Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent
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1 Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent Yūki Naito a and Tokushi Sato b a Department of Mathematics, Ehime University, Matsuyama , Japan b Mathematical Institute, Tohoku University, Sendai , Japan Let R N be a bounded domain with smooth boundary with N 3. We consider the existence of multiple positive solutions of the following semilinear elliptic equations (1.1) λ Δu + κu = u p + λf(x) in, on, where λ>0 is a parameter, κ R is a constant, p =(N +2)/(N 2) is the critical Sobolev exponent, and f(x) is a non-homogeneous perturbation satisfying f H 1 () and f 0, f 0 in. Let κ 1 be the first eigenvalue of Δ with zero Dirichlet condition on. Since (1.1) λ has no positive solution if κ κ 1 (see Remark 1 below), we will consider the case κ> κ 1. Let us recall the results for the homogeneous semilinear elliptic problem involving critical Sobolev exponent (1.2) Δu + κu = u p in, on. It is well known that (1.2) admits no nontrivial solution for each κ 0 via Pohozaev identity, provided that is star-shaped. On the other hand, when κ<0, Brezis and Nirenberg [2] showed the following remarkable phenomenon: (i) if N 4, then for every κ ( κ 1, 0), there exists a positive solution; (ii) if N = 3 and is a ball, then there exists a positive solution if and only if κ ( κ 1, κ 1 /4). The results in [2] show that the space dimension plays a fundamental role when one seeks solutions of (1.2). Similar phenomena have been shown for more general problems 162
2 involving the p-laplacian [7, 6, 8], higher order elliptic problems [15, 16, 11], elliptic problems with Hardy potential [12], and quasilinear elliptic equations [1]. For the non-homogeneous problem, let us first consider the problem (1.1) λ with κ =0; Δu = u p + λf(x) in, (1.3) on, Tarantello [18] investigated suitable minimization and minimax principles of mountain pass-type, and showed that (1.3) has at least two positive solution for λ>0sufficiently small. The existence of two nontrivial solutions was shown by Cao and Zhou [3] for more general problem Δu = u p + g(x, u)+λf(x) in, on, where g(x, u) is a suitable lower-order perturbation of u p. The achievement, existence of at least two nontrivial solutions for λ>0 sufficiently small, has been extended to the problems involving the p-laplacian by Chabrowski [4] and Zhou [19], and to more general problems by Squassina [17]. In this paper we investigate the multiplicity of positive solutions to the problem (1.1) λ with κ> κ 1, and show that the problem exhibits the phenomenon depending on the space dimension. Precisely, we find that, when κ>0, the situation is drastically different between the cases N =3, 4, 5 and N 6. We show first the existence of the extremal value of λ for the existence of solutions to (1.1) λ. We call a positive minimal solution u λ of (1.1) λ,ifu λ satisfies u λ u in for any positive solution u of (1.1) λ. Theorem 1. Let κ> κ 1. Then there exists λ (0, ) such that (i) if 0 <λ<λ then the problem (1.1) λ has a positive minimal solution u λ H0 1(). Furthermore, if 0 <λ<ˆλ <λ then u λ <uˆλ a.e. in, and u λ 0 in H0 1 () as λ 0; (ii) if λ>λ then the problem (1.1) λ has no positive solution u H 1 0 (). Remark 1. There is no positive solution of (1.1) λ with κ κ 1. In fact, assume to the contrary that there exists a positive solution u of (1.1) λ with κ κ 1. Let ϕ 1 be the eigenfunction corresponding to the first eigenvalue κ 1 with ϕ 1 > 0 on. Then we have 0= ( u ϕ 1 κ 1 uϕ 1 )dx ( u ϕ 1 + κuϕ 1 )dx = (u p ϕ 1 + λfϕ 1 )dx >
3 This is a contradiction. Let us consider the existence of solutions of (1.1) λ at the extremal value λ = λ, so-called extremal solutions. For the semilinear elliptic equation Δu = λf (u) in with Dirichlet condition u = 0 on, the existence of extremal solutions has been well studied, see e.g., [9, 13, 5, 14], and it is well known that the existence and nonexistence of the extremal solutions depend strongly on the dimension, domain, and nonlinearity. For non-homogeneous problem (1.1) λ, we obtain the following. Theorem 2. If λ = λ then the problem (1.1) λ has a unique positive solution in H 1 0 (). Let us consider the existence and nonexistence of second positive solutions to (1.1) λ for 0 <λ<λ. Theorem 3. Assume that either (i) or (ii) holds. (i) κ ( κ 