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Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 845946, 16 pages doi:10.1155/2009/845946 Research Article Entire Solutions for a Quasilinear Problem in the Presence of Sublinear and Super-Linear Terms C. A. Santos Department of Mathematics, University of Bras ´ ılia, 70910–900 Bras ´ ılia, DF, Brazil Correspondence should be addressed to C. A. Santos, csantos@unb.br Received 31 May 2009; Revised 13 August 2009; Accepted 2 October 2009 Recommended by Wenming Zou We establish new results concerning existence and asymptotic behavior of entire, positive, and bounded solutions which converge to zero at infinite for the quasilinear equation −Δ p u  axfuλbxgu,x∈ R N , 1 <p<N,where f, g : 0, ∞ → 0, ∞ are suitable functions and ax,bx ≥ 0 are not identically zero continuous functions. We show that there exists at least one solution for the above-mentioned problem for each 0 ≤ λ<λ  , for some λ  > 0. Penalty arguments, variational principles, lower-upper solutions, and an approximation procedure will be explored. Copyright q 2009 C. A. Santos. 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 In this paper we establish new results concerning existence and behavior at infinity of solutions for the nonlinear quasilinear problem −Δ p u  a  x  f  u   λb  x  g  u  in R N , u>0inR N ,u  x  −→ 0, | x | −→ ∞ , 1.1 where Δ p u  div|∇u| p−2 ∇u,with1<p<N, denotes the p-Laplacian operator; a, b : R N → 0, ∞ and f, g : 0, ∞ → 0, ∞ are continuous functions not identically zero and λ ≥ 0isa real parameter. A solution of 1.1 is meant as a positive function u ∈ C 1 R N  with ux → 0as |x|→∞and  R N | ∇u | p−2 ∇u∇ϕdx   R N  a  x  f  u   λb  x  g  u   ϕdx, ∀ϕ ∈ C ∞ 0  R N  . 1.2 2 Boundary Value Problems The class of problems 1.1 appears in many nonlinear phenomena, for instance, in the theory of quasiregular and quasiconformal mappings 1–3, in the generalized reaction-diffusion theory 4, in the turbulent flow of a gas in porous medium and in the non-Newtonian fluid theory 5. In t he non-Newtonian fluid theory, the quantity p is the characteristic of the medium. If p<2, the fluids are called pseudoplastics; if p  2 Newtonian and if p>2the fluids are called dilatants. It follows by the nonnegativity of functions a, b, f, g of parameter λ and a strong maximum principle that all non-negative and nontrivial solutions of 1.1 must be strictly positive see Serrin and Zou 6. So, again of 6, it follows that 1.1 admits one solution if and only if p<N. The main objective of this paper is to improve the principal result of Yang and Xu 7 and to complement