EXISTENCE OF SOLUTIONS TO NONHOMOGENEOUS DIRICHLET PROBLEMS WITH DEPENDENCE ON THE GRADIENT

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1 Electronic Journal of Differential Equations, Vol (2018), No. 101, pp ISSN: URL: or EXISTENCE OF SOLUTIONS TO NONHOMOGENEOUS DIRICHLET PROBLEMS WITH DEPENDENCE ON THE GRADIENT YUNRU BAI Communicated by Vicentiu D. Radulescu Abstract. The goal of this article is to explore the existence of positive solutions for a nonlinear elliptic equation driven by a nonhomogeneous partial differential operator with Dirichlet boundary condition. This equation a convection term and thereaction term is not required to satisfy global growth conditions. Our approach is based on the Leray-Schauder alternative principle, truncation and comparison approaches, and nonlinear regularity theory. 1. Introduction Given a bounded domain R N with C 2 -boundary, 1 < p < +, a continuous function a : R N R N, and a nonlinear function f : R R N R, we consider the following nonlinear nonhomogeneous Dirichlet problem involving a convection term: div a(du(z)) = f(z, u(z), Du(z)) in, (1.1) u(z) = 0 on, with u(z) > 0 in. In this article, the function a : R N R N is assumed to be continuous and strictly monotone, also satisfies certain regularity and growth conditions listed in hypotheses (H1) below. It is worth to mention that these conditions are mild and incorporate in our framework many classical operators of interest, for example the p-laplacian, the (p, q)-laplacian (that is, the sum of a p-laplacian and a q- Laplacian with 1 < q < p < ) and the generalized p-mean curvature differential operator. The forcing term depends also on the gradient of the unknown function (convection term). For this reason we are not able to apply variational methods directly on equation (1.1). For problems with convection terms we mention the following works: Figueiredo- Girardi-Matzeu [8], Girardi-Matzeu [23] (semilinear equations driven by the Dirichlet Laplacian), Faraci-Motreanu-Puglisi [6], Huy-Quan-Khanh [25], Iturriaga-Lorca- Sanchez [26], Ruiz [36] (nonlinear equations driven by the Dirichlet p-laplacian), 2010 Mathematics Subject Classification. 35J92, 35P30. Key words and phrases. Nonhomogeneous p-laplacian operator; nonlinear regularity; Dirichlet boundary condition; convection term; truncation; Leray-Schauder alternative. c 2018 Texas State University. Submitted December 8, Published May 2,

2 2 Y. BAI EJDE-2018/101 Averna-Motreanu-Tornatore [2], Faria-Miyagaki-Motreanu [7], Tanaka [37] (equations driven by the Dirichlet (p, q)-laplacian) and finally Gasiński-Papageorgiou [22] (Neumann problems driven by a differential operator of the form div(a(u)du)). Unlike the aforementioned works, in this paper, the convection term f does not have any global growth condition. Instead we suppose that f(z,, y) admits a positive root (zero) and all the other conditions refer to the behaviour of the function x f(z, x, y) near zero locally in y R N. Our approach is based on the Leray-Schauder alternative principle, truncation and comparison techniques, nonlinear regularity theory and it is closely related to the paper Bai- Gasiński-Papageorgiou in [3], where the Robin boundary value problem was considered. Finally for other problems with a general nonhomogeneous operator we refer to Gasiński-Papageorgiou [14, 15, 19, 21], Papageorgiou-Rădulescu [30, 31, 33, 34] and for particular cases of a nonhomogeneous operator we refer to Gasiński- Papageorgiou [13, 16] (p(z)-laplacian) and Gasiński-Papageorgiou [17, 18], and for (p, q)-laplacian, Papageorgiou-Rădulescu [32]. 2. Notation and preliminaries In the study of problem (1.1), we will use the Sobolev space W 1,p 0 () as well as the ordered Banach space C 1 0() = {u C 1 ( ) : u(z) = 0 on } which has positive (order) cone The interior of this cone contains the set C 1 0() + = { u C 1 0() : u(z) 0 in }. D + = { u C 1 0() : u(z) > 0 in }. Then, we give the following notation, which will be used in the sequel. For x R, we denote x ± = max{±x, 0}. Likewise, for u W 1,p 0 () fixed, we use the notation u ± ( ) = u( ) ±. We have that u ± W 1,p 0 (), u = u + u, u = u + + u. For u W 1,p 0 () such that u(z) 0 for a.a. z, we define [0, u] = {h W 1,p 0 () : 0 h(z) u(z) for a.a. z }. Now we present the conditions on the map a(y). Assume that ζ C 1 (0, ) is such that 0 < ĉ ζ (t)t ζ(t) c 0 and c 1 t p 1 ζ(t) c 2 (1 + t p 1 ) t > 0, (2.1) for some c 1, c 2 > 0. The hypotheses on the map y a(y) are as follows: (H1) a : R N R N is such that a(y) = a 0 ( y )y for all y R N with a 0 (t) > 0 for all t > 0 and (i) a 0 C 1 (0, ), t a 0 (t)t is strictly increasing on (0, ) and a lim a 0 (t)t = 0 and lim 0(t)t t 0 + t 0 + a 0 (t) = c > 1; (ii) there exists c 3 > 0 such that a(y) c 3 ζ( y ) y for all y R N \ {0};