1, 0] and N 3; (ii) κ>0 and N =3, 4, 5. If 0 <λ<λ then (1.1) λ has a positive solution u λ H 1 0() satisfying u λ >u λ. Theorem 4. Assume that κ>0 and N 6. (i) There exists λ = λ (κ) (0, λ) such that if λ λ<λ then the problem (1.1) λ has a positive solution u λ H 1 0() satisfying u λ >u λ. (ii) Let ={x R N : x <R} with some R>0, and let f = f( x ) be radially symmetric about the origin. Assume that f C α ([0,R]) with some 0 <α<1, and f(r) is nonincreasing in r (0,R). Then there exists λ (0,λ ) such that (1.1) λ has a unique positive solution u λ for λ (0,λ ]. Remark 2. (i) In Theorem 3, the existence of non-minimal solution was shown by Tarantello [18] in the case κ = 0. However, in the case κ>0, the result for the existence of non-minimal solutions depends on the space dimension by Theorem 4 (ii), and seems to be new. (ii) In Theorem 4, it is an open question whether the problem (1.1) λ has a nonminimal solution for λ <λ<λ. (iii) When one of the conditions in Theorem 4 (ii) does not hold, for example, when f is not radially symmetric, we do not know the uniqueness of the positive solution of (1.1) λ for λ>0 sufficiently small. 164
4 In the proof of Theorem 1, we will employ the bifurcation results and the comparison argument for solutions of (1.1) λ to obtain the minimal solutions. We will prove Theorem 2 by establishing a priori bound for the solutions of (1.1) λ at λ = λ. In order to find a second positive solution of (1.1) λ, we introduce the problem (1.4) Δv + κv =(v + u λ ) p u p λ in, v H 1 0(), where u λ is the minimal positive solution of (1.1) λ for λ (0, λ) obtained in Theorem 1. In fact, assume that (1.4) has a positive solution v, and put u λ = v + u λ. Then u λ H0 1() and solves (1.1) λ and satisfies u λ >u λ in. In the proof of Theorem 3, we will show the existence of solutions of (1.4) by using a variational method. To this end we define the corresponding variational functional of (1.4) by I κ (v) = 1 ( v 2 + κv 2) dx G(v, u 2 λ )dx R N for v H0 1 (), where G(t, s) = 1 p +1 (t + + s) p+1 1 p +1 sp+1 s p t +. It is easy to see that I κ : H0 1() R is C1 and the critical point v 0 H0 1 () satisfies ( v 0 ψ + κv 0 ψ + g(v 0,u λ )ψ) dx =0 for any ψ H 1 0 (), where g(t, s) =(t + + s) p s p. Denote by S the best Sobolev constant of the embedding H0 1() Lp+1 (), which is given by u 2 dx S = inf ( ) u H0 1()\{0} 2/(p+1). u p+1 dx We will obtain Theorem 3 as a consequence of the following two propositions. Proposition 1. Let λ (0,λ ). Assume that there exists v 0 H 1 0() with v 0 0, v 0 0 such that (1.5) sup I κ (tv 0 ) < 1 t>0 N SN/2. Then there exists a positive solution v H0 1 () of (1.4). Proposition 2. Assume that either (i) or (ii) holds. (i) κ ( κ 1, 0] and N 3; (ii) κ>0 and N =3, 4,
5 Then there exists a positive function v 0 H 1 0() such that (1.5) holds. In the proof of Proposition 1, we will derive some estimates to establish inequalities relating certain minimizing sequences. In order to prove Proposition 2, for ε>0, we will set φ(x) u ε (x) = (ε + x 2 ), (N 2)/2 where φ C0 (R N ), 0 φ 1, is a cut off function, and will show that (1.5) holds with v 0 = u ε for sufficiently small ε>0. In the proof of Theorem 4 (ii), we will verify the nonexistence of positive solutions of (1.4) in the radial case by the Pohozaev type argument for the associated ODE. In fact, by [10], the solution v of (1.4) must be radially symmetric, and v = v(r), r = x, satisfies the problem of the following ordinary differential equation (r N 1 v r ) r κr N 1 v + r N 1 g(v, u λ )=0, 0 <r<r, (1.6) v r (0) = v(r) =0. For the solution v to (1.6), we will obtain the following Pohozaev type identity: R [ ] 2N r N 1 0 N 2 G(u, u λ) g(u, u λ )u dr + 2 R r N G s (u, u N 2 λ )u λ dr 0 + 2κ N 2 0 r N 1 u 2 dr = 1 N 2 RN v r (R) 2. In the proofs of Theorems 2, 3 and 4, the results concerning the eigenvalue problems to the linearized equations around the minimal solutions play a crucial role. Δφ + φ = μp(u λ ) p 1 φ in, φ H 1 0 (). References [1] G. Arioli and F. Gazzola, Quasilinear elliptic equations at critical growth, NoDEA Nonlinear Differential Equations Appl. 5 (1998) [2] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math. 36 (1983) [3] D.-M. Cao and H.-S. Zhou, On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents, Z. Angew. Math. Phys. 47 (1996)
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