other works see, e.g., 8–20 and references therein for more general nonlinearities in the terms f and g which include the cases considered by them. The principal theorem in 7 considered, in problem 1.1, fuu m ,u>0, and guu n ,u>0with0<m<p− 1 <n. Another important fact is that, in our result, we consider different coefficients, while in 7 problem 1.1 was studied with axbx, ∀x ∈ R N . In order to establish our results some notations will be introduced. We set a  r  : min |x|r a  x  ,  b  r  : min |x|r b  x  ,r≥ 0, a  r  : max | x | r a  x  ,  b  r  : max | x | r b  x  ,r≥ 0. 1.3 Additionally, we consider H 1 i lim s → 0 fs/s p−1 ∞, ii lim s → 0 fs/s p−1 0, H 2 i lim s → 0 gs/s p−1 0, ii lim s → 0 gs/s p−1 ∞. Concerning the coefficients a and b, H 3 i  ∞ 1 r 1/p−1 a 1/p−1 rdr,  ∞ 1 r 1/p−1  b 1/p−1 rdr < ∞, if 1 <p≤ 2, ii  ∞ 1 r p−2N1/p−1 ardr,  ∞ 1 r p−2N1/p−1  brdr < ∞, if p ≥ 2. Our results will be established below under the hypothesis N ≥ 3. Theorem 1.1. Consider H 1 –H 3 , then there exists one λ  > 0 such that for each 0 ≤ λ<λ  there exists at least one u  u λ ∈ C 1 R N  solution of problem 1.1. Moreover, C | x | −N−p/p−1 ≤ u  x  ,x∈ R N , | x | ≥ 1 1.4 for some constant C  Cλ > 0. If additionally f  t  t p−1 is nonincreasing and g  t  t p−1 is nondecreasing for t>0, 1.5 Boundary Value Problems 3 then there is a positive constant D  Dλ   such that u 2  x  f  ux 4  −1/p−1 ≤ D  ∞ | x |  t 1−N  t 0 as  bsds  1/p−1 dt, x ∈ R N . 1.6 Remark 1.2. If we assume 1.5 with ftt m , t>0, where 0 ≤ m<p− 1, then 1.6 becomes 0 <u  x  ≤ C ⎛ ⎝  ∞ | x |  t 1−N  t 0 as  bsds  1/p−1 dt ⎞ ⎠ 1/2−m/p−1 ,x∈ R N . 1.7 In the sequel, we will establish some results concerning to quasilinear problems which are relevant in itself and will play a key role in the proof of Theorem 1.1. We begin with the problem of finding classical solutions for the differential inequality −Δ p v ≥ a  x  f  v   λb  x  g  v  in R N , v>0inR N ,v  x  −→ 0, | x | −→ ∞ . 1.8 Our result is. Theorem 1.3. Consider H 1 –H 3 , then there exists one λ  > 0 such that problem 1.8 admits, for each 0 ≤ λ<λ  , at least one radially symmetric solution v  v λ ∈ C 2 R N \{0}∩ C 1,ν loc R N ,forsome ν ∈ 0, 1. Moreover, if in additionally one assumes 1.5, then there is a positive constant D  Dλ   such that v 2  x  f  vx 4  −1/p−1 ≤ D  ∞ | x |  t 1−N  t 0 as  b sds  1/p−1 dt, x ∈ R N . 