Đang tải... (xem toàn văn)
1 to [7-11], while in ℝN and for p = to [12,13], and for p >1 to [3,14-17], and the references therein In the present paper, our research is mainly related to (1.1) with < q < p < N, the critical exponent and weight functions f, g that change sign on Ω When p = 2, < q ¯ 0 such that (1.1) has at least two positive solutions for all l Ỵ (0, Λ) For the case p ≠ 2, [19] studied (1.1) and obtained the multiplicity of positive solutions when < q < p < N, μ = 0, f, g are sign changing and Ω is bounded However, little has been done for this type of problem (1.1) Recently, Wang et al [11] have studied (1.1) in a bounded domain Ω under the assumptions < q < p < N, N > p , −∞ < μ < μ and f, g are nonnegative They also proved that there existence of Λ0 >0 ¯ such that for l Ỵ (0, Λ0), (1.1) possesses at least two positive solutions In this paper, we study (1.1) and extend the results of [11,18,19] to the more general case < q < p ¯ < N, −∞ < μ < μ, f, g are sign changing and Ω is a smooth domain (not necessarily bounded) in ℝN (N ≥ 3) By extracting the Palais-Smale sequence in the Nehari manifold, the existence of at least two positive solutions of (1.1) is verified The following assumptions are used in this paper: (H)μ < μ, l >0, < q < p < N, N ≥ ¯ p∗ (f1) f ∈ C( ¯ ) ∩ Lq∗ ( ) (q∗ = ∗ )f+ = max{f, 0} ≢ in Ω p −q (f2) There exist b0 and r0 >0 such that B(x0; 2r0) ⊂ Ω and f (x) ≥ b0 for all x Ỵ B(x0; 2r0) (g1) g ∈ C( ¯ ) ∩ L∞ ( ) and g+ = max{g, 0} ≢ in Ω (g2) There exist x0 Ỵ Ω and b >0 such that |g|∞ = g(x0 ) = max g(x), x∈ ¯ g(x) > 0, ∀x ∈ g(x) = g(x0 ) + o(|x − x0 |β ) where | · |∞ denotes the L∞(Ω) norm , as x → Hsu Boundary Value Problems 2011, 2011:37 http://www.boundaryvalueproblems.com/content/2011/1/37 Page of 15 Set = (μ) = p−q (p ∗ −q)|g+ |∞ p−q p∗−p p ∗ −p (p ∗ −q)|f + |q∗ q N p2 (p−q)+ p Sμ (1:3) The main results of this paper are concluded in the following theorems When Ω is an unbounded domain, the conclusions are new to the best of our knowledge Theorem 1.1 Suppose (H), (f1) and (g1) hold Then, (1.1) has at least one positive solution for all l Î (0, Λ1) Theorem 1.2 Suppose (H), (f1) - (g2) hold, and g is the constant defined as in Lemma ¯ 2.2 If ≤ μ < μ, x0 = and b ≥ pg, then (1.1) has at least two positive solutions for q all λ ∈ (0, p ) Theorem 1.3 Suppose (H), (f1) - (g2) hold If μ 0 and < q < p < p2 0, r→+∞ a(μ) δ , lim r a(μ)+1 |U p,μ (r)| = c1 a(μ) ≥ 0, r→0+ c3 ≤ Up,μ (r)(r p∗−p N(μ − μ) ¯ N−p + r→+∞ b(μ) r δ )δ ≤ c4 , δ := N−p , p where ci (i = 1, 2, 3, 4) are positive constants depending on N, μ and p, and a(μ) and b(μ) are the zeros of the function h(t) = (p - 1)tp - (N - p)tp-1 + μ, t ≥ 0, satisfying ≤ a(μ) < N−p p < b(μ) ≤ N−p p−1 Take r >0 small enough such that B(0; r) ⊂ Ω, and