3 EJDE-2018/101 NONHOMOGENEOUS DIRICHLET PROBLEMS 3 (iii) for all y R N \ {0} and ξ R N, ( a(y)ξ, ξ) R N ζ( y ) ξ 2 ; y (iv) denoting G 0 (t) = t 0 a 0(s)s ds, we can find q (1, p) satisfying t G 0 (t 1/q ) is convex on R + = [0, + ), qg 0 (t) lim t 0 + t q = c > 0, 0 pg 0 (t) a 0 (t)t 2 for all t > 0. Remark 2.1. Conditions (H1)(i), (ii) and (iii) are required by the nonlinear regularity theory of Lieberman [28] and the nonlinear strong maximum principle of Pucci-Serrin [35]. Example 2.2. The following maps satisfy hypotheses (H1) (see Papageorgiou- Rădulescu [29]). (a) a(y) = y p 2 y with 1 < p <. The operator div(a(du)) reduces to the p-laplace differential operator p u = div( Du p 2 Du) for all u W 1,p 0 (). (b) a(y) = y p 2 y + y q 2 y with 1 < q < p <. The map div(a(du)) corresponds to the (p, q)-laplace differential operator p u + q u for all u W 1,p 0 (). Such operators arise in problems of mathematical physics (see Cherfils- Il yasov [4]). (c) a(y) = (1 + y 2 ) p 2 2 y with 1 < p <. The operator div(a(du)) corresponds to the generalized p-mean curvature differential operator (d) div((1 + Du 2 ) p 2 2 Du) for all u W 1,p 0 (). a(y) = where 1 < q < p, γ = p+q 2. { 2 y γ 2 y, if y < 1, y p 2 y + y q 2 y if 1 < y, On the other hand, we use hypotheses (H1) to indicate that G 0 is strictly increasing and strictly convex. Also, we denote We have G(y) = G 0 ( y ) for all y R N. G(y) = G 0( y ) y y = a 0( y )y = a(y) for all y R N \ {0}. So, G is the primitive of a, it is convex with G(0) = 0. Hence, one has G(y) = G(y) G(0) (a(y), y) R N for all y R N. (2.2) Such hypotheses were also considered in recent the works of Gasiński-O Regan- Papageorgiou [10], Papageorgiou-Rădulescu [29, 30, 31] and Bai-Gasiński-Papageorgiou [3].

4 4 Y. BAI EJDE-2018/101 Under hypotheses (H1)(i), (ii) and (iii), we have the following lemma, which summarizes some of important properties for the map a( ). Lemma 2.3 ([38, Lemma 3]). Assume that the map a( ) satisfies hypotheses (H1) (i), (ii), (iii). Then the following statements hold (a) y a(y) is continuous and strictly monotone (hence maximal monotone); (b) a(y) c 4 (1 + y p 1 ) for all y R N, for some c 4 > 0; (c) (a(y), y) R N c1 p 1 y p for all y R N, where c 1 is given in (2.1). We have the following bilateral growth restrictions on the primitive G is established. Lemma 2.4. Assume that the map a( ) satisfies hypotheses (H1) (i), (ii), (iii). Then, there exists c 5 > 0 such that c 1 p(p 1) y p G(y) c 5 (1 + y p ) for all y R N. Let W 1,p () be the dual space of the Sobolev space W 1,p 0 (). We denote the duality brackets between W 1,p () and W 1,p 0 () by,. Also, we introduce a nonlinear operator A : W 1,p 0 () W 1,p () corresponding to map a( ) defined by A(u), h = (a(du), Dh) R N dz for all u, h W 1,p 0 (). Next proposition summarizes some properties of the operator A (see Gasiński- Papageorgiou [12] for a more general version). Proposition 2.5. Assume that (H1)(i), (ii) and (iii) are fulfilled. Then, the map A : W 1,p 0 () W 1,p () is continuous, bounded (thus is, maps bounded sets in W 1,p 0 () to bounded sets in W 1,p ()), monotone (hence maximal monotone too) and of type (S) +, i.e., w if u n u in W 1,p 0 () and lim sup n + A(u n ), u n u 0, then u n u in W 1,p 0 (). For 1 < q < +, we consider the nonlinear eigenvalue problem q u(z) = λ u(z) q 2 u(z) u = 0 on. in The number λ such that the above Dirichlet problem admits a nontrivial solution û is called an eigenvalue of q with Dirichlet boundary condition, also the nontrivial solution û is an eigenfunction corresponding to λ. From Faraci-Motreanu-Puglisi [6], we can see that there exists a smallest eigenvalue λ 1 (q) > 0 such that λ 1 (q) is positive, isolated and simple (that is, if û, v are eigenfunctions corresponding to λ 1 (q), then û = ξ v for some ξ R \ {0}). the following variational characterization holds { λ 1 (q) = inf Du q dx } u q dx : u W 1,q 0 () with u 0.

