Boundedness and blow-up of solutions for a nonlinear elliptic system

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1 International Journal of Mathematics Vol. 25, No. 9 (2014) (12 pages) c World Scientific Publishing Company DOI: /S X Boundedness and blow-up of solutions for a nonlinear elliptic system Dragos-Patru Covei Department of Applied Mathematics Bucharest University of Economic Studies Piata Romana, 1st District, , Bucharest, Romania dragos.covei@csie.ase.ro Received 3 April 2014 Accepted 5 September 2014 Published 25 September 2014 The main objective in this paper is to obtain the existence results for bounded and unbounded solutions of some quasilinear elliptic systems. Related results as obtained here have been established recently in [C. O. Alves and A. R. F. de Holanda, Existence of blow-up solutions for a class of elliptic systems, Differ. Integral Eqs. 26(1/2) (2013) ]. Also, we present some references to give the connection between these types of problems with probability and stochastic processes, hoping that these are interesting for the audience of analysts likely to read this paper. Keywords: Quasilinear system; existence; sub/super-solution; boundary blow-up. Mathematics Subject Classification 2010: 35J60, 35J20, 35A05 1. Introduction The question of existence of solutions for elliptic equation of the form p u = f(x, u) in, (1.1) was studied by many researchers (see [2, 4, 11, 15, 16, 18] and references therein). This work is devoted to the study of the more general nonlinear elliptic problems of the type p u 1 = F u1 (x, u 1,...,u d ) in,.. (1.2) p u d = F ud (x, u 1,...,u d ) in, where d 1 is integer, R N (N>1) is a smooth, bounded domain or = R N, p u i := div( u i p 2 u i )(1<p<, )isthep-laplacian operator and F ui () stands for the derivatives of a continuously differentiable

2 D.-P. Covei function F : [R + ] d R + in (u 1,...,u d ). For the case = R N,wealso consider the following class of elliptic systems: p u 1 = a 1 (x)f u1 (x, u 1,...,u d ) in R N,. (1.3) p u d = a d (x)f ud (x, u 1,...,u d ) in R N, u i > 0 in R N,, where a i : R N (0, ) () are suitable functions. Associated with the class of systems (1.3), our main result is concerned with the existence of entire large solutions, that is, solutions (u 1,...,u d ) satisfying u i (x) as x for all. The interest on systems (1.2) (1.3) comes from some problems studied in the works of Lasry Lions [12], Busca Sirakov [3] and Dynkin [6 9] where the authors give the connection between these types of problems with probability and stochastic processes and from the recent work of Alves and Holanda [1] where these systems are considered for the case p = 2, in terms of the pure mathematics. The difference between our work and the paper [1] is that: our systems can have any number of equations, the potential functions a i cover more general properties and that we use in the proofs theories for quasilinear operators instead of the theories for linear operators used by Alves Holanda [1]. We also remark that the authors Alves Holanda [1] extended the results of Bandle Marcus [2], obtained for the scalar equation in bounded domains, to the system of two equations while our proof work for any numbers of equations. To begin with our results we make the following convention: we say that a function h :[0, + ) [0, ) belongs to F if h C 1 ([0, )), h(0) = 0, h (t) 0, t [0, ), h(t) > 0, t (0, ) and the Keller Osserman [10, 17] condition is satisfied, that is, 1 dt <, 1 H(t) 1/p where H(t) = t 0 h(s)ds. Our main result for problem (1.2) on a bounded domain is the following. Theorem 1.1. Suppose is a smooth, bounded domain in R N and that there exist f i,g F satisfying and F ti (x, t 1,...,t i,...,t d ) f i (t i ), x, t i > 0 and (1.4) g(t) max{f t (x,t,...,t)}, x, t > 0. (1.5)

3 Nonlinear elliptic system Then: (1) Problem (1.2) admits a positive solution with boundary condition u i = α i, on,, α i (0, ). (1.6) (2) Problem (1.2) admits a positive solution with the boundary condition u 1 = = u d = on, (1.7) where u i = on () should be understood as u i (x) as dist(x, ) 0. (3) Problem (1.2) admits a positive solution with boundary condition: there are i 1,...,i m,j m+1,...,j d {1,...,d} such that { uip = on with p {1,...,m}, (1.8) u jq < on with q {m +1,...,d} for any 1 m<dfor which i p j q. Our next result is related to the existence of a solution for system (1.3). For expressing the next result, we assume that functions a i ()satisfythe following conditions: a i (x) > 0 for all x R N and a i C 0,ϑ loc (RN ), ϑ (0, 1) (1.9) and that the quasilinear equation p z(x) = d a i (x) forx R N, z(x) 0as x (1.10) i=1 has a C 1 -upper solution, in the sense that d z p 2 z φdx a i (x)φdx, φ C0 (R N ), φ 0, R N R N i=1 z C 1 (R N ), z > 0inR N, z(x) 0as x. (1.11) Theorem 1.2. Assume that (1.4) (1.5), (1.9) (1.11) hold. Then system (1.3) has an entire large C 1 -solution (in the distribution sense). To prepare for proving our theorems, we need some additional results. 2. Preliminary Results Let R N (N 2) be a smooth, bounded domain in R N and 1 <p<. The first auxiliary result can be seen in the paper of Matero [14]