define the function uε (x) = η(x)Vp,μ,ε (x) = ε N−p − p η(x)Up,μ |x| , ε (2:2) where η ∈ C∞ (B(0; ρ) is a cutoff function such that h(x) ≡ in B(0, ρ ) ¯ Lemma 2.2 [9,20]Suppose < p < N and ≤ μ < μ Then, the following estimates hold when ε ® ||uε ||p μ = N p Sμ + O(ε pγ ), N p |uε |p∗ = Sμ + O(ε p∗γ ), ⎧ ⎪ O1 (εθ ), ⎨ q |uε | = O1 (εθ |)lnε|, ⎪ ⎩ O1 (εqγ ), where δ = N−p p , θ =N− < q < p∗, N q = b(μ) , N ≤ q < b(μ) , N b(μ) N−p p qand g = b(μ) - δ We also recall the following known result by Ben-Naoum, Troestler and Willem, which will be employed for the energy functional Lemma 2.3 [21]Let Ω be an domain, not necessarily bounded, in ℝ N , ≤ p 0}, Nλ = {u ∈ Nλ : ψ λ (u), u = 0}, − Nλ = {u ∈ Nλ : ψ λ (u), u < 0} Lemma 3.2 Suppose ul is a local minimizer of Jl on Nλand uλ ∈ Nλ / 1,p Then, Jλ (uλ ) = 0in (D0 ( ))−1 Proof The proof is similar to [[23], Theorem 2.3] and is omitted □ Lemma 3.3 Nλ = ∅for all l Ỵ (0, Λ1) Proof We argue by contradiction Suppose that there exists l Ỵ (0, Λ1) such that 0 Nλ = ∅ Then, the fact u ∈ Nλ and (3.3) imply that ||u||p = μ p ∗ −q p−q g|u|p∗ , and ||u||p = λ μ p∗ − q p∗ − p f |u|q Hsu Boundary Value Problems 2011, 2011:37 http://www.boundaryvalueproblems.com/content/2011/1/37 Page of 15 By (f1), (g1), the Hölder inequality and Sobolev embedding theorem, we have that p∗ −p p−q (p∗ − q)|g+ |∞ ||u||μ ≥ N p2 Sμ , and p−q q p∗ − q + − ||u||μ ≤ λ ∗ |f |q∗ Sμ p p −p Consequently, p−q p∗ −p p−q (p∗ − q)|g+ |∞ λ≥ q N p2 (p−q)+ p Sμ p∗ − p (p∗ − q)|f + |q∗ = 1, which is a contradiction □ ∗ 1,p For each u ∈ D0 ( ) with g|u|p > 0, we set p∗ −p p tmax = (p − q)||u||μ (p∗ − q) g|u|p∗ > 1,p Lemma 3.4 Suppose that l Ỵ (0, Λ1) and u ∈ D0 ( )is a function satisfying with ∗ g|u|p > − f |u|q ≤ 0, then there exists a unique t- >tmax such that t− u ∈ Nλ and (i) If Jλ (t− u) = sup Jλ (tu) t≥0 (ii) If f |u|q ≤ 0, then there exists a unique t ± such that p−q (p∗ − q)|g+ |∞ p∗ −p N p2 Sμ − for all u ∈ Nλ (3:7) By (3.2) and (3.7), we get q −p p∗ − q ||u||p−q − λ ∗ |f + |q∗ Sμ μ N p q ⎡ q qN p∗ −p p2 p−q p−q Sμ ⎣ ∗ − q)|g+ | ∗ − q)|g+ | (p N (p ∞ ∞ Jλ (u) ≥ ||u||q μ > p−q p∗ −p N(p−q) p2 Sμ q −λ −p p∗ − q + |f |q∗ Sμ ∗q p which implies that − Jλ (u) > d0 for all u ∈ Nλ , for some positive constant d0 □ q Remark 3.6 If λ ∈ (0, p ), 1,p then by Lemmas 3.4 and 3.5, for each u ∈ D0 ( )with ∗ g|u|p > 0, we can easily deduce that − − t− u ∈ Nλ and Jλ (t− u) = sup Jλ (tu) ≥ αλ > t≥0 Proof of Theorem 1.1 First, we define the Palais-Smale (simply by (PS)) sequences, (PS)-values and (PS)-con1,p ditions in D0 ( ) for Jl as follows: 1,p Definition 4.1 (i) For c Ỵ ℝ, a sequence {un} is a (PS)c-sequence in D0 ( )for Jl if Jl 1,p (un) = c + o(1) and (Jl)’(un) = o(1) strongly in (D0 ( ))−1as n ® ∞ Hsu Boundary Value Problems 2011, 2011:37 