5 EJDE-2018/101 NONHOMOGENEOUS DIRICHLET PROBLEMS 5 In what follows, we denote by û 1 (q) the positive eigenfunction normalized as û 1 (q) q q = u q dx = 1, which is associated to λ 1 (q). One has û 1 (q) D +. Additionally, we know that if u is an eigenfunction corresponding to an eigenvalue λ λ 1 (q), then u C 1 0() changes sign (see Lieberman [27, 28]). Let f : R R N R. The function f is called to be Carathéodory, if for all (x, y) R R N, z f(z, x, y) is measurable on ; for a.a. z, (x, y) f(z, x, y) is continuous. Such a function is automatically jointly measurable (see Hu-Papageorgiou [24, p. 142]). For the convection term f in problem (1.1), we assume that (H2) f : R R N R is a Carathéodory function such that f(z, 0, y) = 0 for a.a. z, all y R N and (i) there exists η > 0 such that f(z, η, y) = 0 for a.a. z, all y R N, f(z, x, y) 0 for a.a. z, all 0 x η, all y R N, f(z, x, y) c 1 + c 2 y p for a.a. z, all 0 x η, all y R N, with c 1 > 0, c 2 < c1 p 1 ; (ii) for every M > 0, there exists η M L () such that η M (z) c λ1 (q) for a.a. z, η M c λ1 (q), lim inf x 0 + f(z, x, y) x q 1 η M (z) uniformly for a.a. z, all y M (here q (1, p) and c are as in hypothesis (H1)(iv)); (iii) there exists ξ η > 0 such that for a.a. z, all y R N the function x f(z, x, y) + ξ η x p 1 is nondecreasing on [0, η], for a.a. z, all y R N and λ p 1 f(z, 1 x, y) f(z, x, y), (2.3) λ f(z, x, y) λ p f(z, x, 1 λ y) for a.a. z, all 0 x η, all y R N and all λ (0, 1). Remark 2.6. Because the goal of the present paper is to explore the existence of nonnegative solutions, so for x 0, without loss of generality, we may assume that f(z, x, y) = 0 for a.a. z, all y R N. Note that (2.3) is satisfied if for example, for a.a. z, all y R N, the function x f(z,x,y) x is nonincreasing on (0, + ). p 1 Example 2.7. The following function satisfies hypotheses (H2). For the sake of simplicity we drop the z-dependence: { (x r 1 x s 1 ) y p if 0 x 1, f(x, y, z) = (x τ ln x) y p if 1 < x, with 1 < r < s < q < p and τ > 1.

6 6 Y. BAI EJDE-2018/101 Finally we recall the well known Leray-Schauder alternative principle (see e.g., Gasiński-Papageorgiou [11, p. 827]), which will play important role to establish our main results. Theorem 2.8. Let X be a Banach space and C X be nonempty and convex. If ϑ : C C is a compact map, then exactly one of the following two statements is true: (a) ϑ has a fixed point; (b) the set S(ϑ) = {u C : u = λϑ(u), λ (0, 1)} is unbounded. 3. Positive solutions In this section, we explore a positive solution to nonlinear nonhomogeneous Dirichlet problem (1.1). To this end, for v C 1 0() fixed, we first consider the following intermediate Dirichlet problem div a(du(z)) = f(z, u(z), Dv(z)), in, u(z) = 0, on. (3.1) Now, we apply truncation and perturbation approaches to prove that (3.1) has at least one positive solution. So, we turn our attention to consider the following truncation-perturbation Dirichlet problem div a(du(z)) + ξ η u(z) p 1 = f(z, u(z), Dv(z)), in, u(z) = 0, on, (3.2) where f : R R N R is the corresponding truncation-perturbation of convection term f with respect to the second variable, defined by { f(z, x, y) + ξ η (x f(z, + ) p 1 if x η, x, y) = f(z, η, y) + ξ η η p 1 (3.3) if η < x. Remark 3.1. Recall that f is a Carathéodoty function (see hypotheses (H2)). It is obvious that the truncation-perturbation f is a Carathéodoty function as well. It is obvious that if a function u : R with u = 0 on and 0 u(z) η for a.a. z is a solution of problem (3.2), then u is also a solution of problem (3.1). Using this fact, we will now prove the existence of a positive solution for problem (3.1). Proposition 3.2. Assume that (H1) and (H2) are satisfied. Then problem (3.1) has a positive solution u v such that u v [0, η] D +. Proof. To prove the existence of a nontrivial solution, we introduce the C 1 -functional ϕ v : W 1,p 0 () R defined by ϕ v (u) = G(Du) dz + ξ η p u p p F v (z, u) dz for all u W 1,p 0 (), where F v is given by F v (z, x) = x 0 f(z, s, Dv(z)) ds.

