## Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli.(English)Zbl 0878.35043

The existence of many nonequivalent nonradial solutions of the problem $\Delta u+u^p=0,\;u>0\text{ in } \Omega_R, \quad u=0 \text{ on } \partial\Omega_R \tag{1}$ where $$\Omega_R\equiv \{x\in\mathbb{R}^n |R-1\leq |x|\leq R+1\}$$, $$1<p< (h+2)/(n-2)$$ for $$n\geq 3$$, $$1<p<\infty$$ for $$n=2$$. It is shown that for $$n=3$$ the number of nonequivalent nonradial solutions of (1) tends to infinity as $$R\to\infty$$. A constrained minimization problem is formulated to prove the existence of minimal energy solutions of (1) and to prove that for large $$R$$ the minimal energy solutions of (1) in various symmetry classes have the same energy levels.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations

### Keywords:

constrained minimization; minimal energy solution
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### References:

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