On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions
© Yang and Wang; licensee Springer. 2014
Received: 1 March 2014
Accepted: 29 April 2014
Published: 14 May 2014
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.
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 , 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.
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 ); to be precise, for all .
It is well known that the weak solutions of problem (1.3) are the critical points of the energy functional (see Rabinowitz ).
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.
In this section, we prove several lemmas.
Lemma 2.1 For (where is given in Theorem 1.1), we have .
Hence, by (2.5) the desired conclusion yields. □
The proof is completed. □
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 .
From and Lemma 2.1, we have and . So, by (2.6)-(2.7) we get in . □
Then we have the following lemma.
there is a unique such that and ;
is a continuous function for nonzero u;
if , then there is a unique such that and .
Case I. .
Case 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
By Case II of item (i). □
where . Now we prove that problem (2.9) has a positive solution such that .
is continuous and the induced continuous map defines a homeomorphism of the unit sphere of with .
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 . □
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 ), 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.
is coercive and bounded below on for all .
- (ii)For , we have . Then by the Hölder, Sobolev, and Young inequalities,
for all . □
Next, we will use the idea of Tarantello  to get the following results.
for all .
we can get the desired results applying the implicit function theorem at the point . □
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.
- (i)there exists a minimizing sequence such that
- (ii)there exists a minimizing sequence such that
for a suitable positive constant .
moreover, , which contradicts (3.5).
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 .
is a positive solution of problem (1.3);
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).
which implies that as . □
Next, we establish the existence of a local minimum for on .
is a positive solution of problem (1.3).
Connecting with Lemma 2.2, it is easy to see that is a nontrivial weak solution of problem (1.3).
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. □
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).
- Alama S, Tarantello G: On semilinear elliptic equations with indefinite nonlinearities. Calc. Var. Partial Differ. Equ. 1993, 1: 439-475. 10.1007/BF01206962MathSciNetView ArticleGoogle Scholar
- 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.1994.1078MathSciNetView ArticleGoogle Scholar
- Bartsch T, Willem M: On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc. 1995, 123: 3555-3561. 10.1090/S0002-9939-1995-1301008-2MathSciNetView ArticleGoogle Scholar
- 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)00121-9MathSciNetView ArticleGoogle Scholar
- Li S, Wu S, Zhou HS: Solutions to semilinear elliptic problems with combined nonlinearities. J. Differ. Equ. 2001, 185: 200-224.MathSciNetView ArticleGoogle Scholar
- 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.2007.07.030MathSciNetView ArticleGoogle Scholar
- 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/S0308210500002614View ArticleGoogle Scholar
- 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.2005.05.057MathSciNetView ArticleGoogle Scholar
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.Google Scholar
- Willem M: Minimax Theorems. Birkhäuser, Basel; 1996.View ArticleGoogle Scholar
- 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)00273-0MathSciNetView ArticleGoogle Scholar
- 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/s00526-002-0182-9MathSciNetView ArticleGoogle Scholar
- Zhang Y: Positive solutions of semilinear biharmonic equations with critical Sobolev exponents. Nonlinear Anal. 2012, 75: 55-67. 10.1016/j.na.2011.07.065MathSciNetView ArticleGoogle Scholar
- Adams RA: Sobolev Spaces. Academic Press, New York; 1975.Google Scholar
- Rabinowitz PH Reg. Conf. Ser. Math. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.Google Scholar
- Tarantello G: On nonhomogeneous elliptic involving critical Sobolev exponent. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1992, 9: 281-304.MathSciNetGoogle Scholar
- Ekeland I: On the variational principle. J. Math. Anal. Appl. 1974, 17: 324-353.MathSciNetView 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/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.