NONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT AND WEIGHT

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1 Electronic Journal of Differential Equations, Vol. 016 (016), No. 08, pp ISSN: URL: or ftp ejde.math.txstate.edu NONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT AND WEIGHT MOHAMMED BOUCHEKIF, ALI RIMOUCHE Abstract. In this article we consider the problem div `p(x) u = u u + λf u = 0 on Ω in Ω where Ω is a bounded domain in R N, We study the relationship between the behavior of p near its minima on the existence of solutions. 1. Introduction and statement of main results In this article we study the existence of solutions to the problem div ( p(x) u ) = u u + λf u = 0 on Ω, in Ω (1.1) where Ω is a smooth bounded domain of R N, N 3, f belongs to H 1 = W 1, (Ω) \ {0}, p H 1 (Ω) C( Ω) is a positive function, λ is a real parameter and = N N is the critical Sobolev exponent for the embedding of H1 0 (Ω) into L (Ω). For a constant function p, problem (1.1) has been studied by many authors, in particular by Tarantello [8]. Using Ekeland s variational principle and minimax principles, she proved the existence of at least one solution of (1.1) with λ = 1 when f H 1 and satisfies fu dx K N( u ) (N+)/4 for u = 1, Ω Ω Ω with K N = 4 N (N N + )(N+)/4. Moreover when the above inequality is strict, she showed the existence of at least a second solution. These solutions are nonnegative when f is nonnegative. 010 Mathematics Subject Classification. 35J0, 35J5, 35J60. Key words and phrases. Critical Sobolev exponent; Nehari manifold; variational principle. c 016 Texas State University. Submitted June 30, 015. Published January 6,

2 M. BOUCHEKIF, A. RIMOUCHE EJDE-016/08 The following problem has been considered by several authors, div(p(x) u) = u u + λu u > 0 in Ω u = 0 on Ω. in Ω (1.) We quote in particular the celebrate paper by Brezis and Nirenberg [4], and that of Hadiji and Yazidi [6]. In [4], the authors studied the case when p is constant. To our knowledge, the case where p is not constant has been considered in [6] and [7]. The authors in [6] showed that the existence of solutions depending on a parameter λ, N, and the behavior of p near its minima. More explicitly: when p H 1 (Ω) C( Ω) satisfies p(x) = p 0 + β k x a k + x a k θ(x) in B(a, τ), (1.3) where k, β k, τ are positive constants, and θ tends to 0 when x approaches a, with a p 1 ({p 0 }) Ω, p 0 = min x Ω p(x), and B(a, τ) denotes the ball with center 0 and radius τ, when 0 < k, and p satisfies the condition kβ k p(x).(x a) x a k a.e x Ω. (1.4) On the one hand, they obtained the existence of solutions to (1.) if one of the following conditions is satisfied: (i) N 4, k > and λ ]0, λ 1 (p)[; (ii) N 4, k = and λ ] γ(n), λ 1 (p)[; (iii) N = 3, k and λ ]γ(k), λ 1 (p)[; (iv) N 3, 0 < k < and p satisfies (1.4), λ ]λ, λ 1 (p)[; where γ(n) = (N )N(N + ) β, 4(N 1) γ(k) is a positive constant depending on k, and λ [ β k N 4, λ 1(p)[, with β k = β k min[(diam Ω) k, 1]. On the other hand, non-existence results are given in the following cases: (a) N 3, k > 0 and λ δ(p). (b) N 3, k > 0 and λ λ 1 (p). We denote by λ 1 (p) the first eigenvalue of ( div(p.), H) and δ(p) = 1 Ω inf p(x)(x a) u dx. u H0 1(Ω)\{0} Ω u dx Then we formulate the question: What happens in (1.1) when p is not necessarily a constant function? A response to this question is given in Theorem 1.5 below. Notation. S is the best Sobolev constant for the embedding from H 1 0 (Ω) to L (Ω). is the norm of H 1 0 (Ω) induced by the product (u, v) = Ω u v dx. 1 and p = ( Ω. p dx) 1/p are the norms in H 1 and L p (Ω) for 1 p < respectively. We denote the space H 1 0 (Ω) by H and the integral Ω u dx by u. ω N is the area of the sphere S N 1 in R N.