7 EJDE-2018/101 NONHOMOGENEOUS DIRICHLET PROBLEMS 7 Combining Lemma 2.4 and definition of f (see (3.3)), we conclude that the functional ϕ v is coercive. On the other hand, the Sobolev embedding theorem and the convexity of G reveal that the functional ϕ v is sequentially weakly lower semicontinuous. Therefore, it allows us to use the Weierstrass-Tonelli theorem to find u v W 1,p 0 () such that ϕ v (u v ) = inf ϕ v (u). (3.4) u W 1,p 0 () We take M := sup z Dv(z) and then use hypothesis (H2)(ii) to obtain that for any ε > 0 fixed, there exists δ (0, η] satisfying f(z, x, y) (η M (z) ε)x q 1 for a.a. z, all x [0, δ], all y M; this results in f(z, x, Dv(z)) (η M (z) ε)x q 1 + ξ η x p 1 for a.a. z, all x [0, δ] (see (3.3)). Also, we can calculate F v (z, x) 1 q (η M (z) ε)x q + ξ η p xp for a.a. z, all x [0, δ]. (3.5) qg0(t) t q Note that G(y) = G 0 ( y ) for all y R N and lim t 0 + = c > 0 (see (H1)(iv)), so G(y) c + ε y q for all y δ. (3.6) q As û 1 (q) D +, we can take t (0, 1) small enough such that tû 1 (q)(z) [0, δ], t Dû 1 (q)(z) δ for all z. (3.7) Obviously, we can obtain ϕ v (tû 1 (q)) c + ε t q λ1 (q) tq (η M (z) ε)û 1 (q) q dz q q ( ) (3.8) tq (c λ1 (q) η M (z))û 1 (q) q dz + ε( λ 1 (q) + 1) q (recall that û 1 (q) q = 1). From η M (z) c λ 1 (q) for a.a. z, η M c λ 1 (q) (see (H2)(ii)) and û 1 (q) D +, it yields r 0 = (η M (z) c λ1 (q))û 1 (q) q dz > 0. So, (3.8) becomes Now, we pick ε (0, ϕ v (tû 1 (q)) tq q ( r 0 + ε( λ 1 (q) + 1)). r 0 bλ 1(q)+1 ) to obtain ϕ v(tû 1 (q)) < 0. This means that ϕ v (u v ) < 0 = ϕ v (0), hence u v 0. Therefore, we have proved the existence of a nontrivial solution to problem (3.1). Next, we show that u v is nonnegative. Equality (3.4) indicates ϕ v(u v ) = 0, hence A(u v ), h + ξ η u v p 2 u v h dz = f(z, u v, Dv)h dz for all h W 1,p 0 (). (3.9)

8 8 Y. BAI EJDE-2018/101 Inserting h = u v W 1,p 0 () into (3.9) to obtain A(u v ), u v ξ η u v p 2 u v u v dz = thus (see (3.3) and (H2)), A(u v ), u v + ξ η u v p p 0. Combining with Lemma 2.3 and (3.3), we calculate c 1 p 1 Du v p + ξ η u v p p 0, f(z, u v, Dv)u v dz, which gives u v 0 and u v 0. Furthermore, we shall illustrate that u v [0, η]. Putting h = (u v η) + W 1,p 0 () into (3.9), we obtain A(u v ), (u v η) + + ξ η (u v η) + dz, = u p 1 v ( f(z, η, Dv) + ξη η p 1) (u v η) + dz = ξ η η p 1 (u v η) + dz (see (3.3) and condition (H2)(i)). We use the fact that A(η) = 0, to obtain A(u v ) A(η), (u v η) + + ξ η (uv p 1 η p 1 )(u v η) + dz 0. However, the monotonicity of A implies u v η. Until now, we have verified that u v [0, η] \ {0}. (3.10) Finally, we demonstrate the regularity of u v, more precisely we will show that u v D +. It follows from (3.3), (3.9) and (3.10) that A(u v ), h = f(z, u v, Dv)h dz for all h W 1,p 0 (), which gives div a(du v (z)) = f(z, u v (z), Dv(z)) for a.a. z, u v (z) = 0 on. From (3.11) and Papageorgiou-Rădulescu [30], we have u v L (). (3.11) However, using the regularity results from Lieberman [28] (see also Fukagai-Narukawa [9]), we have u v C 1 0() \ {0}. To conclude, we have u v [0, η] C0() 1 \ {0}. Moreover, we can use the nonlinear maximum principle, see Pucci-Serrin [35]), to conclude directly that u v D +. From the proof of Proposition 3.2, we know that problem (3.1) has a solution u v [0, η] D +. Next, we will prove that problem (3.1) has a smallest positive solution in the order interval [0, η]. In what follows, we denote S v = {u W 1,p 0 () : u 0, u [0, η] is a solution of (3.1)}. Proposition 3.2 implies S v [0, η] D +.

