Existence and multiplicity of solutions for nonlocal -Laplacian problems in
© Guo and Zhao; licensee Springer 2012
Received: 13 March 2012
Accepted: 9 July 2012
Published: 26 July 2012
In this paper, we study the nonlocal -Laplacian problem of the following form
By using the method of weight function and the theory of the variable exponent Sobolev space, under appropriate assumptions on f and M, we obtain some results on the existence and multiplicity of solutions of this problem. Moreover, we get much better results with f in a special form.
MSC:35B38, 35D05, 35J20.
where is a function defined on , is a continuous function, satisfies the Caratheodory condition.
The operator is called -Laplacian, which becomes p-Laplacian when (a constant). The -Laplacian possesses more complicated nonlinearities than p-Laplacian; for example, p-Laplacian is -homogeneous, that is, for every ; but the -Laplacian operator, when is not a constant, is not homogeneous. These problems with variable exponent are interesting in applications and raise many difficult mathematical problems. Some of the models leading to these problems of this type are the models of motion of electrorheological fluids, the mathematical models of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium. We refer the reader to [1–7] for the study of -Laplacian equations and the corresponding variational problems.
has been investigated by Autuori, Pucci and Salvatori . In  Fan studied -Kirchhoff type equations with Dirichlet boundary value problems. Many papers are about these problems in bounded domains. According to the information I have, for Kirchhoff-type problems in , the results are seldom, in  Jin and Wu obtained three existence results of infinitely many radial solutions for Kirchhoff-type problems in , and in  Ji established the existence of infinitely many radially symmetric solutions of Kirchhoff-type -Laplacian equations in . The main difficulty here arises from the lack of compactness. Jin  and Ji  investigated these problems in radial symmetric spaces. In this paper, to deal with problem (P), we overcome the difficulty caused by the absence of compactness through the method of weight function. We establish conditions ensuring the existence and multiplicity of solutions for the problem.
This paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. In Section 3, we obtain the solutions with negative energy by the coercivity of functionals, and in Section 4, we obtain the solutions with positive energy by the Mountain Pass Theorem. Finally in Section 5, we obtain the infinity of solutions by the Fountain Theorem and the Dual Fountain Theorem when f satisfies a special form.
In order to discuss problem (P), we need some theories on space which we call variable exponent Sobolev space. Firstly, we state some basic properties of space which will be used later (for details, see [6, 31, 32]).
and becomes a Banach space. We call it a variable exponent Lebesgue space.
where ; and we denote by the closure of in , , , when , and , when .
- (1)If, the spaceis a separable, uniform convex Banach space, and its dual space is, where. For anyand, we have
- (2)If, then for any, , and,
Proposition 2.2 (see )
where, , andis a constant, then the superposition operator fromtodefined byis a continuous and bounded operator.
Proposition 2.3 (see )
Proposition 2.4 (see )
in measure in Ω and.
If, thenandare separable reflexive Banach spaces.
Proposition 2.6 Ifis Lipschitz continuous and, then forwith, there is a continuous embedding.
Proposition 2.7 Let Ω be a bounded domain in, , . Then for anywith, there is a compact embedding.
Proposition 2.8 (Poincare inequality)
So, is a norm equivalent to the normin the space.
3 Solutions with negative energy
In the following sections, we consider problem (P), the nonlocal -Laplacian problem with variational form, where M is a real function satisfying the following condition: (M1) = is continuous and bounded.. And we assume that , is Lipschitz continuous, , satisfies Caratheodory conditions.
For simplicity, we write . Denote by C a general positive constant (the exact value may change from line to line).
Before giving our main results, we first give several lemmas that will be used later.
, , for.
- (2), , , and
Lemma 3.2 (see )
Thenand Φ, are weakly-strongly continuous, i.e., impliesand.
The functionalis sequentially weakly lower semi-continuous, is sequentially weakly continuous, and thus E is sequentially weakly lower semi-continuous.
