Positive weak solutions for some semilinear elliptic equations

Nonlinear Analysis 48 (2002) 939 – 945

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Positive weak solutions for some semilinear elliptic equations Nguyen Bich Huy Department of Mathematics, College of Education, 280 An Duong Vuong dist 5, Ho Chi Minh City, Viet Nam Received 27 May 1999; accepted 1 April 2000

Keywords: Monotone increasing operators in ordered Banach spaces; Weak solution; Elliptic equation; Logistic equation

0. Introduction Fixed point theorems of monotone increasing operators in ordered Banach spaces are a useful tool for studying elliptic and parabolic equations (see [1,2,9,7] and references therein). In the present paper we will give such a theorem and use a Entropy principle of Brezis–Browder [4] to obtain a very simple proof. Then we apply this result to investigate the existence and approximation of positive weak solutions for the following elliptic semilinear boundary value problem: − :u = f(x; u) in <;

u = 0 on @<;

(1)

where < is a bounded open set in RN (N ≥ 3) with su=ciently smooth boundary. Our main assumption on f is that, there exists a Caratheodory function F(x; u; v), increasing on u and decreasing on v such that f(x; u) = F(x; u; u). The useful assumption that there is an increase in u function g(x; u) such that f(x; u) + g(x; u) is increasing, can be reduced to that of ours by setting F(x; u; v) = f(x; u) + g(x; u) − g(x; v). In [11], to study a vector variant of (1), Mitidieri and Sweers assumed f(x; u) = F(x; u; u) with F(x; u; v) increasing on u, but they imposed existence of ordered sub- and supersolutions of (1) and a growth condition for F. We shall assume the existence of a subsolution in combination with a priori estimate for the set of all subsolutions. These assumptions E-mail address: [email protected] (N.B. Huy). 0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 0 ) 0 0 2 2 4 - 8

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hold for the logistic equation − :u = m(x)u − up in <;

u = 0 on @<

(2)

q

with p ¿ 1; ¿ 0; m(x) ∈ L (<) and q satisfying an appropriate condition. Our results in this paper are motivated by work [8] on problems (1) and (2). 1. A xed point theorem for monotone increasing operators There are many Gxed point theorems for monotone increasing operators by Krasnoselskii, Bahtin, Amann, Lakshmikantham–HeikkilIa and others [1,2,9,7]. Our theorem is related to the theorem of Lakshmikantham–HeikkilIa [7, Theorem 1:2:1] where the existence of a smallest Gxed point is proved, and with the theorem of Krasnoselskii [9, Theorem 38:2] in which it is imposed that every increasing sequence {An un } is convergent. A crucial point in our treatment is the use of the following theorem of Brezis–Browder [4]. Theorem A. Let (M; ≤) be an ordered set and S : M →[ − ∞; +∞) be a function such that (i) Every monotone increasing sequence in M has an upper bound; (ii) S is monotone increasing and bounded above. Then there exists an element u ∈ M such that v ∈ M; v ≥ u

implies S(v) = S(u):

Theorem 1. Let real Banach space X be ordered by a cone K; M ⊂ X be a closed subset and A : M →M be an increasing operator such that (H1 ) There exists an element u0 ∈ M satisfying u0 ≤ Au0 ; (H2 ) Sequence {Aun } converges whenever {un } is an increasing sequence from M0 := {u ∈ M | u0 ≤ u ≤ Au}. Then; A has a maximal 4xed point u ≥ u0 in the sense that if v is a 4xed point and v ≥ u then v = u. This maximal 4xed point will be greatest if the following additional hypothesis holds: (H3 ) For every pair of 4xed points u1 ; u2 there exists an element u ∈ M0 such that u1 ≤ u; u2 ≤ u. Proof. We Grst observe that A(M0 ) ⊂ M0 . We shall apply theorem A to the set M0 and the function (−S), where S is deGned by S(u) = sup{Ay − Ax | x; y ∈ M0 ; u ≤ x ≤ y};

u ∈ M0 :