1.9 Remark 1.4. Theorems 1.1 and 1.3 are still true with N  2ifH 3  hypothesis is replaced by H 3    ∞ 1 t 1−N  t 0 as  bsds 1/p−1 dt < ∞. In fact, H 3  implies H 3   ,ifN ≥ 3. see sketch of the proof in the appendix. Remark 1.5. In Theorem 1.3, it is not necessary to assume that f and g are continuous up to 0. It is sufficient to know that f, g : 0, ∞ → 0, ∞ are continuous. This includes terms f, g singular in 0. The next result improves one result of Goncalves and Santos 21 because it guarantees the existence of radially symmetric solutions in C 2 B0,R \{0} ∩ C 1 B0,R ∩ CB0,R for the problem −Δ p u  ρ  x  h  u  in B  0,R  , u>0inB  0,R  ,u  x   0,x∈ ∂B  0,R  , 1.10 4 Boundary Value Problems where ρ : B0,R → 0, ∞, h : 0, ∞ → 0, ∞ are continuous and suitable functions and B0,R ⊂ R N is the ball in R N centered in the origin with radius R>0. Theorem 1.6. Assume ρx ρ|x|,x ∈ R N where ρ : 0, ∞ → 0, ∞,withρ /  0,is continuous. Suppose that h satisfies (H 1  and additionally h  s  s p−1 ,s>0 is nonincreasing. 1.11 then 1.10 admits at least one radially symmetric solution u ∈ C 2 B0,R \{0} ∩ C 1 B0,R ∩ CB0,R. Besides this, uxu|x|,x∈ B0,R, and u satisfies u  r   u  0  −  r 0  t 1−N  t 0 s N−1 ρshusds  1/p−1 dt, r ≥ 0. 1.12 The proof of principal theorem Theorem 1.1 relies mainly on the technics of lower and upper solutions. First, we will prove Theorem 1.3 by defining several auxiliary functions until we get appropriate conditions to define one positive number λ  and a particular upper solution of 1.1 for each 0 ≤ λ<λ  . After this, we will prove Theorem 1.6, motivated by arguments in 21, which will permit us to get a lower solution for 1.1. Finally, we will obtain a solution of 1.1 applying the lemma below due to Yin and Yang 22. Lemma 1.7. Suppose that fx, r is defined on R N1 and is locally H ¨ older continuous (with γ ∈ 0, 1)inx. Assume also that there exist functions w, v ∈ C 1,γ loc R N  such that −Δ p v ≥ f  x, v  ,x∈ R N , −Δ p w ≤ f  x, w  ,x∈ R N , w  x  ≤ v  x  ,x∈ R N , 1.13 and fx, r is locally Lipschitz continuous in r on the set   x, r  /x ∈ R N ,w  x  ≤ r ≤ v  x   . 1.14 Then there exists u ∈ C 1 R N  with wx ≤ ux ≤ vx,x∈ R N satisfying  R N | ∇u | p−2 ∇u∇ϕdx  R N f  x, u  ϕdx, ∀ϕ ∈ C ∞ 0  R N  . 1.15 In the two next sections we will prove Theorems 1.3 and 1.6. Boundary Value Problems 5 2. Proof of Theorem 1.4 First, inspired by Zhang 20 and Santos 16, we will define functions F : 0, ∞ → 0, ∞ and G : 0, ∞ × 0, ∞ → 0, ∞ by F  s   sup t≥s f  t  t p−1 ,s>0,G  τ,s   ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ sup s≤t≤τ g  t  t p−1 ,s≤ τ, g  τ  τ p−1 ,s≥ τ. 2.1 So, for each λ ≥ 0, let F λ : 0, ∞ × 0, ∞ → 0, ∞ given by F λ  τ,s   F 0  s   λF  τ,s  , 2.2 where F 0  s   s p−1 F  s  ,s>0,F  τ,s   s p−1 G  τ,s  ,τ,s>0. 