- Open Access
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.
- Weak Solution
- Previous Lemma
- Energy Solution
- Mountain Pass
- Quasilinear Elliptic Equation
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 .
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).
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.
- Diaz JI: Nonlinear Partial Differential Equations and Free Boundaries. Elliptic Equations. Pitman, Boston; 1986.Google Scholar
- Alves CO, Morais Filho DC, Souto MA:Radially symmetric solutions for a class of critical exponent elliptic problems in . J. Math. Anal. Appl. 1996, 7: 1-12.Google Scholar
- Alves CO, Carrião PC, Miyagaki OH: Nonlinear perturbations of a periodic elliptic problem with critical growth. J. Math. Anal. Appl. 2001, 260: 133-146. 10.1006/jmaa.2001.7442MathSciNetView ArticleGoogle Scholar
- Alves CO, Soares SHM: Existence and concentration of positive solutions for a class of gradient systems. Nonlinear Differ. Equ. Appl. 2005, 12: 437-457.MathSciNetView ArticleGoogle Scholar
- Bartsch T, Wang Z-Q:Existence and multiplicity results for some superlinear elliptic problems on . Commun. Partial Differ. Equ. 1995, 20: 1725-1741. 10.1080/03605309508821149MathSciNetView ArticleGoogle Scholar
- Berestycki H, Lions PL: Nonlinear scalar fields equation, I: existence of a ground state. Arch. Ration. Mech. Anal. 1983, 82: 313-346.MathSciNetGoogle Scholar
- Costa DG:On a class of elliptic systems in . Electron. J. Differ. Equ. 1994., 1994: Article ID 7Google Scholar
- Coti-Zelati V, Rabinowitz PH:Homoclinic type solutions for a semilinear elliptic PDE on . Commun. Pure Appl. Math. 1992, 45(10):1217-1269. 10.1002/cpa.3160451002MathSciNetView ArticleGoogle Scholar
- Kryszewski W, Szulkin A: Generalized linking theorem with an application to semilinear Schrödinger equation. Adv. Differ. Equ. 1998, 3: 441-472.MathSciNetGoogle Scholar
- Pankov AA, Pflüger K: On a semilinear Schrödinger equation with periodic potential. Nonlinear Anal. 1998, 33: 593-609. 10.1016/S0362-546X(97)00689-5MathSciNetView ArticleGoogle Scholar
- Pankov AA: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 2005, 73: 259-287. 10.1007/s00032-005-0047-8MathSciNetView ArticleGoogle Scholar
- Rabinowitz PH: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 1992, 43: 270-291. 10.1007/BF00946631MathSciNetView ArticleGoogle Scholar
- del Pino M, Felmer P: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. 1996, 4: 121-137. 10.1007/BF01189950MathSciNetView ArticleGoogle Scholar
- Alves CO, do Ó JM, Miyagaki OH: On perturbations of a class of a periodic m -Laplacian equation with critical growth. Nonlinear Anal. 2001, 45: 849-863. 10.1016/S0362-546X(99)00421-6MathSciNetView ArticleGoogle Scholar
- Alves CO, Soares SHM: Multiplicity of positive solutions for a class of nonlinear Schrödinger equations. Adv. Differ. Equ. 2010, 15(11-12):1083-1102.MathSciNetGoogle Scholar
- Zhu XP, Yang J: On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded domains. Acta Math. Sci. 1987, 7: 341-359.MathSciNetGoogle Scholar
- Zhu XP, Yang J: The quasilinear elliptic equation on unbounded domain involving critical Sobolev exponent. J. Partial Differ. Equ. 1989, 2: 53-64.MathSciNetGoogle Scholar
- Ambrosetti A, Felli V, Malchiodi A: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. 2005, 7: 117-144.MathSciNetView ArticleGoogle Scholar
- Ambrosetti A, Wang Z-Q: Nonlinear Schrödinger equations with vanishing and decaying potentials. Differ. Integral Equ. 2005, 18: 1321-1332.MathSciNetGoogle Scholar
- Benci V, Grisant CR, Micheletti AM:Existence of solutions of nonlinear Schrödinger equations with . Prog. Nonlinear Differ. Equ. Appl. 2005, 66: 53-65.View ArticleGoogle Scholar