Let {un } ⊂ M0 be an increasing sequence and u = lim Aun . Then u ∈ M and un ≤ Aun ≤ u. By letting n → ∞ in inequality Aun ≤ Au we get u ≤ Au. Therefore, u ∈ M0 is an upper bound for {un }. It is easily seen that function S is decreasing, hence the function (−S) is increasing and bounded above. Therefore, there is an element a ∈ M0 such that u ∈ M0 ; u ≥ a implies S(u) = S(a):

(3)

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We claim that S(a) = 0. Actually, assuming the contrary that S(a) ¿  for some  ¿ 0, we can Gnd some u1 ; u2 in M0 such that a ≤ u1 ≤ u2 , Au2 − Au1  ¿ . Then (3) implies S(u2 ) ¿ . Thus, we can repeatedly apply the above argument to obtain an increasing sequence {un } in M0 such that Au2n − Au2n−1  ¿ . This contradicts hypothesis (H2 ). Now, we prove that b = A(a) will be a maximal Gxed point of A. Indeed, since a ≤ b we have A(b) − A(a) ≤ S(a) = 0, so b = A(b). If c ∈ M is a Gxed point of A such that c ≥ b then A(c) − A(b) ≤ S(b) = 0, hence c = b. If hypothesis (H3 ) holds then b will be greatest. Actually, for a Gxed point c of A we choose an element uJ ∈ M0 such that uJ ≥ b; uJ ≥ c. By the arguments given above, applying to the set {u | uJ ≤ u ≤ Au}, one can Gnd a Gxed point d ≥ u. J Since d ≥ b, we have d = b, hence b ≥ c. Theorem is completely proved. Remark. Assume the hypotheses of Theorem 1, in addition assume that (H) sequence {Aun } converges to Au whenever {un } ⊂ M0 is increasing and convergent to u. Then it is easy to prove that the iterative sequence un = Aun−1 converges to minimal in {u ∈ M | u ≥ u0 } Gxed point of A. 2. Positive weak solutions for semilinear elliptic equations 2.1. First, we consider semilinear boundary value problem (1) with < to be a regular bounded open subset in RN (N ≥ 3) and f : < × R → R be a Caratheodory function, f(x; 0) = 0 for all x ∈ <. Denition (1) We say that u ∈ H01 (<) is a weak solution of (1) if f(x; u) is in L2N=(N +2) (<) and for any ’ ∈ H01 (<) we have

∇u∇’ d x − f(x; u)’ d x = 0: <

<

1

(2) We say that u ∈ H (<) is a subsolution of (1) if f(x; u) is in L2N=(N +2) (<) and for any ’ ∈ H01 (<) such that ’ ≥ 0 in < we have

∇u∇’ d x − f(x; u)’ d x ≤ 0 <

<

and u ≤ 0 on @< in the sense of traces [10]. The main auxiliary tool in our treatment is the following result of Brezis– Browder [5]. Theorem B. Let g : < × R → < be a Caratheodory function satisfying (1) g(x; 0) = 0 for all x ∈ <; (2) sup{|g(x; z)| | |z| ≤ t} ≤ ht (x) ∈ L1 (<) for all t ¿ 0; (3) g(x; u) is increasing in u for all x ∈ <.

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Then for any h ∈ H −1 (<) the boundary value problem −:u + g(x; u) = h in <;

u = 0 on @<

has a unique weak solution u such that ug(x; u) ∈ L1 (<). Theorem 2. Let the following hypotheses be satis4ed (1) Boundary value problem (1) has a subsolution u0 and the set of all subsolutions u ≥ u0 is bounded in Lp (<) for some p ≤ 2N=(N − 2). (2) There exists a Caratheodory function F : < × R2 → R such that f(x; u) = F(x; u; u) and furthermore: (i) F(x; 0; 0) ≡ 0; F(x; u; v) is increasing in u and decreasing in v; (ii) For any u ∈ Lp (<) we have F(x; u; 0) ∈ L2N=(N +2) sup{|F(x; u(x); z)|: |z| ≤ t} ≤ ht (x) ∈ L1 (<)

for all t ¿ 0:

Then problem (1) has a greatest weak solution and a minimal weak solution in the set {u: u ≥ u0 }. Furthermore; let un be a unique weak solution of the boundary value problem − :u = F(x; un−1 ; u) in <;

u = 0 on @<;

(4)

then {un } converges to minimal weak solution of (1). Proof. For any u ∈ Lp (<); u ≥ u0 the boundary value problem − :z = F(x; u; z) in <;

z = 0 on @<

(5)

has a unique weak solution z = Tu ∈ H01 (<). This assertion follows from Theorem B since (5) can be rewriting as −:z + [F(x; u; 0) − F(x; u; z)] = F(x; u; 0) in <; 2N=(N +2)

z = 0 on @<

−1

and that L ⊂H . Every Gxed point of T will be a weak solution of (1). We shall apply Theorem 1 and remark to operator T . We Grst observe that T maps Lp (<) into itself since H01 (<) ⊂ L2N=(N −2) . We claim that T is increasing. Indeed, let u ≤ v in Lp (<). Choosing ’ = (Tu − Tv)+ as a test function in

∇(Tu − Tv)∇’ d x = [F(x; u; Tu) − F(x; v; Tv))]’ d x; <

<

we have by monotonicity of F that
∇(Tu − Tv)∇(Tu − Tv)+ d x <
≤ [F(x; v; Tu) − F(x; v; Tv)](Tu − Tv)+ d x ≤ 0: <

Taking into account that
∇W + ∇W − d x = 0 <

for W ∈H01 (<);

(6)

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we obtain from (6) that ∇(Tu − Tv)+ = 0 and then (Tu − Tv)+ = 0. Thus, u ≤ v implies Tu ≤ Tv. The same arguments prove u0 ≤ Tu0 . Let {vn } be an increasing sequence such that u0 ≤ vn ≤ Tvn . To prove the convergence of {Tvn }, it is su=cient to show its boundedness in Lp (<). We have for all ’ ∈ H01 (<), ’ ≥ 0



∇Tvn ∇’ d x = F(x; vn ; Tvn )’ d x ≤ F(x; Tvn ; Tvn )’ d x = f(x; Tvn )’ d x: <

<

<

<

Consequently, Tvn is a subsolution of (1) and then {Tvn } is bounded by hypothesis 1. Since the maximun of two solutions of (1) is a subsolution [6], hence hypothesis (H3 ) in Theorem 1 holds. It remains to show that, if an increasing sequence {vn } converges in Lp (<) to v, then lim Tvn = Tv in H01 (<). Indeed, letting y = lim Tvn pointwise in < we have from
c:Tv − Tvn 22 ≤ |∇(Tv − Tvn )|2 d x <


=

<

[F(x; v; Tv) − F(x; vn ; Tvn )]:(Tv − Tvn ) d x


=

<

[F(x; v; Tv) − F(x; vn ; Tv)](Tv − Tvn ) d x




+

<

[F(x; vn ; Tv) − F(x; vn ; Tvn )](Tv − Tvn ) d x

(7)

and Beppo-Levi’s theorem that
2 c:Tv − y2 ≤ [F(x; v; Tv) − F(x; v; y)](Tv − y) d x ≤ 0 <

and then y = Tv. Here  · r denotes the norm in Lr (<). Repeatedly using (7) we get lim Tvn = Tv in H01 (<). Theorem is proved. 2.2. Now, we consider logistic equation (2) with unbounded limitation m(x). In [8] Hernandez proved the existence and uniqueness of a positive weak solution for (2) when m(x) ≥ m0 ¿ 0 and m(x) is in Lq (<) for q such that   N 2Np 2N (2p − 1) : q ¿ max ; ; 2 (p − 1)(N + 2) (p − 1)(N + 6) We will impose other restriction for q, allowing q may be smaller than N=2. Theorem 3. Let the following hypotheses be satis4ed (1) m(x) ≥ 0 in < and there exist a positive number m0 ; a regular open subset <  such that
1 where 1 is 4rst eigenvalue of the boundary value problem (3) ¿ m0 − :u = u in < ; u = 0 on @< : (8)

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N.B. Huy / Nonlinear Analysis 48 (2002) 939 – 945