2.3 It is easy to check that F 0  s  ≥ f  s  ,s>0, for each τ>0,F  τ,s  ≥ g  s  , 0 <s≤ τ 2.4 and, as a consequence, F λ  τ,s  ≥ f  s   λg  s  , 0 <s≤ τ. 2.5 Moreover, it is also easy to verify. Lemma 2.1. Suppose that H 1  and H 2  hold. Then, for each τ>0, i Fτ,s/s p−1 ,s>0 is non-increasing, ii F 0 s/s p−1 ,s>0 is non-increasing, iii lim s → 0 Fτ, s/s p−1 sup 0<t≤τ gt/t p−1 , iv lim s → 0 F 0 s/s p−1 ∞, v lim s →∞ Fτ, s/s p−1 gτ/τ p−1 , vi lim s →∞ F 0 s/s p−1 0. By Lemma 2.1iii, iv,and2.2, the function  F λ : 0, ∞ × 0, ∞ → 0, ∞, given by  F λ  τ,s   s 2  s 0  t/F λ  τ,t  1/p−1  dt , 2.6 is well defined and continuous. Again, by using Lemma 2.1i and ii,  F λ  τ,s  ≥ F λ  τ,s  1/p−1 , ∀τ, s > 0. 2.7 6 Boundary Value Problems Besides this,  F λ τ,· ∈ C 1 0, ∞, for each τ>0, and using Lemma 2.1, it follows that  F λ satisfies, for each λ ≥ 0, the following. Lemma 2.2. Suppose that H 1  and H 2  hold. Then, for each τ>0, i  F λ τ,s/s is non-increasing in s>0, ii lim s → 0   F λ τ,s/s∞, iii lim s → 0   F λ τ,s/sλgτ/τ p−1  1/p−1 , if λ>0, iv lim s → 0   F λ τ,s/s0, if λ  0. And, in relation to λ, we have the folowing. Lemma 2.3. Suppose that H 1  and H 2  hold. Then, for each τ, s > 0, i  F λ 1 τ,s <  F λ 2 τ,s, if λ 1 <λ 2 , ii  F λ τ,s/s → F 0 s/s, as λ → 0. Finally, we will define, for each λ ≥ 0, H λ : 0, ∞ → 0, ∞,by H λ  τ   1 τ  τ 0 t  F λ  τ,t  dt. 2.8 So, H λ is a continuous function and we have see proof in the appendix. Lemma 2.4. Suppose that H 1  and H 2  hold. Then, i lim τ → 0 H λ τ0, for any λ ≥ 0, ii lim τ →∞ H λ τ∞, if λ  0, iii lim τ →∞ H λ τ0, if λ>0, iv H λ 1 τ,s >H λ 2 τ,s, if λ 1 <λ 2 , v lim λ → 0 H λ τH 0 τ, for each τ>0. By Lemma 2.4ii, there exists a τ ∞ > 0 such that H 0 τ ∞  >α 1, where by either H 3  or H 3   , we have 0 <α:  ∞ 0  t 1−N  t 0 as  bsds  1/p−1 dt < ∞. 2.9 So, by Lemma 2.4v, there exists a λ  > 0 such that H λ  τ ∞  >α.Thatis, 1 τ ∞  τ ∞ 0 t  F λ   τ ∞ ,t  dt > α. 2.10 Let P : 0, ∞ × 0,τ ∞  → R N by P  t, s   ω  t  − 1 τ ∞  s 0 ς  F λ   τ ∞ ,ς  dς, 2.11 Boundary Value Problems 7 where ω : 0, ∞ → 0, ∞, ω ∈ C 2 0, ∞ ∩ C 1 0, ∞ is given by ωx ω|x|,x∈ R N where ω ∈ C 2 R N \{0} ∩ C 1 R N  is the unique positive and radially symmetric solution of problem −Δ p ω  a  | x |    b  | x |  in R N , ω>0inR N ,ω  x  −→ 0, | x | −→ ∞ . 2.12 More specifically, by DiBenedetto 23, ω ∈ C 2 R N \{0} ∩ C 1,ν loc R N , for some ν ∈ 0, 1.In fact, ω satisfies ω  r   α −  r 0  t 1−N  t 0 as  bsds  1/p−1 dt, r ≥ 0. 2.13 So, by 2.10, 2.11,and2.13, we have for each t>0, P  t, 0   ω  t  > 0,P  t, τ ∞  <α− 1 τ ∞  τ ∞ 0 t  F λ   τ ∞ ,t  dt < 0. 2.14 Hence, after some pattern calculations, we show that there is a ϑ ∈ C 2 0, ∞ ∩ C 1 0, ∞ such that ϑr ≤ τ ∞ ,r ≥ 0and ω  r   1 τ ∞  ϑr 0 t  F λ   τ ∞ ,t  dt, r ≥ 0. 