Existence of solutions for a class of degenerate quasilinear elliptic equation in with vanishing potentials
© Bastos et al.; licensee Springer 2013
Received: 28 November 2012
Accepted: 3 April 2013
Published: 17 April 2013
We establish the existence of positive solution for the following class of degenerate quasilinear elliptic problem
where , , , , , and denote the Hardy-Sobolev’s critical exponent, V is a bounded nonnegative vanishing potential and f has a subcritical growth at infinity. The technique used here is a truncation argument together with the variational approach.
MSC:35B09, 35J10, 35J20, 35J70.
where , , , , , and denote the Hardy-Sobolev’s critical exponent, is a bounded, nonnegative and vanishing potential and a continuous function with a subcritical growth at infinity. Here, is the completion of the with the norm . We impose the following hypotheses on f and V:
is a continuous function verifying
() , uniformly in x.
() There exists such that , uniformly in x.
() There exists such that , for all .
is a continuous function verifying
() , for all .
() There are and such that .
with and α given by ().
Consider first the case , that is is the p-Laplacian operator, and the potential is bounded from below by a positive constant .
Equations involving the p-Laplacian operator appear in many problems of nonlinear diffusion. Just to mention, in nonlinear optics, plasma physics, condensed matter physics and in modeling problems in non-Newtonian fluids. For more information on the physical background, we refer to .
Let us now consider the case and the potential bounded from below by a positive constant .
In this case, the equations arise in problems of existence of stationary waves for anisotropic Schrödinger equation (see ) and others problems (for example, see [6, 23]). We cite  for ; and [24, 25] for . For the case , we cite , for and ; and , for and .
The result presented here for and extends that one in  for and . In , the presence of Hilbertian structure and some compact embeddings provide the convergence of the gradient. In the case studied here, with the absence of this structure, we do not obtain the convergence so directly. To overcome this problem, we use a result found in [28, 29], whose ideas come from [30, 31], when the domain is a smooth and bounded. In addition to this difficulty, there are others. For instance, in the present situation, our space is no longer Hilbert, which forces us to obtain new estimates. Since the problem involves singular terms, the estimates are more refined and for which the principal ingredient is the Caffarelli-Kohn-Nirenberg’s inequality (see ). Now we state the main result of this work.
Theorem 1.3 Suppose that V and f satisfy, respectively, () and () and () to (). Then there is a constantsuch that the problem (P) has a positive solution, for all, beingthe maximum of the f in the ball ofcentered in the origin with radius 1.
In order to prove this theorem, we first build an auxiliary problem (AP), and then we solve the problem (AP) using variational methods. To finish, we show that the solution of (AP) is also a solution of (P). These steps are the content of the next three sections.
Hereafter, C is a positive constant which can change value in a sequence of inequalities. We denote the ball in centered in the origin with radius R. The weak (⇀) and strong (→) convergences are always taken as and means . The weighted spaces are denoted by . When , we denote the usual norm in , with . For , we use .
2 The auxiliary problem
3 Solving the problem (AP)
In this section, we show that the problem (AP) has a least energy solution, but first we define some minimax levels. To begin with we set in the space , the norm and we define the functional given by . Here, is the completion of the with the norm .
such that for .
such that and .
Proof By using the growth of f given in Remark 1.1 and the Caffarelli-Kohn-Nirenberg’s inequality (see ), we get , and hence . Since , there exists such that . Thus, we have for . By (), it follows that there exist and such that . Now implies . Since , there exists a large enough such that, taking , we have and . □
such that for .
such that and .
Proof From the definition of G, we have . Thus, like the previous lemma, we have for . Take the same of the proof of the previous lemma. Thus, and . With the same argument, we have . Since , there exists a large enough such that, taking , we have and . □
respectively. Since in , we have , by their definitions. Now using the above lemma together with the mountain pass theorem [, Theorem 2.2], we conclude that there exists a Palais-Smale sequence ((PS) sequence for short) for J, i.e., satisfies and .
Lemma 3.3 Suppose () and () to () and letbe a (PS) sequence for the functional J. Thenis bounded in E.
Assuming , equation (4) implies that . So , which is a contradiction. Therefore, is bounded in E. □
Lemma 3.4 Suppose () and () to (). Then the functional J satisfies the Palais-Smale condition, i.e., every (PS) sequence has a convergent subsequence.
Claim 1 .
Claim 2 .
Claim 3 .
Claim 4 .
Assuming Claims 1 to 4 for now, we proceed with the proof of lemma.
is such that in and , , for all .
Observe that condition 1 follows by integrability of and . Note that, when one of these conditions holds for some , it also verifies for every . Thus, we can choose an r that satisfies both conditions.
, in ;
, in ;
Verification of claims
In all proofs, except for Claim 4, we consider separately integration in and .
Now, using Remark 3.5, we have .
Therefore, and the proof of Claim 1 is completed.
Claim 2: The proof of this fact is made as in the proof of Claim 1.
From this fact, we get and the proof of this claim is completed.
Claim 4: Let be dual of p. Since ,b we see that and . As is bounded, we conclude that is also bounded. Moreover, satisfies the hypothesis of Lemma 3.6, below. So that we have a.e. , which give us a.e. . Using a theorem [, Theorem 13.44], we conclude that in . Thus, we have , and hence .
Lemma 3.6 Let E and J be respectively the space and functional defined in Section 2. Leta bounded sequence such thatin E and. Then, passing to a subsequence if necessary, we have, a.e. .
Using Lemmas 3.2, 3.3, 3.4 and the mountain pass theorem, in [, Theorem 2.4], we conclude that there exists which is a critical point for the functional J, in the minimax level c. Moreover, u is the least energy solution to the problem (AP).
4 The solution of (AP) is solution of (P)
Now, our aim is to show that the solution found in the previous section is also a solution of the problem (P). It is sufficient to verify , for all .
Lemma 4.1 Any least energy solution u of (AP) satisfies the estimate.
Proof Since u is a critical point in the minimax level , by Lemma 3.3, we have . So that . □
Remark 4.2 The constant depends only on , θ and f.
Then there exists a constantsuch that.
As σ depends on q, we have, by definition of M, that . □
Remark 4.4 In the previous lemma, the constant M does not depend on the potential b of the problem (AP 2).
Lemma 4.5 There exists a constantsuch that, for all u positive solution of (AP).
We will show that, if u is solution of (AP), then u is weak solution of (AP 2).
Then for all . Thus, we have , in , which implies that , in . From this, we conclude that in , and lemma is proved. □
Now, taking it follows that , for every x in , which give us , in . □
The authors would like to thank the referees for valuable comments and suggestions in improving this article. OHM was partially supported by CNPq/Brazil and Fapemig/Brazil (CEX-APQ 00025-11). RSV was partially supported by Capes/Brazil.
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