3 EJDE-016/08 NONHOMOGENEOUS ELLIPTIC EQUATIONS 3 with Put Let E = {u H : f(x)u(x)dx > 0} and Ω α(p) := 1 inf u E f(x) := f(x).(x a) + N + f(x), p 1/ 0 Λ 0 := K N Ω ˆp(x) u(x) dx Ω f(x)u(x)dx, ˆp(x) = p(x).(x a). RN x l+ (1 + x ) N, (S(p)) N/4, A l = (N ) f 1 1 B = R (1 + x ) N, D := w 0(a) (1 + x ) (N+)/, N R N where l 0 and Ω S(p) := inf p(x) u u H\{0} u. (1.5) Definition 1.1. We say that u is a ground state solution of (1.1) if J λ (u) = min{j λ (v) : v is a solution of (1.1)}. Here J λ is the energy functional associate with (1.1). Remark 1.. By the Ekeland variational principle [5] we can prove that for λ (0, Λ 0 ) there exists a ground state solution to (1.1) which will be denoted by w 0. The proof is similar to that in [8]. Remark 1.3. Noting that if u is a solution of the problem (1.1), then u is also a solution of the problem (1.1) with λ instead of λ. Without loss of generality, we restrict our study to the case λ 0. Our main results read as follows. Theorem 1.4. Suppose that Ω is a star shaped domain with respect to a and p satisfies (1.3). Then there is no solution of problem (1.1) in E for all 0 λ α(p). Theorem 1.5. Let p H 1 (Ω) C( Ω) such that p 0 > 0 and p satisfies (1.3) then, for 0 < λ < Λ0, problem (1.1) admits at least two solutions in one of the following condition: (i) k > N, (ii) β (N )/ > D A (N )/ ( A0 B )(6 N)/4. This article is organized as follows: in the forthcoming section, we give some preliminaries. Section 3 and 4 present the proofs of our main results.. Preliminaries A function u in H is said to be a weak solution of (1.1) if u satisfies (p u v u uv λfv) = 0 for all v H. It is well known that the nontrivial solutions of (1.1) are equivalent to the non zero critical points of the energy functional J λ (u) = 1 p u 1 u λ fu. (.1)

4 4 M. BOUCHEKIF, A. RIMOUCHE EJDE-016/08 We know that J λ is not bounded from below on H, but it is on a natural manifold called Nehari manifold, which is defined by N λ = {u H \ {0} : J λ(u), u = 0}. Therefore, for u N λ, we obtain J λ (u) = 1 p u λ N + fu, (.) N N or J λ (u) = 1 p u + N + u. (.3) N It is known that the constant S is achieved by the family of functions U ε (x) = ε (N )/ (ε + x ) (N )/ ε > 0, x R N, (.4) For a Ω, we define U ε,a (x) = U ε (x a) and u ε,a (x) = ξ a (x)u ε,a (x), where ξ a C 0 (Ω) with ξ a 0 and ξ a = 1 in a neighborhood of a. (.5) We start with the following lemmas given without proofs and based essentially on [8]. Lemma.1. The functional J λ is coercive and bounded from below on N λ. Set Ψ λ (u) = J λ(u), u. (.6) For u N λ, we obtain Ψ λ(u), u = p u ( 1) u (.7) = ( ) p u λ(1 ) fu. (.8) So it is natural to split N λ into three subsets corresponding to local maxima, local minima and points of inflection defined respectively by N + λ = {u N λ : Ψ λ(u), u > 0}, N λ = {u N λ : Ψ λ(u), u < 0}, N 0 λ = {u N λ : Ψ λ(u), u = 0}. Lemma.. Suppose that u 0 is a local minimizer of J λ on N λ. Then if u 0 / N 0 λ, we have J λ (u 0) = 0 in H 1. Lemma.3. For each λ (0, Λ 0 ) we have N 0 λ =. By Lemma.3, we have N λ = N + λ N λ for all λ (0, Λ 0). For u H \ {0}, let ( p u t m = t max (u) := ( 1) ) (N )/4. u Lemma.4. Suppose that λ (0, Λ 0 ) and u H \ {0}, then (i) If fu 0, then there exists an unique t + = t + (u) > t m such that t + u N λ and J λ (t + u) = sup t t m J λ (tu).

