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On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions
Boundary Value Problems volume 2014, Article number: 117 (2014)
Abstract
In this paper, we consider semilinear biharmonic equations with concave-convex nonlinearities involving weight functions, where the concave nonlinear term is and the convex nonlinear term is with . By use of the Nehari manifold and the direct variational methods, the existence of multiple positive solutions is established as , here the explicit expression of is provided.
MSC:35J35, 35J40, 35J65.
1 Introduction
In recent years, there has been extensive attention on semilinear second-order elliptic equations,
here Ω is a bounded smooth domain in (), and λ is a positive parameter; see [1–8] and the references therein. As is sublinear, say, , , the monotone iteration scheme or the method of sub-solutions and super-solutions are effective; see [9]. As is superlinear, for example, , , variational methods are applicable; see [10]. In contrast with the pure sublinear case and the pure superlinear case, in [2] Ambrosetti et al. considered problem (1.1) when is, roughly, the sum of a sublinear and a superlinear term. To be precise, they considered the following problem:
with . They proved that problem (1.2) admits at least two positive solutions for λ sufficiently small. In [6], Sun and Li considered a similar problem:
with , the authors studied the value of Λ, the supremum of the set λ, related to the existence and multiplicity of positive solutions and established uniform lower bounds for Λ. In [8], Wu considered the subcritical case of problem (1.2) with replaced by , here is a sign-changing function, and he showed that problem (1.2) has at least two positive solutions as λ is small enough.
Some interesting generalizations of (1.2) have been provided in the framework of quasi-linear elliptic equations or systems, semilinear second-order elliptic systems or fourth-order elliptic equations. More recently, the semilinear fourth-order elliptic equations have been studied by many authors, we refer the reader to [11–13] and the references therein. Motivated by some work in [6, 8, 13], we deal with the following semilinear biharmonic elliptic equation:
where Ω is a bounded smooth domain in (), ( for and for ), is a parameter, is a positive or sign-changing weight function and is a positive weight function.
For convenience and simplicity, we introduce some notations. The norm of u in is denoted by , the norm of u in is denoted by ; is denoted by , endowed with the norm ; S denotes the best Sobolev constant for the embedding of in (see [14]); to be precise, for all .
Now we define
It is well known that the weak solutions of problem (1.3) are the critical points of the energy functional (see Rabinowitz [15]).
Next, we consider the Nehari minimization problem: for ,
where . Define
Then for ,
Similarly to the method used in Tarantello [16], we split into three parts:
Note that all solutions of (1.3) are clearly in the Nehari manifold, . Hence, our approach to solve problem (1.3) is to analyze the structure of , and then to deal with the minimization problems for on and applying the direct variational method.
The following is our main result.
Theorem 1.1 Let with , then problem (1.3) has at least two positive solutions for any .
The paper is organized as follows: in Section 2, we give some lemmas; in Section 3, we prove Theorem 1.1.
2 Preliminaries
In this section, we prove several lemmas.
Lemma 2.1 For (where is given in Theorem 1.1), we have .
Proof Suppose that for all . If , then we have
and
By (2.1)-(2.2), the Sobolev inequality, and the Hölder inequality, we get
and
where . Thus, using (2.3) and (2.4), we have
Hence, by (2.5) the desired conclusion yields. □
Lemma 2.2 If , then
Proof From , it is easy to see that
By the Sobolev inequality, we get
In addition,
The proof is completed. □
By Lemma 2.1, for we write and define
The following lemma shows that the minimizers on are ‘usually’ critical points for .
Lemma 2.3 For , if is a local minimizer for on , then in .
Proof If is a local minimizer for on , then is a solution of the optimization problem
Hence, by the theory of Lagrange multipliers, there exists such that
Thus,
From and Lemma 2.1, we have and . So, by (2.6)-(2.7) we get in . □
For each , we write
Then we have the following lemma.
Lemma 2.4 For each and , we have
-
(i)
there is a unique such that and ;
-
(ii)
is a continuous function for nonzero u;
-
(iii)
;
-
(iv)
if , then there is a unique such that and .
Proof (i) Fix . Let
Then we have , as , is concave and reaches its maximum at . Moreover,
Case I. .
There is a unique such that and . Now,
and
Thus, . In addition,
and
Hence, .