9 EJDE-2018/101 NONHOMOGENEOUS DIRICHLET PROBLEMS 9 Let p be the critical Sobolev exponent corresponding to p, i.e., { Np p N p if p < N, = + if N p. For ε > 0 and r (p, p ) fixed, from hypotheses (H2)(i) and (ii), there exists c 6 = c 6 (ε, r, M) > 0 (recall that M := sup z Dv(z) ) such that f(z, x, Dv(z)) (η M (z) ε)x q 1 c 6 x r 1 (3.12) for a.a. z, and all 0 x η. This unilateral growth restriction on f(z,, Dv(z)) drives us to consider another auxiliary Dirichlet problem as follows: with u(z) > 0 in. div a(du(z)) = (η M (z) ε)u(z) q 1 c 6 u(z) r 1 in, u(z) = 0 on, (3.13) Proposition 3.3. If hypotheses (H1) holds, then for all ε > 0, auxiliary problem (3.13) admits a unique positive solution u D +. Proof. First we show the existence of positive solutions for problem (3.13). To do so, consider the C 1 -functional ψ : W 1,p 0 () R defined by ψ(u) = G(Du) dz + 1 p u p p 1 (η M (z) ε)(u + ) q dz q + c 6 r u+ r r for all u W 1,p 0 (). From the facts G(0) = 0, u = u + u and [11, Proposition ], we have G(Du) dz = G(Du + ) dz + G( Du ) dz. So, from Lemma 2.4 we have c 1 ψ(u) p(p 1) Du+ p p + c 6 r u+ r c 1 r + p(p 1) Du p p + 1 p u p p 1 (η M (z) ε)(u + ) q dz, q hence (see, Poincaré inequality, e.g. [11, Theorem 2.5.4, p.216]) ψ(u) c 7 u p c 8 ( u q + 1), for some c 7, c 8 > 0. Since q < p, it is clear that ψ is coercive. We use the compactness of embedding W 1,p 0 () L p () and the convexity of G again, to conclude that ψ is sequentially weakly lower semicontinuous. By the Weierstrass- Tonelli theorem, we get u W 1,p 0 () such that ψ(u ) = inf ψ(u). (3.14) u W 1,p 0 () Using the same method as in the proof of Proposition 3.2, we can take t (0, 1) and ε > 0 small enough to obtain ψ(tû 1 (q)) < 0. This implies (see (3.14)) so, u 0. ψ(u ) < 0 = ψ(0),

10 10 Y. BAI EJDE-2018/101 The equality (3.14) implies ψ (u ) = 0. For h W 1,p 0 (), one has A(u ), h ((u ) ) p 1 h dz = (η M (z) ε)((u ) + ) q 1 h dz c 6 ((u ) + ) r 1 h dz. Taking h = (u ) W 1,p 0 () into (3.15), we use Lemma 2.3 again to obtain c 1 p 1 D(u ) p p + (u ) p p 0. So, we have u 0 and u 0. Therefore, (3.15) reduces to A(u ), h = (η M (z) ε)(u ) q 1 h dz c 6 (u ) r 1 h dz for all h W 1,p 0 (), this means div a(du (z)) = (η M ε)(u )(z) q 1 c 6 (u )(z) r 1 for a.a. z, u (z) = 0 on. (3.15) (3.16) As in the proof of Proposition 3.2, using the nonlinear regularity theory, we have u C 1 0() + \ {0}. Next we shall verify that u is the unique positive solution to problem (3.13). For this goal, we consider the integral functional j : L 1 () R = R {+ } defined by { j(u) = G(Du1/q ) dz if u 0, u 1/q W 1,p 0 (), + otherwise, where the effective domain of the functional j is denoted by dom j = {u L 1 () : j(u) < + }. We will show that the integral functional j is convex. Let u 1, u 2 dom j and u = (1 t)u 1 +tu 2 with t [0, 1]. [5, Lemma 1] states that the function u Du 1/q q is convex, so we have Du 1/q (z) ((1 t) Du 1 (z) 1/q q + t Du 2 (z) 1/q q) 1/q for a.a. z. The monotonicity of G 0 and the convexity of t G 0 (t 1/q ) (see hypothesis (H1)(iv)) ensure that G 0 ( Du 1/q (( (z) ) G 0 (1 t) Du1 (z) 1/q q + t Du 2 (z) 1/q q) 1/q) for a.a. z. Which leads to (1 t)g 0 ( Du 1 (z) 1/q ) + tg 0 ( Du 2 (z) 1/q ) G(Du 1/q (z)) (1 t)g(du 1 (z) 1/q ) + tg(du 2 (z) 1/q ) for a.a. z, thus the map j is convex. Suppose that ũ is another positive solution of (3.13). As we did for u, we can check that ũ C 1 0() + \ {0}. For h C 1 0() fixed and t small enough, we obtain u + th dom j and ũ + th dom j.

11 EJDE-2018/101 NONHOMOGENEOUS DIRICHLET PROBLEMS 11 Recalling that j is convex, it is evidently Gâteaux differentiable at u and at ũ in the direction h. Further, we apply the chain rule and the nonlinear Green s identity (see Gasiński-Papageorgiou [11, p. 210]) to obtain j (u )(h) = 1 q j (ũ )(h) = 1 q div a(du ) (u ) q 1 h dz for all h C0(), 1 div a(dũ ) (ũ ) q 1 h dz for all h C0(). 1 Putting h = (u ) q (ũ ) q into the above inequalities and then subtracting the resulting equalities, it follows from the monotonicity of j (since j is convex) that 0 1 ( div(du ) q (u ) q 1 div a(dũ ) ) (ũ ) q 1 ((u ) q (ũ ) q ) dz = c 6 ( (ũ ) r q (u ) r q)( (u ) q (ũ ) q) dz q (see (3.13)), so, from q < p < r, we conclude that u = ũ. This proves that u C0() 1 + \ {0} is the unique positive solution for problem (3.13). We are now to apply the nonlinear maximum principle, see Pucci-Serrin [35]), again to obtain u D +. Proposition 3.4. If hypotheses (H1) and (H2) hold, then u u for all u S v. Proof. Let u S v. We now introduce the following Carathéodory function e : R R { (η M (z) ε)(x + ) q 1 c 6 (x + ) r 1 + ξ η (x + ) p 1 if x u(z), e(z, x) = (η M (z) ε)u(z) q 1 c 6 u(z) r 1 + ξ η u(z) p 1 (3.17) if u(z) < x. Also, we denote E(z, x) = x 0 e(z, s) ds and consider the C 1 -functional τ : W 1,p 0 () R defined by τ(u) = G(Du) dz + ξ η p u p p E(z, u) dz for all u W 1,p 0 (). By the definition of e (see (3.17)), we see that τ is coercive. Also, it is sequentially weakly lower semicontinuous. Invoking the Weierstrass-Tonelli theorem, we can find ũ W 1,p 0 () such that τ(ũ ) = As before, since q < p < r, we have inf τ(v). (3.18) v W 1,p 0 () τ(ũ ) < 0 = τ(0), which implies ũ 0. From (3.18), we have τ (ũ ) = 0, which means A(ũ ), h + ξ η ũ p 2 ũ h dz = e(z, ũ )h dz (3.19) for all h W 1,p 0 (). Putting h = (ũ ) W 1,p 0 () into the above equality and then using Lemma 2.3, we have c 1 p 1 D(ũ ) p p + ξ η (ũ ) p p 0