5 EJDE-016/08 NONHOMOGENEOUS ELLIPTIC EQUATIONS 5 (ii) If fu > 0, then there exist unique t = t (u), t + = t + (u) such that 0 < t < t m < t +, t u N + λ, t+ u N λ and Thus we put J λ (t + u) = sup t t m J λ (tu); J λ (t u) = inf 0 t t + J λ(tu). c = inf J λ (u), c + = inf J λ (u), c = inf J λ (u). u N λ u N + λ u N λ Lemma.5. (i) If λ (0, Λ 0 ), then c c + < 0. (ii) If λ (0, Λ0 ), then c > 0. Some properties of α(p). 3. Nonexistence result Proposition 3.1. (1) Assume that p C 1 (Ω) and there exists b Ω such that p(b)(b a) < 0 and f C 1 in a neighborhood of b. Then α(p) =. () If p C 1 (Ω) satisfying (1.3) with k > and p(x)(x a) 0 for all x Ω and f C 1 in a neighborhood of a and f(a) 0, then α(p) = 0 for all N 3. (3) If p H 1 (Ω) C( Ω) and p(x)(x a) 0 a.e x Ω, then α(p) 0. Proof. (1) Set ϕ C 0 (R N ) such that 0 ϕ 1, ϕ(x) = { 1 if x B(0; r) 0 if x / B(0; r), where 0 < r < 1. Set ϕ j (x) = sgn[ f(x)]ϕ(j(x b)) for j N. We have α(p) 1 B(b, r j ) ˆp(x) ϕ j(x) f(x)ϕ. B(b, r j ) j (x) (3.1) Using the change of variable y = j(x b) and applying the dominated convergence theorem, we obtain [ α(p) j ˆp(b) B(0,r) ϕ(y) ] f(b) B(0,r) ϕ(y) + o(1), letting j, we obtain the desired result. () Since p C 1 (Ω) in a neighborhood V of a, we write p(x) = p 0 + β k x a k + θ 1 (x), (3.) where θ 1 C 1 (V ) such that θ 1 (x) lim = 0. (3.3) x a x a k Thus, we deduce that there exists 0 < r < 1, such that θ 1 (x) x a k, for all x B(a, r). (3.4) Let ψ j (x) = sgn[ f(x)]ϕ(j(x a)), ϕ C0 (R N ) defined as in (3.1), we have 0 α(p) 1 p(x).(x a) ψj (x). f(x)ψj (x)

6 6 M. BOUCHEKIF, A. RIMOUCHE EJDE-016/08 Using (3.), we obtain 0 α(p) kβ k B(a, r j ) x a k ψ j (x) f(x)ψ + 1 B(a, r j ) j (x) B(a, r j ) θ 1(x).(x a) ψ j (x) f(x)ψ. B(a, r j ) j (x) Using the change of variable y = j(x a), and integrating by parts the second term of the right hand side, we obtain 0 α(p) kβ k B(0,r) y k ϕ(y) j k B(0,r) f( y j + a) ϕ(y) + j B(0,r) θ 1( y j + a)div(y ϕ(y) ) B(0,r) f( y j + a) ϕ(y). Using (3.4) and applying the dominated convergence theorem, we obtain kβ k B(0,r) 0 α(p) y k ϕ(y) (N + )j k f(a) B(0,r) ϕ(y) 1 B0,r) + y k div(y ϕ(y) ) (N + )j k 1 f(a) B(0,r) ϕ(y) + o(1). Therefore, for k > we deduce that α(p) = 0, which completes the proof. Proof of Theorem 1.4. Suppose that u is a solution of (1.1). We multiply (1.1) by u(x).(x a) and integrate over Ω, we obtain u 1 u(x).(x a) = N u(x), (3.5) λ f(x) u(x).(x a) = λ ( f(x).(x a) + Nf(x))u(x), (3.6) div(p(x) u(x)) u(x).(x a) = N p(x) u(x) 1 p(x).(x a) u(x) (3.7) 1 p(x)(x a).ν u ν. Ω Combining (3.5), (3.6) and (3.7), we obtain N p(x) u(x) 1 p(x).(x a) u(x) 1 p(x)(x a).ν u Ω ν = N u(x) λ ( f(x).(x a) + Nf(x))u(x). (3.8) Multiplying (1.1) by N u and integrating by parts, we obtain N p(x) u(x) = N u(x) + λ N f(x)u(x). (3.9) From (3.8) and (3.9), we obtain 1 p(x).(x a) u(x) 1 Ω p(x)(x a).ν u ν + λ f(x)u(x) = 0.