Case II. .
From (2.8) and
there exist unique and such that ,
and
Similar to the argument in Case I above, we have , , and
-
(ii)
By the uniqueness of and the external property of , we find that is continuous function of .
-
(iii)
For , let . By item (i), there is a unique such that , that is, . Since , we have , which implies
Conversely, let such that . Then . Therefore,
-
(iv)
By Case II of item (i). □
By and changes sign in Ω, we have is an open set in . Without loss of generality, we may assume that Θ is a domain in . Consider the following biharmonic equation:
Associated with (2.9), we consider the energy functional
and the minimization problem
where . Now we prove that problem (2.9) has a positive solution such that .
Lemma 2.5 For any , there exists a unique such that . The maximum of for is reached at , the map
is continuous and the induced continuous map defines a homeomorphism of the unit sphere of with .
Proof For any given , consider the function , . Clearly,
It is easy to verify that , for small and for large. Hence, is reached at a unique such that and . To prove the continuity of , assume that in . It is easy to verify that is bounded. If a subsequence of converges to , it follows from (2.10) that and then . Finally the continuous map from the unit sphere of to , , is inverse to the retraction . □
Define
where .
Lemma 2.6 is a critical value of K.
Proof From Lemma 2.5, we know that . Since for and t large, we obtain . The manifold separates into two components. The component containing the origin also contains a small ball around the origin. Moreover, for all u in this component, because , . Then each has to cross and . Since the embedding is compact (see [14]), it is easy to prove that is a critical value of K and a positive solution corresponding to c. □
With the help of Lemma 2.6, we have the following result.
Lemma 2.7 (i) For , there exists such that
-
(ii)
is coercive and bounded below on for all .
Proof (i) Let be a positive solution of problem (2.9) such that . Then
Set as defined by Lemma 2.4(iv). Hence, and
This implies
-
(ii)
For , we have . Then by the Hölder, Sobolev, and Young inequalities,
here .
Thus, is coercive on and
for all . □
Next, we will use the idea of Tarantello [16] to get the following results.
Lemma 2.8 For and any given , there exist and a differentiable functional such that , the function and
for all .
Proof Define as follows:
Since and by Lemma 2.1, we obtain
we can get the desired results applying the implicit function theorem at the point . □
Lemma 2.9 For and any given , there exist and a differentiable functional such that , the function and
for all .
Proof In view of Lemma 2.8, there exist and a differentiable functional such that , for all and we have (2.12). By use of , we have . In combination with the continuity of the functions and , we get as ϵ sufficiently small, this implies that . □
3 Proof of Theorem 1.1
Firstly, we provide the existence of minimizing sequences for on and as λ is sufficiently small.
Proposition 3.1 Let , then
-
(i)
there exists a minimizing sequence such that
-
(ii)
there exists a minimizing sequence such that
Proof (i) By Lemma 2.7(ii) and the Ekeland variational principle [17], there exists a minimizing sequence such that
and
Taking n large, from Lemma 2.7(i) and (3.1), we have
This implies
that is,
Now, we will show that
Exactly as in Lemma 2.8 we may apply suitable functionals to and obtain
Hence, if and small, substituting in (3.6) and applying (3.2), we have
Dividing by and passing to the limit as we derive
Since
by the boundedness of we get
for a suitable positive constant .
Next, we show that is bounded away from zero. Arguing by contradiction, assume that
Since , we have
and consequently by (3.9),
Then by (3.4), the Hölder inequality, Sobolev inequality and (3.9)-(3.10), we obtain
moreover, , which contradicts (3.5).
Thus, we get from (3.8) that
Hence, by (3.7) it follows that
which implies that , as .
-
(ii)
Similar to the arguments in (i), by Lemma 2.9 and Lemma 2.2, we can prove (ii). □
Now, we establish the existence of a local minimum for on .
Theorem 3.1 Let , then the functional has a minimizer in and it satisfies
-
(i)
;
-
(ii)
is a positive solution of problem (1.3);
-
(iii)
as .
Proof By Proposition 3.1(i), there is a minimizing sequence for on such that
Then by Lemma 2.7 and the compact imbedding theorem, there exist a subsequence and such that
and
First, we claim that
If not, by (3.14) we conclude that
Therefore, as ,
this contradicts as .