- Alves CO, Souto MAS:Existence of solutions for a class of elliptic equations in with vanishing potentials. J. Differ. Equ. 2012, 252: 5555-5568. 10.1016/j.jde.2012.01.025MathSciNetView ArticleGoogle Scholar
- Wang Z-Q, Willem M: Singular minimization problems. J. Differ. Equ. 2000, 2(161):307-320.MathSciNetView ArticleGoogle Scholar
- DiBenedetto E: local regularity of weak solutions of degenerate elliptic equations. J. Partial Differ. Equ. 1983, 8(7):827-850.Google Scholar
- Assunção RB, Carrião PC, Miyagaki OH: Multiplicity results for equations with subcritical Hardy-Sobolev exponents and singularities on a half-space. Mat. Contemp. 2007, 32: 1-13.MathSciNetGoogle Scholar
- Miyagaki OH, Rodrigues RS: On positive solutions for a class of degenerate quasilinear elliptic positone-semipositone systems. J. Math. Anal. Appl. 2007, 8(334):818-833.MathSciNetView ArticleGoogle Scholar
- Catrina F, Wang Z-Q: On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence) and symmetry of extremal functions. Commun. Pure Appl. Math. 2001, 54(2):229-258. 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-IMathSciNetView ArticleGoogle Scholar
- Noussair ES, Swanson CA, Yang J: Quasilinear elliptic problems with critical exponents. Nonlinear Anal. 1993, 20: 285-301. 10.1016/0362-546X(93)90164-NMathSciNetView ArticleGoogle Scholar
- Assunção RB, Carrião PC, Miyagaki OH: Critical singular problems via concentration-compactness lemma. J. Math. Anal. Appl. 2007, 326: 137-154. 10.1016/j.jmaa.2006.03.002MathSciNetView ArticleGoogle Scholar
- El Hamidi A, Rakotoson JM: Compactness and quasilinear elliptics problems with critical exponents. Differ. Integral Equ. 2005, 18: 1201-1220. Dedicated to professor R. Temam for his 65th anniversaryMathSciNetGoogle Scholar
- Boccardo L, Murat F: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 1992, 6(19):581-597.MathSciNetView ArticleGoogle Scholar
- Ghoussoub N, Yuan C: Multiple solutions for quasi-linear partial differential equations involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc. 2000, 12(352):5703-5743.MathSciNetView ArticleGoogle Scholar
- Caffarelli L, Kohn R, Nirenberg L: First order interpolations inequalities with weights. Compos. Math. 1984, 53(3):259-275.MathSciNetGoogle Scholar
- Brezis H, Nirenberg L: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 1983, 36(4):437-477. 10.1002/cpa.3160360405MathSciNetView ArticleGoogle Scholar
- Xuan B: The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights. Nonlinear Anal. 2005, 62: 703-725. 10.1016/j.na.2005.03.095MathSciNetView ArticleGoogle Scholar
- Hewitt E, Stromberg K: Real and Abstract Analysis. Springer, New York; 1975.Google Scholar
- Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7MathSciNetView ArticleGoogle Scholar
- Calzolari E, Filippucci R, Pucci P: Dead cores and bursts for p -Laplacian elliptic equations with weights. Discrete Contin. Dyn. Syst. 2007, 2007: 1-10. suppl.Google Scholar
- Lindqvist P: On the definition and properties of a p -superharmonic functions. J. Reine Angew. Math. 1986, 365: 67-70.MathSciNetGoogle Scholar
- Simon J:Régularité de la solution d’une équation non linéaire dans . Lecture Notes in Math. 665. In Journées d’Analyse Non Linéaire (Proc. Conf., Besançon, 1977). Springer, Berlin; 1978:205-227.View ArticleGoogle Scholar
- Brezis H: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York; 2010.View ArticleGoogle Scholar
- Ben-Naoum AK, Troestlers C, Willem M: Extrema problems with critical Sobolev exponents on unbounded domains. Nonlinear Anal. 1996, 26(4):823-833. 10.1016/0362-546X(94)00324-BMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.