4 D.-P. Covei Lemma 2.1. Assume that g meets the conditions: g is a continuous, positive, increasing function on R + and g(0) = 0. Leth W 1,p () be such that (G h) L 1 () where G(s) = s 0 g(t)dt. Then there exists a unique u W 1,p () which (weakly) solves the problem { p u(x) =g(u(x)), x, u(x) =h(x), x. Furthermore, if u 1 and u 2 are the solutions corresponding to h 1 and h 2 with h 1 h 2 on, then u 1 u 2 in. Finally, there exists an β (0, 1) such that u C 1,β (D) for any compact set D. The following comparison principle is proved in the paper of Sakaguchi [19] (or consult some ideas of the proof in work of Tolksdorf [20]). Lemma 2.2. Let u, v W 1,p () satisfy p u p v for x, in the weak sense. If u v on then u v in. The following lemma can be found [23]. Lemma 2.3. Suppose f F and that u W 1,p loc () C() satisfies u p 2 u ϕdx = f(u)ϕdx, ϕ C0 (). Then, there exists a monotone decreasing function µ :(0, ) (0, ) determined by f such that Moreover, u(x) µ(dist(x, )), x. lim µ(t) =, lim t 0 µ(t) =. t Next, we begin with recalling the definition of sub- and super-solution used in this context. The system that we will study is the following p u i = G ui (x, u 1,...,u i,...,u d ) in, u i = f i on, (2.1), where f i W 1,p () and G(x, t 1,...,t i,...,t d ): [R] d R is measurable in x, continuously differentiable in t i R, and satisfies the following condition: for each T i > 0fixed(), there exists C = C(T i ) > 0 such that G(x, t 1,...,t d ) C, (x, t 1,...,t d ) [ T i,t i ] d. (2.2) Now we introduce the concept of sub- and super-solution in the weak sense

5 Nonlinear elliptic system Definition 2.1. By definition (u 1,...,u d ) [W 1,p ()] d is a (weak) sub-solution to (2.1), if u i f i on, u i p 2 u i φdx + G ui (x, u 1,...,u d )φdx 0 for all φ C0 () with φ 0and. Similarly (u 1,...,u d ) [W 1,p ()] d is a (weak) super-solution to (2.1) ifinthe above the reverse inequalities hold. The following result holds. Lemma 2.4. Suppose (u 1,...,u d ) is a sub-solution while (u 1,...,u d ) is a supersolution to problem (2.1) and assume that there are constants a i, a i R such that a i u i u i a i almost everywhere in. If (2.2) holds, then there exists a weak solution (u 1,...,u d ) [W 1,p ()] d of (2.1), satisfying the condition u i u i u i almost everywhere in. We will not give the proof here since it has been proved in [1] with some ideas from [22]. 3. Proof of Main Results In this section, we will prove the main results of this paper ProofofTheorem Proof of (1) In what follows, we denote by ψ W 1,p () the unique positive solution of the problem ψ p 2 ψ φdx = g(ψ)φdx in, φ C0 () with φ 0, ψ>0 in, ψ = m on, where m = min{α 1,...,α d }, which exists and minimizes the Euler Lagrange functional ( ) 1 J(ψ) = p ψ p + G(ψ(x)) dx on the set K = {v L 1 () v m W 1,p 0 () and (G v) L 1 ()}

6 D.-P. Covei i.e. ψ meets the boundary condition (ψ m) W 1,p 0 () in the weak sense (see Lemma 2.1 and [2]). Then, ψ p 2 ψ φdx = g(ψ)φdx F ψ (x,ψ,...,ψ)φdx in, and so ψ = m α i on, (u 1,...,u d )=(ψ,...,ψ) is a sub-solution for the system p u i = F ui (x, u 1,...,u i,...,u d ) in, u i = α i on,. Clearly, with (u 1,...,u d )=(M,...,M), M =max{α i }, is a super-solution of (3.1). We prove that, u i u i for all. Indeed, p u i = g(u i ) p u i =0 in, u i = m u i = M on, (3.1) andthenwiththeuseoflemma2.2 it follows that u i u i in. Then, there exists acriticalpoint(u 1,...,u d ) [W 1,p ()] d, provided by Lemma 2.4, which minimizes the Euler Lagrange functional I(u 1,...,u d )= 1 p d i=1 u i p dx + F (x, u 1,...,u d )dx and that solve, in the weak sense, the system p u i = F ui (x, u 1,...,u i,...,u d ) in, u i = α i on, (P α ) and satisfying ψ u i M in for all.sinceu i L loc (), by the regularity theory [5, 13, 21], it follows that u i C 1 ()