http://www.boundaryvalueproblems.com/content/2011/1/37 Page of 15 1,p 1,p (ii) c Ỵ ℝ is a (PS)-value in D0 ( )for Jl if there exists a (PS)c-sequence in D0 ( )for Jl 1,p 1,p (iii) Jl satisfies the (PS)c-condition in D0 ( )if any (PS)c-sequence {un} in D0 ( )for Jl contains a convergent subsequence Lemma 4.2 (i) If l Ỵ (0, Λ1), then Jl has a (PS)αλ-sequence {un } ⊂ Nλ q (ii) If λ ∈ (0, p ), − then Jl has a (PS)αλ-sequence {un } ⊂ Nλ Proof The proof is similar to [19,25] and the details are omitted □ Now, we establish the existence of a local minimum for Jl on Nλ ¯ Theorem 4.3 Suppose that N ≥ 3, μ < μ, in Ω Finally, by (3.3), the Hölder inequality and Sobolev embedding theorem, we obtain q ||uλ ||p−q < λ μ −p p∗ − q + |f |q∗ Sμ ∗−p p which implies that ||ul||μ ® as l ® 0+ □ Proof of Theorem 1.1 From Theorem 4.3, it follows that the problem (1.1) has a posi+ tive solution uλ ∈ Nλ for all l Ỵ (0, Λ0) □ Proof of Theorem 1.2 ¯ For
and the positive constants Ci (i = 1, 2) independent of ε, such that sup Jλ (tuε ) = Jλ (tε uε ) and < C1 ≤ tε ≤ C2 < ∞ t≥0 (5:6) Hsu Boundary Value Problems 2011, 2011:37 http://www.boundaryvalueproblems.com/content/2011/1/37 Page 11 of 15 By (g2), we conclude that p∗ ∗ g(x)|uε |p − ≤ g(0)|uε | |g(x) − g(0)| |uε |p =O B(0;ρ) |x|β |uε |p ∗ ∗ = O(εβ ), which together with Lemma 2.2 implies that N p ∗ ∗ g(x)|uε |p = g(0)Sμ + O(ε p γ ) + O(ε β ) (5:7) From the fact l > 0, 0, , and by Lemma 2.2, (5.7) and (f2), we get p∗ p tε tε Jλ (tε uε ) = ||uε ||p − ∗ μ p p ∗ N p ≤ ||uε ||μ N q g|uε |p − λ p∗ g|uε | N p N p Sμ + O(ε pγ ) = N N−p − p tε q f |uε |q q −λ N p g(0)Sμ C1 β0 q + O(ε p∗ γ |uε |q β N−p − p ) + O(ε ) (5:8) q −λ C1 β0 q |uε |q N = q N−p C − p g(0) p Sμ + O(ε pγ ) + O(ε β ) − λ β0 N q |uε |q By (5.6) and (5.8), we have that sup Jλ (tuε ) ≤ c∗ + O(ε pγ ) + O(ε β ) − λ t≥0 (i) If < q < N b(μ), q C1 β0 q |uε |q then by Lemma 2.2 and γ = b(μ) − δ = b(μ) − (5:9) N−p p > we have that |uε |q = O1 (εqγ ) Combining this with (5.9), for any l > 0, we can choose εl small enough such that sup Jλ (tuελ ) < c∗ t≥0 (ii) If N b(μ) ≤ q < p, then by Lemma 2.2 and g > we have that |uε |q = O1 (εθ ), O1 (εθ |lnε|), N q > b(μ) , N q = b(μ) , Hsu Boundary Value Problems 2011, 2011:37 http://www.boundaryvalueproblems.com/content/2011/1/37 Page 12 of 15 and pγ = b(μ)p + p − N > N + (1 − N )q = θ p Combining this with (5.9), for any l > 0, we can choose εl small enough such that sup Jλ (tuελ ) < c∗ t≥0 From (i) and (ii), (5.5) holds by taking vλ = uελ In fact, by (f2), (g2) and the definition of uελ, we have that f |uελ |q > and ∗ g|uελ |p > − From Lemma 3.4, the definition of αλ and (5.5), for any l Ỵ (0, Λ 0), there exists − tελ > such that tελ uελ ∈ Nλ and − αλ ≤ Jλ (tελ uελ ) ≤ sup Jλ (ttελ uελ ) < c∗ t≥0 The