Existence and nonexistence results for a weighted elliptic equation in exterior domains
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Existence and nonexistence results for a weighted elliptic equation in exterior domains Zongming Guo, Xia Huang and Dong Ye
Abstract. We consider positive solutions to the weighted elliptic problem −div(|x|θ ∇u) = |x| up in RN \B,
u = 0 on ∂B,
where B is the standard unit ball of RN . We give a complete answer for the existence question for N := N + θ > 2 and , the only nonnegative p > 0. In particular, for N > 2 and τ := − θ > −2, it is shown that for 0 < p ≤ ps := N N+2+2τ −2 0. For θ = = 0, the classical elliptic equation N
− Δu = up in RN \B,
u = 0 on ∂B
has been studied intensively, see, for instance, [1–4,7,11,13,15] and the references therein. It is well known from [1,2,7,11] that (1.2) does not admit any positive solution provided 0 < p ≤ for N ≥ 3; or p > 0 for N ≤ 2. It is also well known from [3] that for any supercritical exponent p > and N ≥ 3, the problem (1.2) admits a unique positive radial solution u satisfying as |x| → ∞. u(x) = O |x|2−N
(1.2) N N −2 N +2 N −2
N +2 For NN−2 < p ≤ N −2 , N ≥ 3, Theorem 2.2 in [13] showed that (1.2) does not admit any positive radial solution. However, the following natural question remained open:
Is there any nonradial positive solution of (1.2) for
N +2 N
always infinitely many positive solutions such that 2 as |x| → ∞. u(x) = O |x|− p−1
N +2 N −2 ,
there exist
Secondly, by Proposition 6.1 in [15], Zhang showed that for any bounded Lipschitz domain Ω ⊂ RN N + (N ≥ 3), p > N −2 , the problem −Δu = |x| up in RN \Ω,
u = f on ∂Ω
admits a positive solution, if f is a nontrivial nonnegative function in L∞ (∂Ω) and f ∞ is small enough. In this paper, we study the existence of positive solutions to the more general equation (1.1), under the following basic assumption: N := N + θ > 2,
τ := − θ > −2.
(1.3)
The elliptic problem like −div(a(x)∇u) = b(x)f (x, u) can be used to modeling some physical phenomena related to the equilibrium of continuous media, see [5]. We will show that N + 2 + 2τ ps := N − 2 is effectively the critical exponent for (1.1). In particular, when θ = = 0, we prove that the answer to the question (∗) is indeed negative. Our main results are the follows. Theorem 1.1. Let N > 2 and τ > −2. The only nonnegative solution of (1.1) is up ≡ 0 provided 0 < p ≤ ps . The interesting feature here is that we do not require any behavior at infinity or any symmetry assumption for u. Note that by scaling, our results hold obviously on RN \BR , ∀ R > 0. +τ For p ≤ N N −2 , we show a more general nonexistence result inspired by [1,2], see Theorem 4.1. For 1 < p ≤ ps , we apply the moving-sphere method to get a monotonicity property for the eventual positive solution, and then, we conclude by contradiction with integral estimate and stability argument. Comparing to Zhang’s result mentioned above, the homogenous Dirichlet boundary condition plays a crucial role +τ for our nonexistence result whenever N N −2 < p ≤ ps . Theorem 1.2. Let N > 2, τ > −2 and p >
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