7 EJDE-016/08 NONHOMOGENEOUS ELLIPTIC EQUATIONS 7 Then λ > 1 p(x).(x a) u(x) f(x)u(x) α(p). (3.10) Hence the desired result is obtained. We begin by proving that 4. Existence of solutions inf J λ (u) = c < c + 1 u N N (p 0S) N/. (4.1) λ By some estimates in Brezis and Nirenberg [3], we have w 0 + Ru ε,a = w 0 + R u ε,a + R w 0 w 0 u ε,a + R 1 u 1 ε,a w 0 + o(ε (N )/ ), (4.) Put u ε,a = A 0 + O(ε N ), u ε,a = B + O(εN ), (4.3) S = S(1) = A 0 B /. (4.4) Lemma 4.1. Let p H 1 (Ω) C( Ω) satisfying (1.3) Then we have estimate p(x) u ε,a (x) p 0 A 0 + O(ε N ) if N < k, p 0 A 0 + A k ε k + o(ε k ) if N > k, p 0 A 0 + (N ) (β N + M)ω N ε N ln ε + o(ε N ln ε ) if N = k, where M is a positive constant. Proof. by calculations, ε N p(x) u ε,a (x) p(x) ξ a (x) p(x) ξa (x) x a = (ε + x a + (N ) ) N (ε + x a ) N p(x) ξ (N ) a (x)(x a) (ε + x a ) N 1. Suppose that ξ a 1 in B(a, r) with r > 0 small enough. So, we obtain ε N p(x) u ε,a (x) p(x) ξ a (x) p(x) ξa (x) x a = Ω\B(a,r) (ε + x a + (N ) ) N (ε + x a ) N p(x)ξ a (x) ξ a (x)(x a) (N ) (ε + x a ) N 1. Ω\B(a,r)

8 8 M. BOUCHEKIF, A. RIMOUCHE EJDE-016/08 Applying the dominated convergence theorem, p(x) u ε,a (x) = (N ) ε N p(x) ξa (x) x a (ε + x a ) N + O(εN ). Using expression (1.3), we obtain ε N (N ) p(x) u ε,a (x) = + Ω\ p 0 x a + β k x a k+ + θ(x) x a k+ (ε + x a ) N p(x) ξ a (x) x a (ε + x a ) N + O(εN ). Using again the definition of ξ a, and applying the dominated convergence theorem, we obtain ε N (N ) p(x) u ε,a (x) = p 0 + x a R (ε + x a ) N + β k N We distinguish three cases: Case 1. If k < N, ε N (N ) θ(x) x a k+ (ε + x a ) N + O(εN ). p(x) u ε,a (x) x a = p 0 R (ε + x a ) N + N [ RN β k x a k+ + (ε + x a ) N R N \ θ(x) x a k+ (ε + x a ) N x a k+ (ε + x a ) N β k x a k+ ] (ε + x a ) N + O(ε N ) Using the change of variable y = ε 1 (x a) and applying the dominated convergence theorem, we obtain (N ) p(x) u ε,a (x) = p 0 B 0 + ε k β k y k+ RN R (1 + y ) N + θ(a + εy) y k+ εk (1 + y ) N χ B(0, τ N ε ) + o(ε k ). Since θ(x) tends to 0 when x tends to a, this gives us p(x) u ε,a (x) = p 0 A 0 + β k A k ε k + o(ε k ). Case. If k > N, p(x) u ε,a (x) = p 0 A 0 + (N ) ε N [ (β k + θ(x)) x a k+ (ε + x a ) N (β k + θ(x)) x a k+ ] \Ω (ε + x a ) N

