EXISTENCE OF POSITIVE RADIAL SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS AND SYSTEMS

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1 Electronic Journal of Differential Equations, Vol. 26 (26), No. 5, pp. 9. ISSN: URL: or ftp ejde.math.txstate.edu EXISTENCE OF POSITIVE RADIAL SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS AND SYSTEMS ZHIJUN ZHANG Abstract. Under simple conditions on f and g, we show that existence of positive radial solutions for the quasilinear elliptic equation and for the system div(φ ( u ) u) = a( x )f(u) x R N, div(φ ( u ) u) = a( x )f(v) x R N, div(φ 2 ( v ) v) = b( x )g(u) x R N.. Introduction The purpose of this article is to study the existence of positive radial solutions to the quasilinear elliptic equation and for the system φ u := div(φ ( u ) u) = a( x )f(u), x R N, (.) div(φ ( u ) u) = a( x )f(v), x R N, div(φ 2 ( v ) v) = b( x )g(u), x R N. (.2) In this article by a solution we mean a solution on the entire domain, as opposed to a local solution. To emphasize this property some authors use entire solution, while others use global solution. We assume the following assumptions: (A) a, b : R N [, ) are continuous; (A2) f, g : [, ) [, ) are continuous and increasing, (A3) φ i C ((, ), (, )) (i =, 2) satisfy (tφ i (t)) >, for all t > ; (A4) there exist p i, q i > such that where Ψ i (t) = sφ i(s)ds, t > ; (A5) there exist k i, l i > such that p i tψ i (t) Ψ i (t) q i, t >, k i tψ i (t) Ψ i (t) l i, t >. 2 Mathematics Subject Classification. 35J55, 35J6, 35J65. Key words and phrases. Quasilinear elliptic equation; radial solutions; existence. c 26 Texas State University. Submitted November 23, 25. Published February 7, 26.

2 2 Z. ZHANG EJDE-26/5 The function φ appears in mathematical models in nonlinear elasticity, plasticity, generalized Newtonian fluids, and in quantum physics, see e.g., Benci, Fortunato and Pisani [8], Cencelj, Repovš and Virk [9], Fuchs and Li [3], Fuchs and Osmolovski [4], Fukagai and Narukawa [5], Rădulescu [28] and [29], Rădulescu and Repov s [3], Repov s [3], Zhang and Yuan [39] and Fukagai and Narukawa [6]. Positive solutions to (.) were first considered by Santos, Zhou and Santos [32]. Some classical examples of φ -Laplacian functions are: () when φ (t) 2, Ψ (t) = t 2, t >, φ u = u is the Laplacian operator. In this case, p = q = 2 in (A4), and k = l = in (A5); (2) when φ (t) = pt p 2, Ψ (t) = t p, t >, p >, φ u = p u is the p- Laplacian operator. In this case, p = q = p in (A4), and k = l = p in (A5); (3) when φ (t) = pt p 2 + qt q 2, Ψ (t) = t p + t q, t >, < p < q, φ u = p u + q u is called as the (p + q)-laplacian operator, p = p, q = q in (A4), and k = p, l = q in (A5); (4) when φ (t) = 2p( + t 2 ) p, Ψ (t) = ( + t 2 ) p, t >, p > /2, p = min{2, 2p}, q = max{2, 2p} in (A4), and k = min{, 2p }, l = max{, 2p } in (A5); (5) when φ (t) = p( +t 2 ) p +t, Ψ 2 (t) = ( + t 2 ) p, t >, p >, p = p, q = 2p in (A4), and k = p, l = 2p in (A5); (6) when φ (t) = pt p 2 (ln( + t)) q + qtp (ln(+t)) q +t, Ψ (t) = t p (ln( + t)) q, t >, p >, q >, p = p, q = p+q in (A4), and k = p, l = p+q in (A5). We say that u C (R N ) is a solution of (.) if φ ( u ) u ψdx = R N a(x)f(u)ψdx, R N ψ C (R N ). When lim x u(x) = +, we say that u is a large solution to equation (.). For convenience, we denote by h i the inverses of h i (t) = tφ i (t), t > ; (.3) I iρ ( ) := lim r I iρ(r), I iρ (r) := where ρ C([, ), [, )) and h i (Λ ρ (t))dt, r, (.4) Λ ρ (t) := t s N ρ(s)ds, t > ; (.5) θ i (t) := min{t pi, t qi }, Θ i (t) := max{t pi, t qi }, t ; (.6) θ i (t) := min{t /pi, t /qi }, Θ i (t) := max{t /pi, t /qi }, t ; (.7) and, for an arbitrary α > and t α, H α ( ) := lim t H α (t), H α (t) := H 2α ( ) := lim t H 2α (t), H 2α (t) := α Θ dτ α Θ (f(τ)); (.8) dτ (f(τ)) + Θ (g(τ)). (.9) 2

