Existence of positive nonradial solutions for nonlinear elliptic equations in annular domains

Source abstract

We study the existence of positive nonradial solutions of equation Δ u + f ( u ) = 0 \Delta u + f(u) = 0 in Ω a {\Omega _a} , u = 0 u = 0 on ∂ Ω a \partial {\Omega _a} , where Ω a = { x ∈ R n : a > | x | > 1 } {\Omega _a} = \{ x \in {\mathbb {R}^n}:a > |x| > 1\} is an annulus in R n {\mathbb {R}^n} , n ≥ 2 n \geq 2 , and f f is positive and superlinear at both 0 0 and ∞ \infty . We use a bifurcation method to show that there is a nonradial bifurcation with mode k k at a k ∈ ( 0 , 1 ) {a_k} \in (0,1) for any positive integer k k if f f is subcritical and for large k k if f f is supercritical. When f f is subcritical, then a Nehari-type variational method can be used to prove that there exists a ∗ ∈ ( 0 , 1 ) {a^{\ast } } \in (0,1) such that for any a ∈ ( a ∗ , 1 ) a \in ({a^{\ast } },1) , the equation has a nonradial solution on Ω a {\Omega _a} .