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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
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