7 Nonlinear elliptic system Proof of (2) To study this case, we begin considering the system p u i = F ui (x, u 1,...,u i,...,u d ) in, u i = n on,. (3.2) Then, by the finite case above, problem (3.2) has a solution (u n 1,...,un d ). We prove that the sequence of solutions (u n 1,...,u n d ) can be chosen satisfying the inequality u n i u n+1 i for all and n N. (3.3) To prove this, we consider the solution (u 1 1,...,u1 d ) of the problem p u i = F ui (x, u 1,...,u i,...,u d ) in, u i =1 on, and note that it is a sub-solution of p u i = F ui (x, u 1,...,u i,...,u d ) in, u i =2 on, (3.4) (3.5) while the pair (M 1,...,M 1 ) is a super-solution of (3.5) form 1 =2.Once0 u i (x) 2(), x, Lemma 2.4 implies that there exists a solution (u 2 1,...,u2 d )of p u i = F ui (x, u 1,...,u i,...,u d ) in, u i =2 on, satisfying u 1 i (x) u2 i (x). Using the argument above, for each M n = n +1; n = 1, 2,..., we get a solution (u n 1,...,u n d )of(3.2), which is a sub-solution, and the pair (M n,...,m n ) is a super-solution respectively of p u i = F ui (x, u 1,...,u i,...,u d ) in, u i = n +1 on,. Thereby, the sequence of solutions (u n 1,...,un d ) satisfies the inequality (3.3). Finally, we construct an upper bound of the sequence. More exactly, we show that

8 D.-P. Covei {(u n 1,...,u n d )} n 1 is uniformly bounded in any compact subset of. To this end, we begin recalling that by (1.4) p u n i f i(u n i ) in, u n i > 0 in, n on u n i with f i F.Ifũ n i () denote the unique solutions of the problems p u i = f i (u i ) in, u i > 0 in, u i = n on it follows from Lemma 2.2 that u n i ũn i in for all n 1. By Lemma 2.3, there exist non-increasing continuous functions µ i : R + R + such that showing that ũ n i µ i(dist(x, )), n N, x and 0 <u 1 i (x) un i (x) µ i(d(x)), n N, x, (3.6) where d(x) =dist(x, ). Thus there exists a subsequence, still denoted again by u n i, which converges to a function u i in W 1,p (). In other words u i (x) := lim n un i (x) for all x and. The estimates for (3.6) combined with the bootstrap argument show that the sequence {u n i (x)} is uniformly bounded in C1,α (K) for any compact subset K and some α (0, 1) (see [22, Appendix A]), which is relatively compact in C 1 (K), and therefore the sequence {u n i (x)} contains a subsequence that converges uniformly in C 1 (K) to the function u i (x). Furthermore, it is clear that u i (x) C 1 () and (u 1,...,u d ) is a solution of (1.2); that is, p u i = F ui (x, u 1,...,u i,...,u d ) in, u i > 0 in,. To complete the proof, it suffices to prove that (u 1,...,u d ) blows up at the boundary. Supposing for the sake of contradiction that u i does not blow up at the boundary, there exist x 0 and(x k ) such that lim k x k = x 0 and lim k u i(x k )=L i (0, )