- (2)For any open setwith, the mappingsandare bounded, and are of type, namely,
Proof Since the function is increasing and the functional is sequentially weakly lower semi-continuous, we conclude that the functional is sequentially weakly lower semi-continuous. From Lemma 3.2, we know that and are sequentially weakly-strongly continuous. Now let . It is clear that the mapping and are bounded. To prove that is of type , assuming that , in X and , then there exist positive constants and such that . Noting that . It follows from that , where . Since is of type . Moreover, since is sequentially weakly-strongly continuous, the mapping is of type . □
Definition 3.1 Let . A -functional satisfies condition if and only if every sequence in X such that , and in has a convergent subsequence.
Lemma 3.4 (see )
Suppose f satisfies the hypotheses in Lemma 3.2, and let (M1) hold. Then, for any, every boundedsequence for E, i.e., a bounded sequencesuch thatand, has a strongly convergent subsequence.
Lemma 3.5 (see )
Theorem 3.1 Suppose f satisfies the hypotheses in Lemma 3.2, let (M1) hold and the following conditions hold: (M2) = There are positive constants, M and C such thatfor.; (H1) = ..Then the functional E is coercive and attains its infimum in X at some. Therefore, is a solution of (P) if E is differentiable at, and in particular, if.
and hence E is coercive. Since E is sequentially weakly lower semi-continuous and X is reflexive, E attains its infimum in X at some . In the case where E is differentiable at , is a solution of (P). □
where, , , , .; (H2) = ..Then (P) has at least one nontrivial solution which is a global minimizer of the energy functional E.
Hence which shows . □
Theorem 3.3 Let the hypotheses of Theorem 3.2 hold, and f satisfy the following condition: (f2) = forand..Then (P) has a sequence of solutionssuch that, andas.
we have .
As , we can find and such that , , which implies , . Since , we get the conclusion .
By the genus theory, each is a critical value of E, hence there is a sequence of solutions of problem (P) such that .
we know as . When and , we have , and then , which concludes as . □
Theorem 3.4 Let the hypotheses of Lemma 3.2, (f1), (M1), (M2), (M3), (H1), (H2) and the following condition hold, (f+) = forand..Then (P) has at least one nontrivial nonnegative solution with negative energy.
Then, like in the proof of Theorem 3.2, using truncation functions above, similarly to the proof of Theorem 3.4 in , we can prove that has a nontrivial global minimizer and is a nontrivial nonnegative solution of (P). □
4 Solution with positive energy
In this section we will find the Mountain Pass type critical points of the energy functional E associated with problem (P).
withhold.; (M4) = , such that
the notation of this conclusion can be seen in .
we conclude that is bounded, since . By Lemma 3.4, E satisfies condition for . □
Lemma 4.2 Under the hypotheses of Lemma 4.1, for any, as.
and then as . □
uniformly in.; (H4) = ..
Then there exist positive constants ρ and δ such thatfor.
Thus by (H4), we obtain the assertion of Lemma 4.3. □
By the famous Mountain Pass lemma, from Lemmas 4.1-4.3, we have the following:
Theorem 4.1 Let all hypotheses of Lemmas 4.1-4.3 hold. Then (P) has a nontrivial solution with positive energy.
5 The case of concave-convex nonlinearity
In this section, we will obtain much better results with f in a special form. We have the following theorem:
Theorem 5.1 Let, where
If (M1), (M2)′, (M4), (H3) hold and we also assume thatand, then problem (P) has solutionssuch thatas.
If (M1), (M4), (M5), (H3) hold and we also assume thatand, then problem (P) has solutionssuch that, as.
We will use the following ‘Fountain Theorem’ and the ‘Dual Fountain Theorem’ to prove Theorem 5.1.
Proposition 5.1 (Fountain Theorem, see )
Assume(A1) = X is a Banach space, is an even functional, the subspaces, andare defined by (3.2)..
If for each, there existssuch that(A2) = as.; (A3) = .; (A4) = E satisfies thecondition for every. Then E has a sequence of critical values tending to +∞..
Proposition 5.2 (Dual Fountain Theorem, see )
Assume (A1) is satisfied and there is aso as to for each, there existssuch that(B1) = .; (B2) = .; (B3) = as.; (B4) = E satisfiescondition for every. Then E has a sequence of negative critical values converging to 0..