9 EJDE-016/08 NONHOMOGENEOUS ELLIPTIC EQUATIONS 9 + (N ) ε N By the change of variable y = x a, we obtain p(x) u ε,a (x) = p 0 A 0 + (N ) ε N + (N ) ε N θ(x) x a k+ (ε + x a ) N + O(εN ). B(0,τ) Put M := max θ(x) where θ(x) is given by (1.3). Then x Ω p(x) u ε,a (x) (β k + θ(a + y)) y k+ (ε + y ) N θ(a + y) y k+ (ε + y ) N + O(εN ). = p 0 A 0 + ε N (N ) y k+ (β k + M) B(0,τ) (ε + y ) N dy + O(εN ). Applying the dominated convergence theorem, p(x) u ε,a (x) = p 0 A 0 + O(ε N ). Case 3. If k = N, following the same previous steps, we obtain p(x) u ε,a (x) = p 0 A 0 + (N ) ε N θ(x) x a k+ (ε + x a ) N + (N ) ε N [ β N x a N (ε + x a ) N Therefore, + O(ε N ). Then p(x) uε,a(x) p(x) u ε,a (x) = p 0 A 0 + (N ) ε N + (N ) ε N \Ω (β N + θ(x)) x a N (ε + x a ) N θ(x) x a k+ (ε + x a ) N + O(εN ). β N x a N ] (ε + x a ) N p 0 A 0 + (N ) ε N x a N (β N + M) (ε + x a ) N + O(εN ). On the other hand ε N x a N (ε + x a ) N = ω Nε N τ 0 = 1 N ω Nε N r N 1 (ε + r ) N dr + O(εN ) τ 0 ((ε + r ) N ) (ε + r ) N dr + O(εN ),

10 10 M. BOUCHEKIF, A. RIMOUCHE EJDE-016/08 and ε N Therefore, p(x) u ε,a (x) p 0 A 0 + x a N (ε + x a ) N = 1 ω N ε N ln ε + o(ε N ln ε ), (4.5) (N ) (β N + M)ω N ε N ln ε + o(ε N ln ε ). Knowing that w 0 0, we set Ω Ω as a set of positive measure such that w 0 > 0 on Ω. Suppose that a Ω (otherwise replace w 0 by w 0 and f by f). Lemma 4.. For each R > 0 and k > N, there exists ε 0 = ε 0 (R, a) > 0 such that J λ (w 0 + Ru ε,a ) < c + 1 N (p 0S) N/, for all 0 < ε < ε 0. Proof. We have J λ (w 0 + Ru ε,a ) = 1 p w 0 + R p w 0 u ε,a + R p u ε,a 1 w 0 + Ru ε,a λ fw 0 λr fu ε,a. Using (4.), (4.3) and the fact that w 0 satisfies (1.1), we obtain J λ (w 0 + Ru ε,a ) c + R p u ε,a R A R 1 u 1 ε,a w 0 + o ( ε (N )/). Taking w = 0 the extension of w 0 by 0 outside of Ω, it follows that u 1 ε (N+)/ ε,a w 0 = w(x)ξ a (x) R (ε N + x a ) (N+)/ = ε (N )/ w(x)ξ a (x) 1 R ε N ψ(x ε ) N where ψ(x) = (1 + x ) (N+)/ L 1 (R N ). We deduce that R N w(x)ξ a (x) 1 ε N ψ(x ε ) D as ε 0. Then Consequently u 1 ε,a w 0 = ε (N )/ D + o(ε (N )/ ). J λ (w 0 + Ru ε,a ) c + R p u ε,a R B R 1 ε (N )/ D + o ( ε (N )/). Replacing p u ε,a by its value in (4.6), we obtain J λ (w 0 + Ru ε,a ) (4.6)

