Resultados de existência, de não existência e de simetria de solução para o operador p-Laplaciano fracionário com múltiplas singularidades críticas e potencial de Hardy

Abstract

In this doctoral thesis we consider a problem involving the fractional p-Laplacian operator (−∆p) su −µ |u| p−2u |x| p s = |u| p ∗ s (β)−2u |x| β + |u| p ∗ s (α)−2u |x| α (x ∈ R N ) (0.7) where 0 < s < 1, 1 < p < +∞, N > sp, 0 < α < sp, 0 < β < sp, β 6= α, µ is a real parameter, and p ∗ s (α) = (p(N −α)/(N − p s) is the critical Hardy-Sobolev exponent; in particular, if α = 0 then p ∗ s (0) = p ∗ s = N p/(N − sp) is the critical Sobolev exponent. The fractional p-Laplacian operator is a nonlinear and nonlocal operator defined for differentiable functions by (−∆p) su(x) := 2 lim ²→0 + Z RN \B²(x) |u(x)−u(y)| p−2 (u(x)−u(y)) |x − y| N+sp d y (x ∈ R N ). (0.8) We prove that for the parameters in the above specified intervals and with 0 6 µ < µH := inf u∈D s,p (R N ) u6=0 [u] p s,p Z RN u p |x| p s d x , there exists a weak solution u ∈ D s,p (R N ) to problem (0.7). The function space where we look for solution is the fractional homogeneous Sobolev space D s,p (R N ) := n u ∈ L p ∗ s (R N ): [u]s,p < ∞o , where [u]s,p denotes the Gagliardo seminorm, u ∈C ∞ 0 (R N ) 7−→ [u]s,p := µÏ R2N |u(x)−u(y)| p |x − y| N+sp d x d y¶ 1 p . A fundamental step to prove the existence result to problem (0.7) is the proof of the independent result relative to the best Hardy constant, given by 1 K(µ,α) := inf u∈D s,p (R N ) u6=0 [u] p s,p −µ Z RN |u| p |x| p s d x ÃZ RN |u| p ∗ s (α) |x| α d x! p p ∗ s (α) , (0.9) which is achieved by a nontrivial function u ∈ D s,p (R N ), under the condition µ ∈ (0,µH ). In the case p = 2 the fractional 2-Laplacian operator defined in (0.8) is denoted by (−∆) su(x) := (−∆2) su(x). In this case, we consider the function space H s (R N ), defined as the closure of the space C ∞ 0 (R N ) with respect to the norm kukHs (RN ) := µZ RN |(−∆) s/2u| 2 d x¶ 1 2 = µÏ R2N |u(x)−u(y)| 2 |x − y| N+2s d x d y¶ 1 2 . We also show that if u ∈ H s (R N ) is a weak solution to problem (−∆) su −µ u |x| 2s = |u| q−2u + |u| 2 ∗ s (α)−2u |x| α (x ∈ R N ), (0.10) where 0 < s < 1, 0 < α < 2s < N, 2∗ s (α) = 2(N − α)/(N − 2s), µ is a real parameter and q 6= 2 ∗ s , then u ≡ 0. Therefore, problem (0.10) does not have nontrivial solution when q 6= 2 ∗ s . The proof of this non-existence result is an immediate consequence of a Pohozaev-type identity for problem (0.10) that we state in the following way: Suppose that u ∈ H s (R N ) is a weak solution to problem (0.10). Then the harmonic extension of u on the half-space R N+1 + , denoted by w = E(u), verifies the identity (N −2s) 2 Ï R N+1 + y 1−2s |∇w| 2 d x d y = 1 ks Z RN à NF(x,u)+ X N i=1 xi Z u 0 fxi (x,t)d t! d x, (0.11) where ks = Γ(s) 2 1−2sΓ(1− s) , u = w(·, 0), f (x,u) = µ u |x| 2s + |u| q−2u + |u| 2 ∗ s (α)−2u |x| α and F(x,s) = Z s 0 f (x,t)d t. Finally, still in the case p = 2 of the fractional Laplacian operator, let 0 6 µ < µ¯ := 2 2s Γ 2 ¡ N+2s 4 ¢ Γ 2 ¡ N−2s 4 ¢, where µ¯ is the best constant of the continuous embedding H s (R N ) ,→ L 2 (R N ,|x| −2s ). We prove that every positive solution u ∈ H s (R N ) to problem (−∆) su −µ u |x| 2s = |u| 2 ∗ s −2u + |u| 2 ∗ s (β)−2u |x| β (x ∈ R N ) (0.12) is radially symmetric and decreasing with respect to some point x0 ∈ R N , that is, for every positive solution to problem (0.12) there exists an strictly decreasing function v : (0,+∞) → (0,+∞) such that u(x) = v(r ), r = |x − x0|.