Definition 5.1 We say that E satisfies the condition (with respect to ), if any sequence such that , , and , contains a subsequence converging to a critical point of E.
As is of type, we can conclude ; furthermore, we have .
- (1)We will prove that if k is large enough, then there exist such that (A2) and (A3) are satisfied. (A2) For , denote
Since , we have . (A2) is satisfied.
so (A2) is satisfied.
We use the Dual Fountain Theorem to prove conclusion (2), and now it remains for us to prove that there exist such that if k is large enough (B1), (B2) and (B3) are satisfied.
Hence (B1) is satisfied.
with small enough. Hence (B2) is satisfied.
hence . Hence (B3) is satisfied.
Conclusion (2) is reached by the Dual Fountain Theorem. □
The authors thank the two referees for their careful reading and helpful comments on the study. Research was supported by the National Natural Science Foundation of China (10971088), (10971087) and the Fundamental Research Funds for the Central Universities (lzujbky-2012-180).
- Fan XL:On the sub-supersolution method for -Laplacian equations. J. Math. Anal. Appl. 2007, 330: 665-682. 10.1016/j.jmaa.2006.07.093MathSciNetView ArticleGoogle Scholar
- Fan XL, Han XY:Existence and multiplicity of solutions for -Laplacian equations in . Nonlinear Anal. 2004, 59: 173-188.MathSciNetGoogle Scholar
- Fan XL, Shen JS, Zhao D:Sobolev embedding theorems for space . J. Math. Anal. Appl. 2001, 262: 749-760. 10.1006/jmaa.2001.7618MathSciNetView ArticleGoogle Scholar
- Fan XL, Zhang QH:Existence of solutions for -Laplacian Dirichlet problems. Nonlinear Anal. 2003, 52: 1843-1852. 10.1016/S0362-546X(02)00150-5MathSciNetView ArticleGoogle Scholar
- Fan XL, Zhang QH, Zhao D:Eigenvalues of -Laplacian Dirichlet problem. J. Math. Anal. Appl. 2005, 302: 306-317. 10.1016/j.jmaa.2003.11.020MathSciNetView ArticleGoogle Scholar
- Fan XL, Zhao D:On the spaces and . J. Math. Anal. Appl. 2001, 263: 424-446. 10.1006/jmaa.2000.7617MathSciNetView ArticleGoogle Scholar
- Fan XL, Zhao YZ, Zhang QH:A strong maximum principle for -Laplacian equations. Chin. Ann. Math., Ser. A 2003, 24: 495-500. (in Chinese); Chinese Contemp. Math. 24, 277-282 (2003)MathSciNetGoogle Scholar
- Lions JL: On some questions in boundary value problems of mathematical physical. In Proceedings of International Symposium on Continuum Mechanics and Partial Differential Equations, Rio de Janerro 1977. Edited by: de la Penha GM, Medeiros LAJ. North-Holland, Amsterdam; 1978:284-346.Google Scholar
- Chipot M, Lovat B: Some remarks on non local elliptic and parabolic problems. Nonlinear Anal. 1997, 30: 4619-4627. 10.1016/S0362-546X(97)00169-7MathSciNetView ArticleGoogle Scholar
- Ma TF: Remarks on an elliptic equation on Kirchhoff type. Nonlinear Anal. 2005, 63: 1967-1977. 10.1016/j.na.2005.03.021View ArticleGoogle Scholar
- Perera K, Zhang ZT: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 2006, 221: 246-255. 10.1016/j.jde.2005.03.006MathSciNetView ArticleGoogle Scholar
- Alves CO, Corrêa FJSA: On existence of solutions for a class of problem involving a nonlinear operator. Commun. Appl. Nonlinear Anal. 2001, 8: 43-56.Google Scholar
- Alves CO, Corrêa FJSA, Ma TF: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 2005, 49: 85-93. 10.1016/j.camwa.2005.01.008MathSciNetView ArticleGoogle Scholar