3 EJDE-26/5 EXISTENCE OF POSITIVE RADIAL SOLUTIONS 3 We see that for t > α, H α(t) = Θ >, (f(t)) H 2α(t) = >, (f(t)) + Θ (g(t)) Θ and that H α, H 2α have the inverse functions H α [, H 2α ( )), respectively. First, let us review the model 2 and H 2α on [, H α( )) and u = a( x )f(u), x R N. (.) For a(x) on R N : when f satisfies (A2), Keller [8] and Osserman [27] supplied a necessary and sufficient condition dt 2F (t) =, F (t) = f(s)ds, (.) for the existence of positive radial large solutions to (.). For N 3, f(u) = u γ, γ (, ], and a satisfies (A) with a(x) = a( x ), Lair and Wood [9] first showed that equation (.) has infinitely many positive radial large solutions if and only if ra(r)dr =. (.2) The above results have been extended by many authors and in many contexts, see, for instance, [2, 4, 5,, 2, 2, 23, 33, 35, 36] and the references therein. Next we review the system u = a(x)f(v), x R N, v = b(x)g(u), x R N. (.3) When N 3, f(v) = v γ, g(u) = u γ2, < γ γ 2, and a(x) = a( x ), b(x) = b( x ), Lair and Wood [2] have considered the existence and nonexistence of positive radial solutions to system (.3). For further results, see for instance, [, 3, 6, 7,, 7, 24, 25, 26, 34, 37, 38] and the references therein. Now we return to equation (.). Recently, Santos, Zhou and Santos [32] considered the existence of positive radial and nonradial large solutions to equation A basic result read as follows. div(φ ( u ) u) = a(x)f(u), x R N. Lemma. ([32, Corollary.2]). Let (A3) (A5) hold, f satisfy (A2), and a satisfy (A) with a(x) = a( x ) for x R N. If I a ( ) =, then (.) admits a sequence of symmetric radial large solutions u m ( x ) C (R N ) with u m () as m if and only if f satisfies dt Ψ =, (F (t)) where Ψ is the inverse of Ψ which is given in (A4).

4 4 Z. ZHANG EJDE-26/5 Inspired by the above works, by using a monotone iterative method and Arzela- Ascoli theorem, we show existence of positive radial solutions to equation (.) and system (.2) under simple conditions on f and g. Our main results for equation (.) read as follows. Theorem.2. Let (A) (A5) hold. If (A6) H α ( ) =, then (.) has a positive radial solution u C (R N ). Moreover, if I a ( ) <, then u is bounded, and lim r u(r) = provided I a ( ) =. Theorem.3. Under assumptions (A) (A5) and (A7) I a ( ) < H α ( ) <, equation (.) has a positive radial bounded solution u C (R N ) satisfying α + θ (f(α))i a(r) u(r) Hα (I a(r)), r, where θ is given in (.7). Remark.4. When dτ =, there exists α > sufficiently small such Θ (f(τ)) that (A7) holds provided I a ( ) < and H α ( ) <. Remark.5. For f(s) = s γ with s, γ >, since Θ (t) = p, t, one can see that when γ > p, H α ( ) <, and H α ( ) = provided γ p, where p is given as in (A4). Remark.6. For f(s) = ( + s) γ (ln( + s)) µ with s, µ, γ >, one can see that when γ > p or γ = p and µ > p, H α ( ) <, and H α ( ) = provided γ < p or γ = p and µ p. Remark.7. For f(s) = exp(c s), s, c >, one can see that H α ( ) <. Our main results for system (.2) are as follows. Theorem.8. Let (A) (A5) hold. If (A8) H 2α ( ) =, then (.2) has a positive radial solution (u, v) in C (R N ) C (R N ). Moreover, when I a ( ) + I 2b ( ) <, u and v are bounded; when I a ( ) = I b ( ) =, lim r u(r) = lim r v(r) =. Theorem.9. Under hypotheses (A) (A5) and (A9) I a ( ) + I 2b ( ) < H 2α ( ) <, system (.2) has a positive radial bounded solution (u, v) in C (R N ) C (R N ) satisfying α/2 + θ (f(α/2))i a(r) u(r) H 2α (I a(r) + I 2b (r)), r ; α/2 + θ 2 (g(α/2))i 2b(r) v(r) H 2α (I a(r) + I 2b (r)), r. Remark.. By a similar proof, we can see extend Theorems.8 and.9 to the more general system where f i, g i (i =, 2) satisfy (A2). div(φ ( u ) u) = a( x )f (v)f 2 (u), x R N, div(φ 2 ( v ) v) = b( x )g (v)g 2 (u), x R N, (.4)