12 12 Y. BAI EJDE-2018/101 (see (3.17)), so ũ 0 and ũ 0. On the other hand, inserting h = (ũ u) + W 1,p 0 () into (3.19), we obtain A(ũ ), (ũ u) + + ξ η (ũ ) p 1 (ũ u) + dz ( = (ηm (z) ε)u q 1 c 6 u r 1 + ξ η u p 1) (ũ u) + dz f(z, u, Dv)(ũ u) + dz + ξ η u p 1 (ũ u) + dz = A(u), (ũ u) + + ξ η u p 1 (ũ u) + dz (see (3.12), (3.17), and recall that u S v ). Therefore, we have A(ũ ) A(u), (ũ u) + ( + ξ η (ũ ) p 1 u p 1) (ũ u) + dz 0. Using the monotonicity of A, we deduce ũ u. So, we have verified that ũ [0, u] \ {0}. (3.20) Taking into account (3.17) and (3.20), we rewrite (3.19) as A(ũ ( ), h = (ηm (z) ε)(ũ ) q 1 c 6 (ũ ) r 1) h dz for all h W 1,p 0 (). This combined with Proposition 3.3 gives ũ = u, so u u, which completes the proof. Applying Proposition 3.4, we shall show that problem (3.1) admits a smallest positive solution û v [0, η] D +. Proposition 3.5. If (H1) and (H2) are fulfilled, then problem (3.1) admits a smallest positive solution û v [0, η] D +. Proof. Invoking [24, Lemma 3.10 p. 178], we can find a decreasing sequence {u n } n 1 S v such that inf S v = inf u n. (3.21) n 1 For all n 1, we have A(u n ), h = however, from Proposition 3.4, one has f(z, u n, Dv)h dz for all h W 1,p 0 (), (3.22) u u n η. (3.23) Then by hypothesis (H2)(i) and Lemma 2.3, we have that the sequence {u n } n 1 W 1,p 0 () is bounded. Passing to a subsequence, we may assume that u n w ûv in W 1,p 0 () and u n û v in L p (). (3.24) Choosing h = u n û v W 1,p 0 () for (3.22), we pass to the limit as n and then apply (3.24) to get lim A(u n), u n û v = 0, n +

13 EJDE-2018/101 NONHOMOGENEOUS DIRICHLET PROBLEMS 13 but the (S) + -property of A (see Proposition 2.5), results in u n û v in W 1,p 0 (). (3.25) Passing to the limit as n + in (3.22) and using (3.25) to reveal A(û v ), h = f(z, û v, Dv)h dz for all h W 1,p 0 (). On the other hand, taking the limit as n + in (3.23), we conclude that u û v η. From the above inequality, it follows that which completes the proof. Now, we consider the set û v S v and û v = inf S v, C = {u C 1 0() : 0 u(z) η for all z } and introduce the mapping ϑ : C C given by ϑ(v) = û v. It is obvious that a fixed point of map ϑ is also a positive solution to problem (1.1). Therefore, next, we focus our attention to produce a fixed point for ϑ. Here our approach will apply the Leray-Schauder alternative principle (see Theorem 2.8). To do so, we will need the following lemma. Lemma 3.6. If (H1) and (H2) are satisfied, then for any sequence {v n } n 1 C with v n v in C0(), 1 and u S v, there exists a sequence {u n } C0() 1 with u n S vn for n 1, such that u n u in C0(). 1 Proof. Let {v n } n 1 C be such that v n v in C 1 0(), and u S v. First, we consider the nonlinear Dirichlet problem div a(dw(z)) + ξ η w(z) p 2 w(z) = f(z, u(z), Dv n (z)) in, w(z) = 0 on. Since u S v [0, η] \ {0}, from (3.3) and hypothesis (H2)(i), we see that f(, u( ), Dv n ( )) 0 for all n 1, f(z, u(z), Dv n (z)) 0 for a.a. z and all n 1. (3.26) It is obvious that problem (3.26) has a unique positive solution u 0 n D +. It follows from (3.3), the fact that u S v [0, η] \ {0}, and hypotheses (H2)(i), (iii) that A(u 0 n), (u 0 n η) + + ξ η (u 0 n) p 1 (u 0 n η) + dz = (f(z, u, Dv n ) + ξ η u p 1 )(u 0 n η) + dz (f(z, η, Dv n ) + ξ η η p 1 )(u 0 n η) + dz = ξ η η p 1 (u 0 n η) + dz,

