Existence and nonexistence of nontrivial solutions for degenerate elliptic equations in a solid torus

Abstract

In this paper, we study the existence and non existence of nontrivial solutions to the Dirichlet boundary value problem for the following degenerate elliptic equation

$$\begin{aligned} -div (s^{\alpha } \nabla u)=s^{\ell }\left| u\right| ^{p-1} u&\text { in } T(R,a), \end{aligned}$$

(1)

$$\begin{aligned} {u=0} \text { on } \partial T(R,a) \end{aligned}$$

(2)

where

$$\begin{aligned} T(R,a)&=\left\{ \left( x_1,x_2,x_3\right) \in {\mathbb {R}}^3: x_3 ^{2}+(r-R)^2< a^2 \right\} ,\\ r&=\sqrt{x_1^2+x_2^2}, 0<a<R \end{aligned}$$

is a solid torus in ${\mathbb {R}}^3, s=\sqrt{x_3^2+(r-R)^2}$ and $\alpha \ge 0, \ell \ge -2,1< p<\infty .$ The main results show that when p is small then the problem has a nontrivial positive solution. On the other hand, when p is big there is not a nontrivial soltion. To obtain the existence of nontrivial solutions we use the variational method and the symmetric property of the torus. To obtain the nonexistence of nontrivial solutions we derive a Pohozaev’s type identity and then apply it.