5 EJDE-26/5 EXISTENCE OF POSITIVE RADIAL SOLUTIONS 5 Remark.. For f(s) = s γ, g(s) = s γ2, s, γ, γ 2 >, when γ > p or γ 2 > p 2, H 2α ( ) <, and H 2α ( ) = provided γ p and γ 2 p 2, where p and p 2 are given as in (A4). Remark.2. For f(s) = ( + s) γ (ln( + s)) µ, g(s) = ( + s) γ2 (ln( + s)) µ2, s, γ i, µ i > (i =, 2), when γ > p or γ 2 > p 2 ; or γ = p and η > p ; or γ 2 = p 2 and η 2 > p 2, H 2α ( ) <, and H 2α ( ) = provided γ < p and γ 2 < p 2 ; or γ = p, η p and γ 2 = p 2, η 2 p 2. Remark.3. For f(s) = exp(c s) or g(s) = exp(c 2 s), s, c, c 2 >, one can see that H 2α ( ) <. 2. Proof of Theorems.2 and.3 Lemma 2. ([32, Lemma 2.2]). Let (A3) (A5) hold, θ i, Θ i and θ i, Θ i (i =, 2) be given as in (.6) and (.7). We have (i) θ i, Θ i, θ i and Θ i are strictly increasing on (, ); (ii) θ i (β)h i (t) h i (βt) Θ i (β)h i (t), for all β, t >. Let us consider the initial value problem ( r N φ (u (r))u (r) ) = r N a(r)f(u), r >, by a simple calculation, u (r) = h ( r and thus u(r) = α + u() = α, u () =, h ( t (2.) s N a(s)f(u(s))ds ), r >, u() = α, (2.2) s N a(s)f(u(s))ds ) dt, r. (2.3) Note that solutions in C[, ) to problem (2.3) are solutions in C [, ) to problem (2.). Let {u m } m be the sequence of positive continuous functions defined on [, ) by Obviously, u m (r) = α + h ( t u m(r) = h ( r u (r) = α, s N a(s)f(u m (s))ds ) dt, r. (2.4) s N a(s)f(u m (s))ds ), r >, (2.5) and, for all r and m N, u m (r) α, and u u. Then (A) (A3) and Lemma 2. yield u (r) u 2 (r) for all r. Continuing this line of reasoning, we obtain that the sequence {u m } is non-decreasing on [, ). Moreover, we obtain by (A) (A3) and Lemma 2. that for each r >, u m(r) = h ( r r s N a(s)f(u m (s))ds ) h ( r f(um (r))r s N a(s)ds )

6 6 Z. ZHANG EJDE-26/5 Θ (f(u m(r)))h ( r and um(r) dτ a Θ (f(τ)) I a(r). Consequently, for an arbitrary R >, (i) When (A6) holds, we see that Hα ( ) =, u s N a(s)ds ), H α (u m (r)) I a (r) I a (R), r [, R]. (2.6) m(r) H α (I a(r)) H α (I a(r)), r [, R], (2.7) i.e., the sequence {u m } is bounded on [, R] for an arbitrary R >. It follows from (2.5) that {u m} is bounded on [, R]. By the Arzela-Ascoli theorem, {u m } has a subsequence converging uniformly to u on [, R]. Since {u m } is non-decreasing on [, ), we see that {u m } itself converges uniformly to u on [, R]. By the arbitrariness of R, we see that u is a positive radial solution to equation (.). Moreover, when I a ( ) <, we see by (2.7) that u(r) H α (I a( )), r ; when I a ( ) =, we see by (A2) and Lemma 2. that u(r) α + θ (f(α))i a(r), r. Thus lim r u(r) =. (ii) When (A7) holds, we see by (2.6) that Since H α H α (u m (r)) I a ( ) < H α ( ) <. (2.8) is strictly increasing on [, H α( )), we have u m (r) Hα a( )) <, r. (2.9) The rest of the proof follows from (i). 3. Proof of Theorems.8 and.9 Let us consider the initial value problem ( r N φ (u (r))u (r) ) = r N a(r)f(v), r >, which is equivalent to u(r) = α/2 + v(r) = α/2 + ( r N φ 2 (v (r))v (r) ) = r N b(r)g(u), r >, u() = v() = α/2, u () = v () =, h ( t h ( 2 t s N a(s)f(v(s))ds ) dt, r, s N b(s)g(u(s))ds ) dt, r. Let {u m } m and {v m } m be the sequences of positive continuous functions defined on [, ) by u m (r) = α/2 + h ( t v (r) = α/2, s N a(s)f(v m (s))ds ) dt, r,