14 14 Y. BAI EJDE-2018/101 hence, from A(η) = 0, we have A(u 0 n) A(η), (u 0 n η) + + ξ η ((u 0 n) p 1 η p 1 )(u 0 n η) + dz 0. However, the monotonicity of A implies u 0 n η. So, we conclude that u 0 n [0, η] \ {0} n 1. Moreover the nonlinear regularity theory of Lieberman [28], and the nonlinear maximum principle of Pucci-Serrin [35]) imply that u 0 n [0, η] D + n 1. (3.27) We have div a(du 0 n(z)) + ξ η ((u 0 n(z)) p 1 u(z) p 1 ) = f(z, u(z), Dv n (z)) for a.a. z, u 0 n(z) = 0 on. (3.28) From (3.27) (3.28), Lemma 2.3 and hypothesis (H2)(i), we conclude that the sequence {u 0 n} n 1 is bounded in W 1,p 0 (). So, on account of the nonlinear regularity theory of Lieberman [28], we can find β (0, 1) and c 9 > 0 such that u 0 n C 1,β () and u 0 n C 1,β () c 9 n 1. The compactness of the embedding C 1,β () C 1 () implies that there exists a subsequence {u 0 n k } k 1 of the sequence {u 0 n} n 1 such that Using this fact and (3.28), we have u 0 n k ũ 0 in C 1 () as k +. div a(dũ 0 (z)) + ξ η ((ũ 0 (z)) p 1 u(z) p 1 ) = f(z, u(z), Dv(z)) for a.a. z, ũ 0 (z) = 0 on. (3.29) Recall that u S v, so (3.1) holds. Taking into account (3.1) and (3.29), we have A(ũ 0 ) A(u), h + ξ η (ũ 0 (z) p 1 u(z) p 1 )h dz = 0 for all h W 1,p 0 (). Additionally, we insert h = (ũ 0 u) + and h = (u ũ 0 ) + into the above equality to obtain ũ 0 = u S v. So, for the original sequence {u 0 n} n 1, one has u 0 n u in C 1 0() as n +. Next, we consider the nonlinear Dirichlet problem div a(dw(z)) + ξ η w(z) p 2 w(z) = f(z, u 0 n(z), Dv n (z)) in, w(z) = 0 on. As before, we verify that the above problem admits a unique solution such that u 1 n [0, η] D + n 1. We apply nonlinear regularity theory of Lieberman [28] again to obtain u 1 n u in C 1 0() as n +.

15 EJDE-2018/101 NONHOMOGENEOUS DIRICHLET PROBLEMS 15 Repeating this procedure, we construct a sequence {u k n} k,n 1 such that div a(du k n(z)) + ξ η u k n(z) p 1 = f(z, u k 1 n (z), Dv n (z)) in, for all n, k 1 with u k n(z) = 0 on (3.30) u k n [0, η] D + n, k 1, (3.31) u k n u in C 1 0() as n + k 1. (3.32) For n 1 fixed, as above, we know that the sequence {u k n} k 1 C0() 1 is relatively compact. Therefore, there has a subsequence {u km n } m 1 of the sequence {u k n} k 1 satisfying u km n ũ n in C0() 1 as m +. This and (3.32) imply div a(dũ n (z)) + ξ η ũ n (z) p 1 = f(z, ũ n (z), Dv n (z)) for a.a. z, ũ n (z) = 0 on. (3.33) The uniqueness of the solution of (3.33) deduces that for the original sequence we have u k n ũ n in C0() 1 as k +. However, from (3.31), we obtain ũ n [0, η] D + n 1, but from (3.32) and the double limit lemma (see Aubin-Ekeland [1] or Gasiński- Papageorgiou [20, p. 61]), we have ũ n [0, η] D + n n 0. Consequently, ũ n S v n n 0 and ũ n u in C 1 0(), which completes the proof of the Lemma. Remark 3.7. Actually, if we introduce the set-valued mapping S : C 1 () 2 C1 () by S(v) = S v, then by the above lemma, we conclude that the mapping S is lower semicontinuous. Applying this lemma, we will prove that the map ϑ : C C defined by ϑ(v) = û v is compact. Proposition 3.8. If hypotheses (H1) and (H2) are fulfilled, then the map ϑ : C C is compact. Proof. To end this, we shall show that ϑ is continuous and maps bounded sets in C to relatively compact subsets of C. First, for the part of continuity of ϑ, let v C and {v n } n 1 C be such that v n v in C 1 0(), and denote û n = ϑ(v n ) for n 1. So, we get div a(dû n (z)) = f(z, û n (z), Dv n (z)) for a.a. z, û n (z) = 0 on, (3.34) with û n [0, η] for all n 1. It is easy to check that {û n } n 1 W 1,p 0 () is bounded. So, it follows from Lieberman [28] that there exist β (0, 1) and c 10 > 0 satisfying û n C 1,β () and û n C 1,β () c 10 n 1.

