s inequality, for every , there exists such that
where and (>1). As a result, because of , according to Hölder’s inequality and the monotonicity of with respect to r on , taking a and setting , we obtain
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 7Suppose (f1) or (f2). If is a solution of ( ) for , then .
Proof Taking as a test function in ( ), we have
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 8Suppose (f1) or (f2). Then, for every , ( ) has a positive solution .
Proof Fix any and let be a Schauder basis of (refer to  for the existence). For each , we define the m-dimensional subspace of by . Moreover, set a linear isomorphism by , and let be a dual map of . By identifying and , we may consider that maps from to . Define maps and from to as follows:
for u, . We claim that for every , there exists such that in . Indeed, by the growth condition of f, Remark 3(iii) and Hölder’s inequality, we easily have
for every , where . This implies that is coercive on by . Set a homotopy for and . By recalling that is coercive on , we see that there exists an such that for every and because and the norm of are equivalent on . Therefore, we have
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.
Because (11) with leads to the boundedness of by , we may assume, by choosing a subsequence, that converges to some weakly in and strongly in . Let be a natural projection onto , that is, for . Since and in , by noting that on for a map A defined in Proposition 4, we obtain
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 .
Finally, we shall prove that is a solution of ( ). Fix any and . For each , by letting in , we have
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
Lemma 9Let . Then
holds, where is the positive constant defined by (2).
Proof Because of , there exist such that in . Thus, and in Ω. Hence, hold. Therefore, we have
in Ω by (ii) and (iii) in Remark 3 and Young’s inequality. □
Lemma 10Assume that and let . Then
Proof First, we note that hold by the same reason as in Lemma 9. Applying Young’s inequality to the second term of the right-hand side in (14) (refer to (13) with ), we obtain
in Ω. Similarly, we also have
The conclusion follows from (15) and (16). □
Under (f1) or (f2), we denote a solution of ( ) for each obtained by Proposition 8.
Lemma 11Assume (f1) or (f2). Let . Then is bounded in .
Proof Taking as a test function in ( ), we have
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 12Assume (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.
Proof Fix any and choose any . By taking as a test function, we obtain
by Lemma 5 and . On the other hand, by Remark 3(iii) and , we have
Therefore, (17) and (18) imply the inequality for every . As a result, by letting , our conclusion is shown. □
Lemma 13Assume (f2) and (AH0). Let . If in as , then
holds, where is a continuous function as in (AH0).
Proof Note that hold (as in the proof of Lemma 9). Because we easily see that for every with some independent of u (see (6)), it is sufficient to show as . Here, we fix any . By the property of (see (6)) and because we are assuming that in as , we have for every provided sufficiently small . Therefore, for such sufficiently small , we obtain
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 .
Proof of Theorem 1 Let be an eigenfunction corresponding to the first positive eigenvalue (cf.[14,15], it is well known that we can obtain φ as the minimizer of (4)), namely, φ is a positive solution of in Ω and on ∂Ω. Since p-Laplacian is -homogeneous, we may assume that φ satisfies , and hence holds by taking φ as a test function. Choose satisfying (note that as in (f1)). Due to (f1), there exists a such that for every and . Since we are assuming in as , occurs for sufficiently large n. Then, for such sufficiently large n, according to Lemma 9, (1) and , we obtain
This is a contradiction. □
Proof of Theorem 2 Since holds, by the standard argument as in the p-Laplacian, we see that and it is the first positive eigenvalue of in Ω and on ∂Ω. Therefore, by the well-known argument, there exists a positive eigenfunction corresponding to (we can obtain as the minimizer of (7)). Hence, by taking as a test function, we have . Thus, follows. Because is a solution of and is an eigenfunction corresponding to , according to Lemma 11 and Lemma 13 (note as in (AH0)), we obtain
as since we are assuming in , where we use the facts that and in Ω. Furthermore, by Fatou’s lemma and (3), we have
As a result, by taking a limit superior with respect to n in (19), we have . This is a contradiction. □
The author declares that she has no competing interests.
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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