9 Nonlinear elliptic system In what follows, fix n>4l i and δ>0 such that u n i (x) n/2 for all x δ, where δ = {x dist(x, ) δ}. Then, for k large enough, x k δ and u n i (x k) n/2 > 4L i /2=2L i.since u n i (x k ) u n+1 i (x k ) u n+j i (x k ) u i (x k ), j, we have that u i (x k ) 2L i, which is a contradiction. Therefore, u i blows up at the boundary. This solution (u 1,...,u d ) dominates all other solutions and is therefore commonly called blow-up/large solution Proof of (3) Without loss of generality, assume that i 1 < <i m <j m+1 < <j d. Let (...,u n i p,...,u n j q,...) C 1 () be the solution of the problem (P α )withα ip = n, n N, andα jq fixed. As in the previous case, the sequence u n i p is bounded on a compact subset contained in, implying that there exists function u ip satisfying u n i p u ip in C 1 (K) for any compact subset K (see[22, Appendix A]). Moreover, the arguments used in the previous cases yield that u ip blows up at the boundary, that is, u n i p = on. Related to the sequence (u jq ), we recall that { p u n j q = F (x, uj un q 1,...,un j q,...,u n d ) in, u n j q = α jq on. Then, by the comparison principle u n j q α jq, x andn 1. Passing to the limit as n, we obtain that u jq α jq for all x. Claim. Let x 0 and (x k ) be a sequence with x k x 0.Thenu jq (x k ) α jq as k. Indeed, if the limit does not hold, there exist ε>0 and a subsequence of (x k ), still denoted by itself, such that x k x 0 and u jq (x k ) α jq ε, k N. (3.7) Since u 1 j q = α jq on and is continuous, there is some δ>0 such that u 1 j q (x k ) α jq ε 2, x δ. Hence, for k large enough, x k δ and u jq (x k ) u 1 j q (x k ) α jq ε 2 >α j q ε which contradicts (3.7). From this claim, we can continuously extend the function u jq from to by considering u jq (x) =α jq on, concluding this way the proof of the finite and infinite case

10 D.-P. Covei 3.2. ProofofTheorem1.2 First, we provide a sub-solution for the problem (1.3). To do this we consider the function w : R N [0, ) implicitly defined by 1 z(x) = w(x) g 1/(p 1) (t) dt, x RN, where z is the C 1 -upper solution to (1.10). Note that w C 1 (R N, (0, )),w(x) + as x and z(x) = g 1/(p 1) (w(x)) w(x), (3.8) w(x) p 2 w(x) = g(w(x)) z(x) p 2 z(x). (3.9) Given φ C0 (RN ), φ 0wehave w(x) p 2 w(x) φdx = g(w(x)) z(x) p 2 z(x) φdx R N R N = div[g(w(x)) z(x) p 2 z(x)]φdx. R N Computing the derivatives in the integrand of the expression just above, in the distribution sense, using (3.8) weget, w(x) p 2 w(x) φdx = g(w(x)) p z(x)φdx R N R N g (w(x))g 1 p 1 (w(x)) z(x) p φdx. R N Using the fact that g Fand that z(x) is an upper solution of (1.10) wederive the inequality w(x) p 2 w(x) φdx R N g(w(x)) p z(x)φdx 0, R N and so ( d ) w(x) p 2 w(x) φdx + g(w(x)) a i (x) φdx 0, R N R N i=1 which together with (1.5) leadsto w(x) p 2 w(x) φdx R N a i (x)f ui (x, w(x),...,w(x))φdx R N for all. In the next, we consider the system p u i = a i (x)f ui (x, u 1,...,u i,...,u d ) in B n, u i = w n in B n, (3.10),

11 Nonlinear elliptic system where B n is the open ball of radius n centered at the origin and w n = max x Bn w(x). Clearly, (w,...,w)and(w n,...,w n ) are a sub-solution and supersolution for (3.10) respectively. Thus, by Theorem 1.1, there is a solution (u n 1,...,un d ) [W 1,p (B n )] d of (3.10) satisfying w(x) u n i w n for all x B n and.form 1andn m + 1 consider the family of systems { p u n i = aa i f i(u n i ) in B m+1,, where a a i = min x Bn a i (x) > 0. Arguing as in the previous sections, there are monotone decreasing functions µ a i :(0, ) (0, ) determined by f i such that w(x) u n i (x) µ a i (dist(x, B m+1 )), x B m+1 from which it follows that w(x) u n i (x) Mi m for all n N, x B m, and for some positive constants Mi m. Now using the fact that u n i W 1,p (B m ) L (B m ) it follows from the results of DiBenedetto [5] and Lieberman [13] that there exist some constants C i := C i (p, N, u n i,b m ) > 0 such that u n i C1,α (B m )and u n i C 1,α (B m) C i, and α (0, 1). Arguing as in [22], there is u i C 1 (B m )() such that for some subsequence of u n i, still denoted by itself, we get u n i u i () pointwisely in B m ( m>1). Therefore, (u 1,...,u d ) C 1 (R N ) and is a solution for the system p u i = a i (x)f ui (x, u 1,...,u i,...,u d ) in R N u i > 0 in R N, satisfying w(x) u i (x) for all x R N and. (3.11) Letting x in (3.11) it follows that (u 1,...,u d ) is a large entire solution for (1.3). Acknowledgments The author would like to thank the anonymous reviewers for the insightful and constructive comments on the manuscript. This research was supported by CNCS UEFISCDI, Project number IDEI 303, code PN-II-ID-PCE

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