Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 147308, 19 pages doi:10.1155/2009/147308 Research Article On Multiple Solutions of Concave and Convex Nonlinearities in Elliptic Equation on R N Kuan-Ju Chen Department of Applied Science, Naval Academy, 90175 Zuoying, Taiwan Correspondence should be addressed to Kuan-Ju Chen, kuanju@mail.cna.edu.tw Received 18 February 2009; Accepted 28 May 2009 Recommended by Martin Schechter We consider the existence of multiple solutions of the elliptic equation on R N with concave and convex nonlinearities. Copyright q 2009 Kuan-Ju Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction First, we look for positive solutions of the following problem: −Δu  u  a  x  u p−1  λb  x  u q−1 , in R N , u>0, in R N , u ∈ H 1  R N  , 1.1 where λ>0 is a real parameter, 1 <p<2 <q<2 ∗  2N/N − 2, N ≥ 3. We will impose some assumptions on ax and bx. Assume a1 ax ≥ 0,ax ∈ L α/α−1 R N  ∩ L ∞ R N , where 1 <α<2 ∗ /p, b1 bx ∈ CR N ,bx → b ∞ > 0as|x|→∞,bx ≥ b ∞ for all x ∈ R N , Such problems occur in various branches of mathematical physics and population dynamics, and sublinear analogues or superlinear analogues of problem 1.1 have been considered by many authors in recent years see 1–4. Little information is known about the combination of sublinear and superlinear case of problem 1.1.In5, 6, they deal with the analogue of problem 1.1 when R N is replaced by a bounded domain Ω. For the R N case, the existence of positive solutions for problem 1.1 was proved by few people. 2 Boundary Value Problems In the present paper, we discuss the Nehari manifold and examine carefully the connection between the Nehari manifold and the fibrering maps, then using arguments similar to those used in 7, we will prove the existence of the two positive solutions by using Ekeland’s Variational Principle 8. In 5, Ambrosetti et al. showed that for λ>0 small with respect t o μ>0 there exist infinitely many solutions u ∈ H 1 0 Ω of the semilinear elliptic problem: −Δu  λ | u | p−2 u  μ | u | q−2 u, in Ω, u  0on∂Ω, 1.2 with negative energy: ψ  u   1 2  Ω | ∇u | 2 − λ p  Ω | u | p − μ q  Ω | u | q , 1.3 and infinitely many solutions with positive energy, where Ω ⊂ R N is an open bounded domain. In 9, Bartsch and Willem obtained infinitely many solutions of problem 1.2 with negative energy for every λ>0. For the R N case, the existence of multiple solutions was proved by few people. Finally we propose herein a result similar to 9 or 10 for the existence of infinitely many solutions possibly not positive of −Δu  u  μa  x  | u | p−2 u  λb  x  | u | q−2 u, in R N , u ∈ H 1  R N  , 1.4 by taking advantage of the oddness of the nonlinearity. Our main results state the following. Theorem 1.1. Under the assumptions (a1) and (b1), there exists λ ∗ > 0, such that for all λ ∈ 0,λ ∗ , problem 1.1 has at least two positive solutions u 0 and u 1 , u 0 is a local minimizer of I λ and I λ u 0  < 0, where I λ is the energy functional of problem 1.1. Theorem 1.2. Under the assumptions (a1) and (b1), for every λ>0 and μ ∈ R, the problem 1.4 has infinitely many solutions with positive energy and for every μ>0 and λ ∈ R, infinitely many solutions with negative energy. 