proof is thus complete □ − Now, we establish the existence of a local minimum of Jl on Nλ ¯ Theorem 5.3 Suppose (H)and (f ) - (g ) hold If < μ < μ, x = 0, b ≥ pg and λ ∈ (0, q p ), − then there exists Uλ ∈ Nλ such that − (i) Jλ (Uλ ) = αλ , (ii) Ul is a positive solution of (1.1) q Proof If λ ∈ (0, p − {un } ⊂ Nλ -sequence − on Nλ (see Lemma ), then by Lemmas 3.5 (ii), 4.2 (ii) and 5.2, there exists a − 1,p − {un } ⊂ Nλ in D0 ( ) for Jl with αλ ∈ (0, c∗ ) Since Jl is coercive 1,p 3.1), we get that {un} is bounded in D0 ( ) From Lemma 5.1, 1,p there exists a subsequence still denoted by {un} and a nontrivial solution Uλ ∈ D0 ( ) 1,p of (1.1) such that un ⇀ Ul weakly in D0 ( ) − − + First, we prove that Uλ ∈ Nλ On the contrary, if Uλ ∈ Nλ , then by Nλ ∪ {0} is 1,p closed in D0 ( ), we have ||Ul||μ < lim infn®∞ ||un||μ From (g2) and Ul ≢ in Ω, we have ∗ g|Uλ |p > Thus, by Lemma 3.4, there exists a unique t l such that − tλ Uλ ∈ Nλ If u ∈ Nλ, then it is easy to see that Jλ (u) = p∗ − q ||u||p − λ μ N p∗ q f |u|q (5:10) − From Remark 3.6, un ∈ Nλ and (5.10), we can deduce that − − αλ ≤ Jλ (tλ Uλ ) < lim Jλ (tλ un ) ≤ lim Jλ (un ) = αλ n→∞ n→∞ − This is a contradiction Thus, Uλ ∈ Nλ Next, by the same argument as that in Theorem 4.3, we get that un ® Ul strongly in q − 1,p D0 ( ) and Jλ (Uλ ) = αλ > for all λ ∈ (0, p |Uλ | ∈ − Nλ , ) Since J l (U l ) = J l (|U l |) and by Lemma 3.2, we may assume that Ul is a nontrivial nonnegative solution Hsu Boundary Value Problems 2011, 2011:37 http://www.boundaryvalueproblems.com/content/2011/1/37 Page 13 of 15 of (1.1) Finally, by Harnack inequality due to Trudinger [26], we obtain that Ul is a positive solution of (1.1) □ + Proof of Theorem 1.2 From Theorem 4.3, we get the first positive solution uλ ∈ Nλ + for all l Ỵ (0, Λ0) From Theorem 5.3, we get the second positive solution Uλ ∈ Nλ q for all λ ∈ (0, p − − Since Nλ ∩ Nλ = ∅, this implies that ul and Ul are distinct □ ) Proof of Theorem 1.3 In this section, we consider the case μ ≤ In this case, it is well-known Sμ = S0 where Sμ is defined as in (1.2) Thus, we have ∗ c = N−p N − p p + S0 N |g |∞ when μ ≤ Lemma 6.1 Suppose (H)and (f ) - (g ) hold If N ≤ p , μ < 0, x ≠ and β p ≥ γ := ˜ N−p p(p−1), 1,p then for any l > and μ < 0, there exists vλ,μ ∈ D0 ( )such that sup Jλ (tvλ,μ ) < c∗ (6:1) t≥0 − In particular, αλ < c∗for all l Ỵ (0, Λ1) Proof Note that S0 has the following explicit extremals [27]: ⎛ N−p ¯ − Cε p ⎝1 Vε (x) = |x − x0 | ε + N−p p ⎞− p p−1 ⎠ , ∀ε > 0, x0 ∈ RN , ¯ where C > is a particular constant Take r > small enough such that B(x0; r) ⊂ ˜ Ω\{0} and set uε (x) = ϕ(x)Vε (x), where ϕ(x) ∈ C∞ (B(x0 ; ρ) is a cutoff function such that (x) ≡ in B(x0; r/2) Arguing as in Lemma 2.2, we have N p ˜ ˜ |∇ uε |p = S0 + O(ε pγ ), N ∗ (6:2) ∗ ˜ |˜ ε |p = S0P + +O(εp γ ), u (6:3) ⎧ ⎪ O1 (εθ ), ⎨ q |˜ ε | = O1 (εθ |lnε|), u ⎪ ˜ ⎩ O1 (εqγ ), where θ = N − N−p p q N(p−1) N−p < q < p∗ , N(p−1) N−p , N(p−1) N−p , q= 1≤q< (6:4) ˜ ˜ ˜ Note that β ≥ pγ , p∗ γ > pγ Arguing as in Lemma 5.2, we deduce that there exists ˜ε satisfying < C1 ≤ ˜ε ≤ C2, such that t t ˜ ˜ Jλ (tuε ) ≤ sup Jλ (tuε ) = Jλ (˜ε uε ) t ˜ t≥0 p ˜ε t = p ∗ ˜ |∇ uε |p − p ˜ε t p∗ ∗ g|˜ ε |p − λ u N ≤ q ˜ε t q p C2 p |˜ ε |p u p ˜ε t p |˜ ε |p u |x|p (6:5) q N−p C − p ˜ g(x0 ) p Sμ + O(ε pγ ) − λ β0 N q − μ||x0 | − ρ|−p f |˜ ε |q − μ u |˜ ε |q u Hsu Boundary Value Problems 2011, 2011:37 http://www.boundaryvalueproblems.com/content/2011/1/37 Page 14 of 15 From (H), N ≤ p2 and (6.4), we can deduce that < qγ < pγ = ˜ ˜ N−p N(p − 1) ≤p≤ p−1 N−p and ˜ |˜ ε |q = O1 (εqγ ) and u O1 (εp |lnε|), ˜ O1 (εpγ ), |˜ ε |p = u p= 1 and μ < 0, we can choose εl,μ small enough such that N ˜ sup Jλ (tuελ,μ ) < t≥0 N−p − p g(x0 ) p S0 = c∗ N ˜ Therefore, (6.1) holds by taking vλ,μ = uελ,μ ˜ In fact, by (f2), (g2) and the definition of uελ,μ, we have that f |˜ ελ,μ |q > u and ∗ g|˜ ελ,μ |p > u − From Lemma 3.4, the definition of αλ and (6.1), for any l Î (0, Λ0) and μ < 0, there − ˜ exists tελ,μ > such that tελ,μ uελ,μ ∈ Nλ and − ˜ ˜ αλ ≤ Jλ (tελ,μ uελ,μ ) ≤ sup Jλ (ttελ,μ uελ,μ ) < c∗ t≥0 The proof is thus complete □ Proof of Theorem 1.3 Let Λ1(0) be defined as in 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doi:10.1002/cpa.3160200406 27 Talenti, G: Best constant in Sobolev inequality Ann Mat Pura Appl 110, 353–372 (1976) doi:10.1007/BF02418013 doi:10.1186/1687-2770-2011-37 Cite this article as: Hsu: Multiple positive solutions for a class of quasi-linear elliptic equations involving concave-convex nonlinearities and Hardy terms Boundary Value Problems 2011 2011:37 Submit your manuscript to a journal and benefit from: Convenient online submission Rigorous peer review Immediate publication on acceptance Open access: articles freely available online High visibility within the field Retaining the copyright to your article Submit your next manuscript at springeropen.com ... http://www.boundaryvalueproblems.com/content/2011/1/37 Page 15 of 15 11 Wang, L, Wei, Q, Kang, D: Multiple positive solutions for p-Laplace elliptic equations involving concave-convex nonlinearities and a Hardytype term Nonlinear Anal 74, 626–638... Multiple positive solutions for a class of quasi-linear elliptic equations involving concave-convex nonlinearities and Hardy terms Boundary Value Problems 2011 2011:37 Submit your manuscript to a journal... operator J Math Anal Appl 352, 99–111 (2009) doi:10.1016/j.jmaa.2008.06.021 16 Kang, D: Solution of the quasilinear elliptic problem with a critical Sobolev -Hardy exponent and a Hardy term J Math