Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential

Advances in Nonlinear Analysis

article

Sergio Rolando¹ 

[1] Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano bicocca

关键词: Semilinear elliptic PDE;    singular vanishing potential;    symmetry breaking;   

DOI  :  10.1515/anona-2017-0177

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

Many existence and nonexistence results are known for nonnegative radial solutions to the equation - △ ⁢ u + A | x | α ⁢ u = f ⁢ ( u )   in  ⁢ ℝ N , N ≥ 3 , A , α > 0 , u ∈ D 1 , 2 ⁢ ( ℝ N ) ∩ L 2 ⁢ ( ℝ N , | x | - α ⁢ d ⁢ x ) , -\triangle u+\frac{A}{|x|^{\alpha}}u=f(u)\quad\text{in }\mathbb{R}^{N},\,N\geq 3% ,\,A,\alpha>0,\,u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},|x|^{-% \alpha}\,dx), with the nonlinearities satisfying | f ( u ) | ≤ ( const . ) u p - 1 {|f(u)|\leq(\mathrm{const.})u^{p-1}} for some p > 2 {p>2} . The existence of nonradial solutions, by contrast, is known only for N ≥ 4 {N\geq 4} , α = 2 {\alpha=2} , f ⁢ ( u ) = u ( N + 2 ) / ( N - 2 ) {f(u)=u^{(N+2)/(N-2)}} and A large enough. Here we show that the above equation has multiple nonradial solutions as A → + ∞ {A\rightarrow+\infty} for N ≥ 4 {N\geq 4} , 2 / ( N - 1 ) < α < 2 ⁢ N - 2 {2/(N-1)<\alpha<2N-2} and α ≠ 2 {\alpha\neq 2} , with the nonlinearities satisfying suitable assumptions. Our argument essentially relies on the compact embeddings between some suitable functional spaces of symmetric functions, which yields the existence of nonnegative solutions of mountain-pass type, and the separation of the corresponding mountain-pass levels from the energy levels associated to radial solutions.

【 授权许可】

CC BY