2. The Existence of Two Positive Solutions The variational functional of problem 1.1 is I λ  u   1 2   | ∇u | 2  u 2  − 1 p  a  x  | u | p − λ q  b  x  | u | q , 2.1 here and from now on, we omit “dx”and“R N ” in all the integrations if there is no other indication. Boundary Value Problems 3 Through this paper, we denote the universal positive constant by C unless some special statement is given. Let ·, · denote the usual scalar product in H 1 R N .Easy computations show that I λ is bounded from below on the Nehari manifold, Λ λ   u ∈ H 1  R N  :  I  λ  u  ,u   0  . 2.2 Thus u ∈ Λ λ if and only if || u || 2 −  a  x  | u | p − λ  b  x  | u | q  0. 2.3 In particular, on Λ λ , we have I λ  u    1 2 − 1 p  || u || 2 − λ  1 q − 1 p   b  x  | u | q   1 2 − 1 q  || u || 2 −  1 p − 1 q   a  x  | u | p . 2.4 The Nehari manifold is closely linked to the behavior of the functions of the form φ u : t → I λ tut>0. Such maps are known as fibrering maps and were introduced by Dr ´ abek and Pohozaev in 11 and are discussed by Brown and Zhang 12.Ifu ∈ H 1 R N , we have φ u  t   t 2 2 || u || 2 − t p p  a  x  | u | p − λ t q q  b  x  | u | q , φ  u  t   t || u || 2 − t p−1  a  x  | u | p − λt q−1  b  x  | u | q , φ  u  t   || u || 2 −  p − 1  t p−2  a  x  | u | p − λ  q − 1  t q−2  b  x  | u | q . 2.5 Similarly to the method used in 7, we split Λ λ into three parts corresponding to local minima, local maxima, and points of inflection, and so we define Λ  λ   u ∈ Λ λ : φ  u  1  > 0  , Λ − λ   u ∈ Λ λ : φ  u  1  < 0  , Λ 0 λ   u ∈ Λ λ : φ  u  1   0  , 2.6 and note that if u ∈ Λ λ ,thatis,φ  u 10, then φ  u  1    2 − p  || u || 2 − λ  q − p   b  x  | u | q   2 − q  || u || 2 −  p − q   a  x  | u | p . 2.7 4 Boundary Value Problems This section will be devoted to prove Theorem 1.2. To prove Theorem 1.2, several preliminary results are in order. Lemma 2.1. Under the assumptions (a1), (b1), there exists λ ∗ > 0 such that when 0 <λ<λ ∗ ,for every u ∈ H 1 R N , u / ≡ 0, there exist unique t   t  u > 0, t −  t − u > 0 such that t  u ∈ Λ − λ , t − u ∈ Λ  λ . In particular, one has t  >   2 − q   u  2  p − q   a  x  | u | p  1/p−2  t max >t − , 2.8 I λ t − umin t∈0,t   I λ tu < 0 and I λ t  umax t≥t − I λ tu. Proof. Given u ∈ H 1 R N  \{0},setϕ u tt 2−q ||u|| 2 − t p−q  ax|u| p . Clearly, for t>0, tu ∈ Λ λ if and only if t is a solution of ϕ u  t   λ  b  x  | u | q . 2.9 Moreover, ϕ  u  t    2 − q  t 1−q || u || 2 −  p − q  t p−q−1  a  x  | u | p , 2.10 easy computations show that ϕ u is concave and achieves its maximum at t max    2 − q  || u || 2  p − q   a  x  | u | p  1/p−2 . 2.11 If λ>0issufficiently large, 2.9 has no solution, and so φ u tI λ tu hasnocritical points, in this case φ u is a decreasing function, hence no multiple of u lies in Λ λ . If, on the other hand, λ>0issufficiently small, then there exist exactly two solutions t  u >t − u > 0of2.9, where t   t  u, t −  t − u, ϕ  u t −  > 0, and ϕ  u t   < 0. It follows from 2.7 and 2.10 that φ  tu 1t q1 ϕ  u t,andsot  u ∈ Λ − λ , t − u ∈ Λ  λ ; moreover φ u is decreasing in 0,t − ,increasingint − ,t  , and decreasing in t  , ∞. Next, we will discussion the sufficiently small λ ∗ , such that when 0 <λ<λ ∗ , there exist exactly two solutions of problem 2.9 for all u ∈ H 1 R N  \{0},thatis, λ  b  x  | u | q <ϕ u  t max    2 − q p − q  2−q/p−2  p − 2 p − q  || u || 2p−2q/p−2   a  x  | u | p  2−q/p−2 . 