2.15 As consequences of 2.9, 2.13 and 2.15, we have ϑr → 0,r→∞and  r N−1   ω  r   p−1 ω  r    1 τ p−1 ∞  ϑr  F λ  τ ∞ ,ϑr  p−1  r N−1   ϑ  r   p−1 ϑ  r    p − 1 τ p−1 ∞  ϑr  F λ  τ ∞ ,ϑr  p−2 d ds  s  F λ   τ ∞ ,s   r N−1   ϑ  r   p 2.16 and hence, by Lemma 2.2 i, 2.7 and ϑr ≤ τ ∞ ,r≥ 0, we obtain −  r N−1   ϑ  r   p−1 ϑ  r   ≥  τ ∞ ϑr  p−1  F λ   τ ∞ ,ϑr  p−1  −  r N−1   ω  r   p−1 ω  r     r N−1 F λ   τ ∞ ,ϑ  r   a  r    b  r   , 2.17 8 Boundary Value Problems that is, by using 2.2, we have −  r N−1   ϑ  r   p−1 ϑ  r   ≥ r N−1 a  r  F 0  ϑ  r   λ  r N−1  b  r  F  τ ∞ ,ϑ  r  ,r≥ 0. 2.18 In particular, making vxϑ|x|,x∈ R N ,wegetfrom2.15, Lemma 2.2i and ω ∈ C 2 R N \{0}∩C 1,ν loc R N  that v ∈ C 2 R N \{0}∩C 1,ν loc R N  and satisfies 1.8, for each 0 ≤ λ ≤ λ  . That is, v is an upper solution to 1.1. To prove 1.9, first we observe, using Lemma 2.2 i and 2.15,that ω  r  ≥ 1 τ ∞  ϑr/2 0 t F λ   τ ∞ ,t  1/p−1 dt ≥ 1 τ ∞  ϑr/2 ϑr/4 t F λ   τ ∞ ,t  1/p−1 dt ≥ 1 τ ∞  1 Fϑr/4λ  Gτ ∞ ,ϑr/4  1/p−1  ϑ  r  /4  ,r≥ 0. 2.19 So, by definition of F, Gτ ∞ , · and hypothesis 1.5, we have F  ϑ  r  4   λ  G  τ ∞ , ϑ  r  4   f  ϑ  r  /4   ϑr/4  p−1  λ  g  τ ∞  τ p−1 ∞ ,r≥ 0. 2.20 Thus,  ϑr/4  2p−1 f  ϑ  r  /4  ≤ τ p−1 ∞  1  λ  g  τ ∞  τ p−1 ∞  ϑr/4  p−1 f  ϑ  r  /4   ω  r  p−1 ,r≥ 0. 2.21 Recalling that ϑr ≤ τ ∞ ,r ≥ 0andusing1.5 again, we obtain ϑ  r  2  f  ϑr 4  −1/p−1 ≤ 16τ ∞  1  λ  gτ ∞  τ p−1 ∞  τ ∞ /4  p−1 fτ ∞ /4  1/p−1 ω  r  ,r≥ 0. 2.22 Thus by 2.9, 2.13,andvxϑr,r |x|, for all x ∈ R N , there is one positive constant D  Dλ   such that 1.9 holds. This ends the proof of Theorem 1.3. 3. Proof of Theorem 1.5 To prove Theorem 1.5, we will first show the existence of a solution, say u k ∈ C 2 B0,R \ {0} ∩ C 1 B0,R ∩ CB0,R, for each k  1, 2, ,for the auxiliary problem −Δ p u  ρ  x  h k  u  in B  0,R  , u>0inB  0,R  ,u  x   0,x∈ ∂B  0,R  , 3.1 where h k shs  1/k,s≥ 0. In next, to get a solution for problem 1.10, we will use a limit process in k. Boundary Value Problems 9 For this purpose, we observe that i lim inf s → 0 h k sh1/k > 0, ii lim s →∞ h k s/s p−1 lim s →∞ hs  1/k/s  1/k p−1 1  1/ks p−1  0, by H 1  and by 1.11, it follows that iii h k s/s p−1 hs1/k/s1/k p−1 11/ks p−1 ,s>0 is non-increasing, for each k  1, 2, By items i–iii above, ρ and h k fulfill the assumptions of Theorem 1.3 in 21.Thus3.1 admits one solution u k ∈ C 2 B0,R \{0} ∩ C 1 B0,R ∩ CB0,R, for each k  1, 2, Moreover, u k xu k |x|,x∈ R N with u k ∈ C 2 0,R ∩ C 1 0,R ∩ C0,R satisfying u k  r   u k  0  −  r 0  t 1−N  t 0 s N−1 ρshu k s1/kds  1/p−1 dt, 0 ≤ r ≤ R. 3.2 Adapting the