- Andrade D, Ma TF: An operator equation suggested by a class of stationary problems. Commun. Appl. Nonlinear Anal. 1997, 4: 65-71.MathSciNetGoogle Scholar
- Chipot M, Valente V, Vergara Caffarelli G: Remarks on a nonlocal problem involving the Dirichlet energy. Rend. Semin. Mat. Univ. Padova 2003, 110: 199-220.MathSciNetGoogle Scholar
- Corrêa FJSA: On positive solutions of nonlocal and nonvariational elliptic problems. Nonlinear Anal. 2004, 59: 1147-1155.MathSciNetView ArticleGoogle Scholar
- Corrêa FJSA, Figueiredo GM: On an elliptic equation of p -Kirchhoff type via variational methods. Bull. Aust. Math. Soc. 2006, 74: 263-277. 10.1017/S000497270003570XView ArticleGoogle Scholar
- Corrêa FJSA, Figueiredo GM: On a p -Kirchhoff equation via Krasnoselskii’s genus. Appl. Math. Lett. 2009, 22: 819-822. 10.1016/j.aml.2008.06.042MathSciNetView ArticleGoogle Scholar
- Corrêa FJSA, Menezes SDB, Ferreira J: On a class of problems involving a nonlocal operator. Appl. Math. Comput. 2004, 147: 475-489. 10.1016/S0096-3003(02)00740-3MathSciNetView ArticleGoogle Scholar
- He XM, Zou WM: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 2009, 70: 1407-1414. 10.1016/j.na.2008.02.021MathSciNetView ArticleGoogle Scholar
- Ricceri B: On an elliptic Kirchhoff-type problem depending on two parameters. J. Glob. Optim. 2010, 46: 543-549. 10.1007/s10898-009-9438-7MathSciNetView ArticleGoogle Scholar
- Arosio A, Panizzi S: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 1996, 348: 305-330. 10.1090/S0002-9947-96-01532-2MathSciNetView ArticleGoogle Scholar
- Cavalcanti MM, Domingos Cavalcanti VN, Soriano JA: Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation. Adv. Differ. Equ. 2001, 6: 701-730.MathSciNetGoogle Scholar
- D’Ancona P, Spagnolo S: Global solvability for the degenerate Kirchhoff equation with real analytic date. Invent. Math. 1992, 108: 447-462.MathSciNetGoogle Scholar
- Dreher M: The Kirchhoff equation for the p -Laplacian. Rend. Semin. Mat. (Torino) 2006, 64: 217-238.MathSciNetGoogle Scholar
- Dreher M: The wave equation for the p -Laplacian. Hokkaido Math. J. 2007, 36: 21-52.MathSciNetView ArticleGoogle Scholar
- Autuori G, Pucci P, Salvatori MC: Asymptotic stability for anistropic Kirchhoff systems. J. Math. Anal. Appl. 2009, 352: 149-165. 10.1016/j.jmaa.2008.04.066MathSciNetView ArticleGoogle Scholar
- Fan XL:On nonlocal -Laplacian Dirichlet problems. Nonlinear Anal. 2010, 72: 3314-3323. 10.1016/j.na.2009.12.012MathSciNetView ArticleGoogle Scholar
- Jin J, Wu X:Infinitely many radial solutions for Kirchhoff-type problems in . J. Math. Anal. Appl. 2010, 369: 564-574. 10.1016/j.jmaa.2010.03.059MathSciNetView ArticleGoogle Scholar
- Ji C:Infinitely many radial solutions for the Kirchhoff-type equation with oscillatory nonlinearities in . J. Math. Anal. Appl. 2012, 388: 727-738. 10.1016/j.jmaa.2011.09.065MathSciNetView ArticleGoogle Scholar
- Edmunds DE, Rákosník J:Density of smooth functions in . Proc. R. Soc. Lond. A 1992, 437: 229-236. 10.1098/rspa.1992.0059View ArticleGoogle Scholar
- Edmunds DE, Rákosník J: Sobolev embedding with variable exponent. Stud. Math. 2000, 143: 267-293.Google Scholar
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