2.12 Boundary Value Problems 5 Since  a  x  | u | p ≤ || a || L α/α−1 || u || p L αp ≤ || a || L α/α−1 S p αp || u || p , 2.13 where S αp denotes the Sobolev constant of the embedding of H 1 R N  into L αp R N , hence, ϕ u  t max  ≥  2 − q p − q  2−q/p−2  p − 2 p − q  || u || 2p−2q/p−2  || a || L α/α−1 S p αp || u || p  2−q/p−2   2 − q p − q  2−q/p−2  p − 2 p − q  || u || q  || a || L α/α−1 S p αp  2−q/p−2 , 2.14 and then  b  x  | u | q ≤ M || u || q L q ≤ MS q q || u || q ≤ MS q q  p − q 2 − q  2−q/p−2  p − q p − 2   || a || L α/α−1 S p αp  2−q/p−2 ϕ u  t max   cϕ u  t max  , 2.15 where S q denotes the Sobolev constant of the embedding of H 1 R N  into L q R N , c is independent of u, hence ϕ u  t max  − λ  b  x  | u | q ≥ ϕ u  t max  − λcϕ u  t max   ϕ u  t max  1 − λc  , 2.16 and so λ  bx|u| q <ϕ u t max  for all u ∈ H 1 R N  \{0} provided λ<1/2c  λ ∗ . Hence when 0 <λ<λ ∗ , φ u has exactly two critical points—a local minimum at t −  t − u and a local maximum at t   t  u; moreover I λ t − umin t∈0,t   I λ tu < 0andI λ t  u max t≥t − I λ tu. In particular, we have the following result. Corollary 2.2. Under the assumptions (a1), (b1), when 0 <λ<λ ∗ , for every u ∈ Λ λ , u / ≡ 0, one has  2 − q  || u || 2 −  p − q   a  x  | u | p / ≡ 0 2.17 (i.e., Λ 0 λ  ∅). 6 Boundary Value Problems Proof. Let us argue by contradiction and assume that there exists u ∈ Λ λ \{0} such that 2 − q||u|| 2 − p − q  ax|u| p  0, this implies λ  b  x  | u | q  || u || 2 −  a  x  | u | p   p − 2 2 − q   a  x  | u | p   p − 2 2 − q   a  x  | u | p  p−q/p−2   a  x  | u | p  q−2/p−2   p − 2 2 − q  1 p − q  p−q/p−2   p − q   a  x  | u | p  p−q/p−2   a  x  | u | p  q−2/p−2   p − 2 p − q  2 − q p − q  2−q/p−2 || u || 2p−q/p−2   a  x  | u | p  q−2/p−2  ϕ u  t max  2.18 which contradicts 2.12 for 0 <λ<λ ∗ . As a consequence of Corollary 2.2, we have the following lemma. Lemma 2.3. Under the assumptions (a1), (b1), if 0 <λ<λ ∗ , for every u ∈ Λ λ , u / ≡ 0, then there exist a >0 and a C 1 -map t  tw > 0, w ∈ H 1 R N , ||w|| <satisfying that t  0   1,t  w  u − w  ∈ Λ λ , for || w || <,  t   0  ,w   2   ∇u∇w  uw  − p  a  x  | u | p−2 uw − λq  b  x  | u | q−2 uw  2 − q  || u || 2 −  p − q   a  x  | u | p . 2.19 Proof. We define F : R × H 1 R N  → R by F  t, w   t || u − w || 2 − t p−1  a  x  | u − w | p − λt q−1  b  x  | u − w | q . 2.20 Since F1, 00andF t 1, 0||u|| 2 − p − 1  ax|u| p − λq − 1  bx|u| q 2 − q||u|| 2 − p − q  ax|u| p / ≡ 0 by Corollary 2.2, we can apply the implicit function theorem at the point 1, 0 and get the result. Apply Lemma 2.1, Corollary 2.2, Lemma 2.3, and Ekeland variational principle 8,we can establish the existence of the first positive solution. Proposition 2.4. If 0 <λ<λ ∗ , then the minimization problem: c 0  inf I λ Λ λ  inf I λ Λ  λ 2.21 Boundary Value Problems 7 is achieved at a point u 0 ∈ Λ  λ which is a critical point for I λ with u 0 > 0 and I λ u 0  < 0. Furthermore, u 0 is a local minimizer of I λ . Proof. First, we show that I λ is bounded from below in Λ λ . Indeed, for u ∈ Λ λ ,from2.13, we have I λ  u   1 2 || u || 2 − 1 p  a  x  | u | p − λ q  b  x  | u | q   1 2 − 1 q  || u || 2 −  1 p − 1 q   a  x  | u | p ≥  1 2 − 1 q  || u || 2 −  1 p − 1 q  || a || L α/α−1 S p αp || u || p 2.22 and so I λ is bounded from below in Λ λ . Then we will claim that c 0 < 0, indeed if v ∈ H 1 R N \{0},fromLemma 2.1, there exist 0 <t − v <t  v such that t − vv ∈ Λ λ .Thus, c 0 ≤ I λ  t −  v  v   min t∈0,t  v I λ  tv  < 0. 2.23 By Ekeland’s Variational Principle 8, there exists a minimizing sequence {u n }⊂Λ λ of the minimization problem 2.21 such that c 0 ≤ I λ  u n  <c 0  1 n , 2.24 I λ  v  ≥ I λ  u n  − 1 n || v − u n || , ∀ v ∈ Λ λ . 