arguments of the proof of Theorem 1.3 in 21,weshow cϕ 1  r  ≤ u k1  r   1 k  1 ≤ u k  r   1 k , 0 ≤ r ≤ R, 3.3 where ϕ 1 ∈ C 2 B0,R is the positive first eigenfunction of problem −  r N−1   ϕ    p−2 ϕ     λr N−1 ρ  r    ϕ   p−2 ϕ in B  0,R  , ϕ  0on∂B  0,R  , 3.4 and c>0, independent of k,ischosenusing H 1  such that h  cϕ 1  ∞   cϕ 1  ∞  p−1 >λ 1 , 3.5 with λ 1 > 0 denoting the first eigenvalue of problem 3.4 associated to the ϕ 1 . Hence, by 3.3, u k  r  −→ u  r  with cϕ 1  r  ≤ u  r  ≤ | u 1 | ∞  1, 0 ≤ r ≤ R. 3.6 Using H 1 , 3.3, the above convergence and Lebesgue’s theorem, we have, making k →∞ in 3.2,that u  r   u  0  −  r 0  t 1−N  t 0 s N−1 ρshusds  1/p−1 dt, 0 ≤ r ≤ R. 3.7 So, making uxu|x|,x∈ R N , after some calculations, we obtain that u ∈ C 2 B0,R \ {0} ∩ C 1 B0,R ∩ CB0,R. This completes the proof of Theorem 1.6. 10 Boundary Value Problems 4. Proof of Main Result: Theorem 1.1 To complete the proof of Theorem 1.1, we will first obtain a classical and positive lower solution for problem 1.1,sayw, such that w ≤ v, where v is given by Theorem 1.3.After this, the existence of a solution for the problem 1.1 will be obtained applying Lemma 1.7. To get a lower solution for 1.1, we will proceed with a limit process in u n , where u n is a classical solution of problem 1.10given by Theorem 1.6 with ρ  a, h is a suitable function and R  n for n ≥ n 0 and n 0 is such that a /  0in0,n 0 . Let f ∞  s   s p−1  f ∞  s  ,s>0, where  f ∞  s   inf 0<t≤s f  t  t p−1 ,s>0. 4.1 Thus, it is easy to check the following lemma. Lemma 4.1. Suppose that H 1  and H 2  hold. Then, i 0 <f ∞ s ≤ fs ≤ F 0 sλ  Fτ ∞ ,s,s>0, ii f ∞ s/s p−1 ,s>0 is non-increasing, iii lim s → 0 f ∞ s/s p−1 ∞ and lim s →∞ f ∞ s/s p−1 0. Hence, Lemma 4.1 shows that f ∞ fulfills all assumptions of Theorem 1.6.Thus,for each n ∈ N such that n ≥ n 0 there exists one  n ∈ C 2 B0,n \{0} ∩ C 1 B0,n ∩ CB0,n with  n x  n |x|,x ∈ B0,n and  n satisfying −  r N−1 |   n | p−2   n    r N−1 a  r  f ∞   n  r  in 0 <r<n,  n > 0in  0,n  ,  n  n   0, 4.2 equivalently,  n  r    n  0  −  r 0  t 1−N  t 0 s N−1 asf ∞   n sds  1/p−1 dt, 0 ≤ r ≤ n. 4.3 Consider  n extended on n, ∞ by 0. We claim that 0 ≤··· ≤  n ≤  n1 ≤··· ≤ ϑ. 4.4 Indeed, first we observe that f ∞ satisfies Lemma 4.1ii. So, with similar arguments to those of 21,weshow  n ≤  n1 ,n≥ n 0 . To prove  n ≤ ϑ, first we will prove that  n 0 ≤ ϑ0, for all n ∈ N. In fact, if  n 0 > ϑ0 for some n, then there is one T n > 0 such that ϑ  r  <  n  r  ,r∈  0,T n  ,ϑ  T n    n  T n  > 0, 4.5 because  n n0andϑ>0withϑr → 0asr →∞. [...]... 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