Then problem (2) has a unique positive weak solution; which can be approximated by iterative sequence −:un + unp = m(x)un−1 in <;

un = 0 on @<



with u0 = 0 on <\< and u0 = c’ on < ; where ’ is a positive eigenfunction of (8) corresponding to 1 and c ¿ 0 is su7ciently small. Proof. Arguing as in [3] we can prove that u0 is a weak subsolution for (2). For any u ∈ Lp+1 (<) we have by hypothesis 2 that mu ∈ Lq(p+1)=(q+p+1) ⊂ L2N=(N +2) ⊂ H −1 : Therefore, by Theorem B boundary value problem − :z + z p = m(x)u in

<; z = 0 on @<

(9)

has a unique weak solution z = Tu such that (Tu)p+1 ∈ L1 (<). By the same arguments used in proof of Theorem 2 we see that T is an increasing operator from {u ∈ Lp+1 (<): u ≥ u0 } into itself. To apply Theorem 1 and remark, it remains to prove the boundedness of the set of weak subsolutions u ≥ u0 . In the following, we will denote by C the diLerent numbers, independing on u. Let u ≥ u0 such that u ≤ Tu. Choosing Tu as test function in (9) and using HIolder inequality, we obtain

(10) [|∇Tu|2 + (Tu)p+1 ] d x ≤ m(x)(Tu)2 d x ≤ mq :Tu22q
; <

<

where q = q=(q − 1). If 2q ≤ p + 1 then from (10) we get 2 Tup+1 p+1 ≤ mq :CTup+1 ;

which shows the boundedness of Tu in Lp+1 . Now we consider the case, when p + 1 ¡ 2q . Rewriting the inequality in hypothesis 2 as 2N=(N + 2) ≤ q(p + 1)=(q + p + 1) and taking account of monotonicity of the function qt=(q + t) we get 2q 2N q:2q = ¡ : N +2 q + 2q 2q − 1 This implies that q ¡ N=(N − 2). Therefore, we can apply Gagliardo–Nirenberg inequality to obtain 1−* Tu2q
≤ CTu*1; 2 :Tup+1

(11)

with * such that   1 1 1 1 + (1 − *) =* − :  2q 2 N p+1 From (10) and (11) we deduce that *+(1−*)2=(p+1) : Tu2q
≤ CTu2q



Consequently, the set {Tu | u0 ≤ u ≤ Tu} is bounded in L2q and in Lp+1 . The uniqueness has been proved in [8]. Theorem is proved.

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Acknowledgements The author wishes to thank Prof. J. Hernandez for the presentation of theme. The author is very grateful to the referee for reading the paper carefully and making several corrections and remarks. References [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976) 620–709. [2] I.A. Bahtin, The existence of a common Gxed point for abelian family of operators, Siberian Math J. 13 (2) (1972) (in Russian). [3] L. Boccardo, L. Orsina, Sublinear elliptic equations in Ls , Houston Math. J. 20 (1994) 99–114. [4] H. Brezis, F. Browder, A general ordering principle in nonlinear functional analysis, Adv. Math. 21 (1976) 355–364. [5] H. Brezis, F. Browder, Some properties of higher order Sobolev spaces, J. Math. Pures Appl. 61 (1982) 245–259. [6] E.N. Dancer, G. Sweers, On the existence of a maximal weak solution for a semilinear elliptic equation, DiLerential Integral Equations 2 (1989) 540–553. [7] S. HeikkilIa, V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear DiLerential Equations, Marcel Dekker, New York, 1994. [8] J. Hernandez, Positive solutions for the logistic equation with unbounded weights, in: G. Caristi, E. Mitidieri (Eds.), Reaction–DiLusion Systems, Marcel Dekker, New York, 1998, pp. 183–197. [9] M. Krasnoselskii, P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984. [10] J.L. Lions, E. Magenes, ProblSems aux Limites non HomogSenes et Applications, Vol. 1, Dunod, Paris, 1968. [11] E. Mitidieri, G. Sweers, Existence of a maximal solution for quasimonotone elliptic systems, DiLerential Integral Equations 7 (1994) 1495–1510.