2.25 Taking n large enough, from 2.7 we have I λ  u n    1 2 − 1 q  || u n || 2 −  1 p − 1 q   a  x  | u n | p <c 0  1 n < 0, 2.26 from which we deduce that for n large  a  x  | u n | p ≥ pq p − q c 0 , || u n || 2 ≤ 2  q − p  p  q − 2   a  x  | u n | p , 2.27 which yields b 1 ≤ || u n || ≤ b 2 2.28 for suitable b 1 , b 2 > 0. 8 Boundary Value Problems Now we will show that   I  λ  u n    −→ 0asn −→ ∞ . 2.29 Since u n ∈ Λ λ ,byLemma 2.3, we can find a  n > 0andaC 1 -map t n  t n w > 0, w ∈ H 1 R N , ||w|| < n satisfying that v n  t n  w  u n − w  ∈ Λ λ , for || w || < n . 2.30 By the continuity of t n w and t n 01, without loss of generality, we can assume  n satisfies that 1/2 ≤ t n w ≤ 3/2for||w|| < n . It follows from 2.25 that I λ  t n  w  u n − w  − I λ  u n  ≥− 1 n || t n  w  u n − w  − u n || ; 2.31 that is,  I  λ  u n  ,t n  w  u n − w  − u n   o  || t n  w  u n − w  − u n ||  ≥− 1 n || t n  w  u n − w  − u n || . 2.32 Consequently, t n  w   I  λ  u n  ,w    1 − t n  w   I  λ  u n  ,u n  ≤ 1 n ||  t n  w  − 1  u n − t n  w  w ||  o  || t n  w  u n − w  − u n ||  . 2.33 By the choice of  n ,weobtain  I  λ  u n  ,w  ≤ C n    t  n  0  ,w     o  || w ||   C n || w ||  o     t  n  0  ,w     || u n ||  || w ||   . 2.34 By Lemma 2.3, Corollary 2.2, and the estimate 2.28, we have  t  n  0  ,w   2   ∇u n ∇w  u n w  − p  a  x  | u n | p−2 u n w − λq  b  x  | u n | q−2 u n w  2 − q  || u n || 2 −  p − q   a  x  | u n | p ≤ C || w || , 2.35 then from 2.34 we get  I  λ  u n  ,w  ≤ C n || w ||  C n || w ||  o  || w ||  , for || w || ≤  n . 2.36 Boundary Value Problems 9 Hence, for any  ∈ 0, n , we have   I  λ  u n     1  sup || w ||   I  λ  u n  ,w  ≤ C n  1  o    , 2.37 for some C>0 independent of  and n. Taking  → 0, we obtain 2.29. Let u 0 ∈ H 1 R N  be the weak limit in H 1 R N  of u n .From2.29,  I  λ  u 0  ,w   0, ∀w ∈ H 1  R N  ; 2.38 that is, u 0 is a weak solution of problem 1.1 and consequently u 0 ∈ Λ λ . Therefore, c 0 ≤ I λ  u 0  ≤ lim n →∞ I λ  u n   c 0 ; 2.39 that is, c 0  I λ  u 0   inf Λ λ I λ . 2.40 Moreover, we have u 0 ∈ Λ  λ .Infact,ifu 0 ∈ Λ − λ ,byLemma 2.1, there exists only one t  > 0 such that t  u 0 ∈ Λ − λ , we have t   t  u 0 1, t −  t − u 0  < 1. Since dI λ  t − u 0  dt  0, d 2 I λ  t − u 0  dt 2 > 0, 2.41 there exists t  ≥ t>t − such that I λ tu 0  >I λ t − u 0 .ByLemma 2.1, I λ  t − u 0  <I λ  tu 0  ≤ I λ  t  u 0   I λ  u 0  ; 2.42 this is a contradiction. To conclude that u 0 is a local minimizer of I λ , notice that for every u ∈ H 1 R N  \{0}, we have from Lemma 2.1, I λ  su  ≥ I λ  t − u  ∀0 <s<   2 − q  || u || 2  p − q   a  x  | u | p  1/p−2 . 2.43 In particular, for u  u 0 ∈ Λ  λ , we have t −  u 0   1 <   2 − q  || u 0 || 2  p − q   a  x  | u 0 | p  1/p−2 . 2.44 10 Boundary Value Problems Let >0sufficiently small to have 1 <   2 − q  || u 0 − w || 2  p − q   a  x  | u 0 − w | p  1/p−2 , for || w || <. 2.45 From Lemma 2.3,lettw > 0satisfytwu 0 − w ∈ Λ λ for every ||w|| <. By the continuity of tw and t01, we can always assume that t  w  <   2 − q  || u 0 − w || 2 p − q  ax | u 0 − w | p  1/p−2 , for || w || <. 2.46 Namely, twu 0 − w ∈ Λ  λ and for 0 <s<   2 − q  || u 0 − w || 2  p − q   a  x  | u 0 − w | p  1/p−2 , 2.47 we have I λ  s  u 0 − w  ≥ I λ  t  w  u 0 − w  ≥ I λ  u 0  . 2.48 Taking s  1, we conclude I λ  u 0 − w  ≥ I λ  t  w  u 0 − w  ≥ I λ  u 0  , for || w || <, 2.49 which means that u 0 is a local minimizer of I λ . Furthermore, taking t − |u 0 | > 0witht − |u 0 ||u 0 |∈Λ  λ , therefore, I λ  u 0  ≤ I λ  t −  | u 0 |  | u 0 |  ≤ I λ  | u 0 |  ≤ I λ  u 0  . 