7 EJDE-26/5 EXISTENCE OF POSITIVE RADIAL SOLUTIONS 7 v m (r) = α/2 + h ( 2 t s N b(s)g(u m (s))ds ) dt, r. Obviously, for all r and m N, u m (r) α/2, v m (r) α/2 and v v. Assumptions (A) (A3) and Lemma 2. yield u (r) u 2 (r), for all r, then v (r) v 2 (r), for all r. Continuing this line of reasoning, we obtain that the sequences {u m } and {v m } are increasing on [, ). Moreover, by (A)-(A3) and Lemma 2. that for each r >, we obtain u m(r) = h ( r r s N a(s)f(v m (s))ds ) and Consequently, h ( r f(vm (r))r s N a(s)ds ) Θ (f(v m(r)))h ( r s N a(s)ds ) Θ (f(u m(r) + v m (r)))(h (Λ a(r)) + h 2 (Λ b(r))); v m(r) = h ( 2 r s N b(s)g(u m (s))ds ) Θ 2 (g(u m(r)))h ( 2 r s N b(s)ds ) Θ 2 (g(u m(r) + v m (r))) ( h (Λ a(r)) + h 2 (Λ b(r)) ). u m(r) + v m(r) ( Θ (f(v m(r) + u m (r))) and um(r)+v m(r) a + Θ 2 (g(v m(r) + u m (r))) )( h (Λ a(r)) + h 2 (Λ b(r)) ), r >, dτ Θ (f(τ)) + Θ 2 (g(τ)) I a(r) + I 2b (r), r >, H 2α (u m (r) + v m (r)) I a (r) + I 2b (r), r. (3.) The remaining proofs are similar to that for Theorems.2 and.3. Here we omit their proof. Acknowledgment. This work is supported in part by NSF of China under grant References [] R. Alsaedi, H. Mâagli, V. Rădulescu, N. Zeddini; Positive bounded solutions for semilinear elliptic systems with indefinite weights in the half-space, Electronic J. Diff. Equations 25 (25), No. 77, -8. [2] I. Bachar, N. Zeddini; On the existence of positive solutions for a class of semilinear elliptic equations, Nonlinear Anal. 52 (23), [3] S. Barile, A. Salvatore; Existence and multiplicity results for some Lane-Emden elliptic systems: subquadratic case, Adv. Nonlinear Anal. 4 (25), [4] N. Belhaj Rhouma, A. Drissi, W. Sayeb; Nonradial large solutions for a class of nonlinear problems, Complex Var. Elliptic Equations 59 (5) (24), [5] N. Belhaj Rhouma, A. Drissi; Large and entire large solutions for a class of nonlinear problems, Appl. Math. Comput. 232 (24),

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9 EJDE-26/5 EXISTENCE OF POSITIVE RADIAL SOLUTIONS 9 [34] X. Wang, A. W. Wood; Existence and nonexistence of entire positive solutions of semilinear elliptic systems, J. Math. Anal. Appl. 267 (22), [35] H. Yang; On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in R N, Comm. Pure Appl. Anal. 4 (25), [36] D. Ye, F. Zhou; Invariant criteria for existence of bounded positive solutions, Discrete Contin. Dyn. Syst. 2 (3) (25), [37] Z. Zhang; Existence of entire positive solutions for a class of semilinear elliptic systems, Electronic J. Diff. Equations 2 (2), No. 6, -5. [38] Z. Zhang, Y. Shi, Y. Xue; Existence of entire solutions for semilinear elliptic systems under the Keller-Osserman condition, Electronic J. Diff. Equations 2 (2), No. 39, -9. [39] Zhitao Zhang, R. Yuan; Infinitely-many solutions for subquadratic fractional Hamiltonian systems with potential changing sign, Adv. Nonlinear Anal. 4 (25), Zhijun Zhang School of Mathematics and Information Science, Yantai University, Yantai 2645, Shandong, China address: chinazjzhang22@63.com, zhangzj@ytu.edu.cn