16 16 Y. BAI EJDE-2018/101 Without loss of generality, we may assume that Passing to the limit in (3.34), it yields û n û in C 1 0() as n +. (3.35) div a(dû(z)) = f(z, û(z), Dv(z)) for a.a. z, û(z) = 0 on. By taking M > sup n 1 v n C1 (), we apply Proposition 3.4 to obtain u û n 1, hence, convergence (3.35) implies (3.36) n u û C 1 0() +. (3.37) We now assert that û = ϑ(v). Invoking Lemma 3.6, we can take a sequence {u n } C 1 0() with u n S vn, n 1 and By the definition of ϑ, we have u n ϑ(v) in C 1 0() as n +. (3.38) û n = ϑ(v n ) u n n 1. This combined with (3.35) and (3.38) gives û ϑ(v). Recalling that (3.37), we obtain û = ϑ(v), therefore, ϑ is continuous. Next we will verify that ϑ maps bounded sets in C to relatively compact subsets of C. Assume that B C is bounded in C0(). 1 As before, we know that the set ϑ(b) W 1,p 0 () is bounded. On the other hand, we apply the nonlinear regularity theory of Lieberman [28] and the compactness of the embedding C 1,s 0 () C1 0() (with 0 < s < 1) to reveal that the set ϑ(b) C0() 1 is relatively compact, thus ϑ is compact. Now we give the main result of this article. Theorem 3.9. If (H1) and (H2) are satisfied, then problem (1.1) admits a positive solution û, more precisely, û [0, η] D +. Proof. Let U(ϑ) be the set defined by U(ϑ) = {u C : u = λϑ(u), 0 < λ < 1}. For any u U(ϑ), we have 1 λu = ϑ(u), so A( 1 λ u), h = f(z, u 1,p, Du)h dz for all h W0 (). (3.39) λ Inserting h = u λ W 1,p 0 () into (3.39) and taking into account Lemma 2.3, we calculate c 1 p 1 D( u λ ) p p f(z, u λ, Du) u λ dz f(z, u, Du) u λ p dz f(z, u, D( u λ ))u dz ( c1 + c 2 D( u λ ) p) dz

17 EJDE-2018/101 NONHOMOGENEOUS DIRICHLET PROBLEMS 17 where the last three inequalities are obtained by using (2.3), (H2)(iii), and (H2)(i), respectively. Considering the inequality c 2 < c1 p 1 (see hypothesis (H2)(i)), one has for some c 11 > 0. Hence, we have From (3.39) we have D( u λ ) p c 11 for all λ (0, 1), { u λ } u U(ϑ) W 1,p 0 () is bounded. (3.40) div a(d( u λ )(z)) = f(z, u (z), Du(z)) for a.a. z, λ u = 0 on. However, condition (H2)(iii) ensures that (3.41) f(z, u λ, Du) λp f(z, u λ, D( u )) for a.a. z. (3.42) λ Then from (3.40) (3.42) and the nonlinear regularity theory of Lieberman [28], we have u λ C0 1() c 12 for all u U(ϑ), for some c 12 > 0, thus U(ϑ) C 1 0() is bounded. Recall that ϑ is compact, see Proposition 3.8, we are now in a position to apply the Leray-Schauder alternative theorem (see Theorem 2.8), to look for a function û C such that û = ϑ(û). Consequently, we know that û [0, η] D + is a positive solution of (1.1). References [1] J.-P. Aubin, I. Ekeland; Applied Nonlinear Analysis, Wiley, New York, [2] D. Averna, D. Motreanu, E. Tornatore; Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence, Appl. Math. Lett., 61 (2016), [3] Y. R. Bai, L. Gasiński, N. S. Papageorgiou; Nonlinear nonhomogeneous Robin problems with dependence on the gradient, Bound Value Probl., 2018:17 (2018), pages 24. [4] L. Cherfils, Y. Il yasov; On the stationary solutions of generalized reaction diffusion equations with p&q-laplacian, Commun. Pure Appl. Anal., 4 (2005), [5] J.I. Díaz, J. E. Saá; Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), [6] F. Faraci, D. Motreanu, D. Puglisi; Positive solutions of quasi-linear elliptic equations with dependence on the gradient, Calc. Var. Partial Differential Equations, 54 (2015), [7] L. F. O. Faria, O. H. Miyagaki, D. Motreanu; Comparison and positive solutions for problems with the (p, q)-laplacian and a convection term, Proc. Edinb. Math. Soc. (2), 57 (2014), [8] D. de Figueiredo, M. Girardi, M. Matzeu; Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differential Integral Equations, 17 (2004), [9] N. Fukagai, K. Narukawa; On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. Mat. Pura Appl., (4), 186 (2007), [10] L. Gasiński, D. O Regan, N. Papageorgiou; Positive solutions for nonlinear nonhomogeneous Robin problems, Z. Anal. Anwend., 34 (2015), [11] L. Gasiński, N. S. Papageorgiou; Nonlinear Analysis. Chapman & Hall/CRC, Boca Raton, FL, [12] L. Gasiński, N. S. Papageorgiou; Existence and multiplicity of solutions for Neumann p- Laplacian-type equations, Adv. Nonlinear Stud., 8 (2008),

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