2.50 So we can always take u 0 ≥ 0. By the maximum principle for weak solutions see 13,we can show that u 0 > 0inR N . Since u 0 ∈ Λ  λ and c 0  inf Λ λ I λ  inf Λ  λ I λ , thus, in the search of our second positive solution, it is natural to consider the second minimization problem: c 1  inf Λ − λ I λ . 2.51 Let us now introduce the problem at infinity associated with 1.1: −Δu  u  λb ∞ u q−1 , in R N , u>0, in R N , u ∈ H 1  R N  . 2.52 [...]... 3555–3561, 1995 10 S B Tshinanga, On multiple solutions of semilinear elliptic equation on unbounded domains with concave and convex nonlinearities, ” Nonlinear Analysis: Theory, Methods & Applications, vol 28, no 5, pp 809–814, 1997 11 P Dr´ bek and S I Pohozaev, “Positive solutions for the p-Laplacian: application of the fibering a method,” Proceedings of the Royal Society of Edinburgh Section A, vol 127, no... and Y Zhang, “The Nehari manifold for a semilinear elliptic equation with a signchanging weight function,” Journal of Differential Equations, vol 193, no 2, pp 481–499, 2003 13 D Gilbarg and N S Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, Germany, 2nd edition, 1993 14 P.-L Lions, On positive solutions of semilinear elliptic equations in unbounded domains,” in. .. nonhomogeneous elliptic equations involving critical Sobolev exponent,” Annales de l’Institut Henri Poincar´ Analyse Non Lin´ aire, vol 9, no 3, pp 281–304, 1992 e e 8 I Ekeland, On the variational principle,” Journal of Mathematical Analysis and Applications, vol 47, pp 324–353, 1974 9 T Bartsch and M Willem, On an elliptic equation with concave and convex nonlinearities, ” Proceedings of the American Mathematical... 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Proposition 2.8 If 0 < λ < λ∗ , for c1 such that a Iλ un tω|p − infΛ− Iλ , one can find a minimizing sequence {un } ⊂ Λ− λ λ o1, o 1 strongly in H −1 RN , S∞ λ Proof Set Σ {u ∈ H 1 RN : ||u|| 1} and define the map Ψ : Σ → Λ− given by Ψ u λ t u u Since the continuity of t u follows immediately from its uniqueness and extremal u/||u|| Clearly Λ− property, thus Ψ is continuous with continuous inverse given... the assumptions of in 9, Theorem 2 are satisfied Then, there exists k0 such that for each k ≥ k0 , I u has a critical value ck ∈ bk , dk , so that ck −→ 0 as k −→ ∞ 3.22 This completes the proof of Theorem 1.2, since observe that B3 and B4 imply bk ≤ dk < 0 Proof of Theorem 1.2 The proof follows from Propositions 3.1 and 3.2 References 1 H Brezis and S Kamin, “Sublinear elliptic equations in RN ,” Manuscripta... Academic Press, New York, NY, USA, 1981 16 H Brezis and E Lieb, “A relation between pointwise convergence of functions and convergence of functionals,” Proceedings of the American Mathematical Society, vol 88, no 3, pp 486–490, 1983 17 T Bartsch, In nitely many solutions of a symmetric Dirichlet problem,” Nonlinear Analysis: Theory, Methods & Applications, vol 20, no 10, pp 1205–1216, 1993 . S. B. Tshinanga, On multiple solutions of semilinear elliptic equation on unbounded domains with concave and convex nonlinearities, ” Nonlinear Analysis: Theory, Methods & Applications, vol Martin Schechter We consider the existence of multiple solutions of the elliptic equation on R N with concave and convex nonlinearities. Copyright q 2009 Kuan-Ju Chen. This is an open access article. Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 147308, 19 pages doi:10.1155/2009/147308 Research Article On Multiple Solutions of Concave and Convex Nonlinearities