In combination with (3.11)-(3.14), it is easy to verify that is a nontrivial weak solution of problem (1.3).
Now we prove that strongly in . Supposing the contrary, then and so
this contradicts . Hence, strongly in . This implies
Moreover, we have . In fact, if , by Lemma 2.4, there exist unique and such that and , we get . Since
and
there exists such that . By Lemma 2.4,
which is a contradiction. Since and , by Lemma 2.3 we may assume that is a nonnegative weak solution to problem (1.3). Applying the regularity theory and strong maximum principle of elliptic equations, we find that is one positive solution of problem (1.3). In addition, by Lemma 2.7,
which implies that as . □
Next, we establish the existence of a local minimum for on .
Theorem 3.2 Let , then the functional has a minimizer in and it satisfies
-
(i)
;
-
(ii)
is a positive solution of problem (1.3).
Proof By Proposition 3.1(ii), there is a minimizing sequence for on such that
Then by Lemma 2.7 and the compact imbedding theorem, there exist a subsequence and such that
and
Connecting with Lemma 2.2, it is easy to see that is a nontrivial weak solution of problem (1.3).
Next we prove that strongly in . Supposing the contrary, then and so
this contradicts . Hence, strongly in . This implies
In addition, from Lemma 2.4(ii)-(iii), we have . Since and , by Lemma 2.3 we may assume that is a nonnegative weak solution to problem (1.3). Applying the regularity theory and strong maximum principle of elliptic equations, we see that is one positive solution of problem (1.3). □
Proof of Theorem 1.1 It is an immediate consequence of Theorems 3.1 and 3.2. □
References
Alama S, Tarantello G: On semilinear elliptic equations with indefinite nonlinearities. Calc. Var. Partial Differ. Equ. 1993, 1: 439-475. 10.1007/BF01206962
Ambrosetti A, Brezis H, Cerami G: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 1994, 122: 519-543. 10.1006/jfan.1994.1078
Bartsch T, Willem M: On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc. 1995, 123: 3555-3561. 10.1090/S0002-9939-1995-1301008-2
Brown KJ, Zhang Y: The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J. Differ. Equ. 2003, 193: 481-499. 10.1016/S0022-0396(03)00121-9
Li S, Wu S, Zhou HS: Solutions to semilinear elliptic problems with combined nonlinearities. J. Differ. Equ. 2001, 185: 200-224.
Sun Y, Li S: A nonlinear elliptic equation with critical exponent: estimates for extremal values. Nonlinear Anal. 2008, 69: 1856-1869. 10.1016/j.na.2007.07.030
Tang M: Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities. Proc. R. Soc. Edinb., Sect. A 2003, 133: 705-717. 10.1017/S0308210500002614
Wu TF: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 2006, 318: 253-270. 10.1016/j.jmaa.2005.05.057
Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.
Willem M: Minimax Theorems. Birkhäuser, Basel; 1996.
Ebobisse F, Ahmedou MO: On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent. Nonlinear Anal. 2003, 52: 1535-1552. 10.1016/S0362-546X(02)00273-0
Gazzola F, Grunau H-C, Squassina M: Existence and nonexistence results for critical growth biharmonic elliptic equations. Calc. Var. Partial Differ. Equ. 2003, 18: 117-143. 10.1007/s00526-002-0182-9
Zhang Y: Positive solutions of semilinear biharmonic equations with critical Sobolev exponents. Nonlinear Anal. 2012, 75: 55-67. 10.1016/j.na.2011.07.065
Adams RA: Sobolev Spaces. Academic Press, New York; 1975.
Rabinowitz PH Reg. Conf. Ser. Math. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.
Tarantello G: On nonhomogeneous elliptic involving critical Sobolev exponent. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1992, 9: 281-304.
Ekeland I: On the variational principle. J. Math. Anal. Appl. 1974, 17: 324-353.
Acknowledgements
This work is partly supported by NNSF (11101404, 11201204, 11361053) of China and the Young Teachers Scientific Research Ability Promotion Plan of Northwest Normal University (NWNU-LKQN-11-5).
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Yang, L., Wang, X. On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions. Bound Value Probl 2014, 117 (2014). https://doi.org/10.1186/1687-2770-2014-117
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DOI: https://doi.org/10.1186/1687-2770-2014-117