11 EJDE-016/08 NONHOMOGENEOUS ELLIPTIC EQUATIONS 11 c + R p 0A 0 R B ε (N )/ DR 1 + o(ε (N )/ ) if k > N, c + R p 0A 0 R B + β k A k ε k + o(ε k ) if k < N, c + R p ( 0A 0 R B ε (N )/ R β (N )/A (N )/ ) DR 1 + o(ε (N )/ ) if k = N. Using that the function R Φ(R) = R R B A 0 attains its maximum 1 N (p 0S) N/ at the point R 1 := ( A0 B )(N )/4, we obtain J λ (w 0 + Ru ε,a ) c + 1 N (p 0S) N/ ε (N )/ DR o(ε (N )/ ) if k > N, c + 1 N (p 0S) N/ + A k ε k + o(ε k ) if k < N (, c + 1 N (p 0S) N/ ε (N )/ R 1 β (N )/A (N )/ ) DR o(ε (N )/ ) if k = N. So for ε 0 = ε 0 (R, a) > 0 small enough, k > N or k = N and we conclude that for all 0 < ε < ε 0. β (N )/ > DR 3 1, B (N )/ J λ (w 0 + Ru ε,a ) < c + 1 N (p 0S) N/, (4.7) Proposition 4.3. Let {u n } N λ be a minimizing sequence such that: (a) J λ (u n ) c and (b) J λ (u n) 1 0. Then for all λ (0, Λ 0 /), {u n } admits a subsequence that converges strongly to a point w 1 in H such that w 1 N λ and J λ(w 1 ) = c. Proof. Let u H be such that u = 1. Then t + (u)u N λ and J λ (t + (u)u) = max t t m J λ (tu). The uniqueness of t + (u) and its extremal property give that u t + (u) is a continuous function. We put U 1 = {u = 0 or u H \ {0} : u < t + ( u u )}, U = {u H \ {0} : u > t + ( u u )}. Then H \ N λ = U 1 U and N + λ U 1. In particular w 0 U 1. As in [8], there exists R 0 > 0 and ε > 0 such that w 0 + R 0 u ε,a U. We put F = {h : [0, 1] H continuous, h(0) = w 0 and h(1) = w 0 + R 0 u ε,a }. It is clear that h : [0, 1] H with h(t) = w 0 + tr 0 u ε,a belongs to F. Thus by Lemma 4., we conclude that c 0 = inf h F max t [0,1] J λ(h(t)) < c + 1 N (p 0S) N/. (4.8)

12 1 M. BOUCHEKIF, A. RIMOUCHE EJDE-016/08 Since h(0) U 1, h(1) U and h is continuous, there exists t 0 ]0, 1[ such that h(t 0 ) N λ Hence c 0 c = inf J λ (u). (4.9) u N λ Applying again the Ekeland variational principle, we obtain a minimizing sequence (u n ) N λ such that (a) J λ(u n ) c and (b) J λ (u n) 1 0. Thus, we obtain a subsequence (u n ) such that u n w 1 strongly in H. This implies that w 1 is a critical point for J λ, w 1 N λ and J λ(w 1 ) = c. Proof of Theorem 1.5. From the facts that w 0 N + λ, w 1 N λ and N + λ N λ = for λ (0, Λ0 ), we deduce that problem (1.1) admits at least two distinct solutions in H. References [1] A. Ambrosetti, P. H. Rabinowitz; Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), [] H. Brezis, E. Lieb; A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), [3] H. Brezis, L. Nirenberg; A minimization problem with critical exponent and non zero data, in symmetry in nature, Scuola Norm. Sup. Pisa, (1989) [4] H. Brezis, L. Nirenberg; Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math., 36 (1983), [5] I. Ekeland; On the variational principle, J. Math. Anal. Appl., 47 (1974), [6] R. Hadiji, H. Yazidi; Problem with critical Sobolev exponent and with weight, Chinese Ann. Math, Series B, 3 (007), [7] A. Rimouche; Problème elliptique avec exposant critique de Sobolev et poids, Magister, Université de Tlemcen (01). [8] G. Tarantello; On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non linéaire, 9 (199), Mohammed Bouchekif Laboratoire Systèmes Dynamiques et Applications, Faculté des Sciences, Université de Tlemcen BP 119 Tlemcen 13000, Algérie address: m bouchekif@yahoo.fr Ali Rimouche (corresponding author) Laboratoire Systèmes Dynamiques et Applications, Faculté des Sciences, Université de Tlemcen BP 119 Tlemcen 13000, Algérie address: ali.rimouche@mail.univ-tlemcen.dz