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Concentration and dynamic system of solutions for semilinear elliptic equations. (English) Zbl 1109.35317

The author deals with the semilinear elliptic equation \[ \begin{cases} -\Delta_x u+ u= u| u|^{p-2}\quad \text{in }\Omega,\\ u\in H^1_0(\Omega),\end{cases}\tag{1} \] where \(\Omega\) is a domain in \(\mathbb{R}^N\), \(N\geq 2\), \(2^*= {2N\over N-2}\) for \(N\geq 3\) and \(2^*=\infty\) for \(N= 2\), \(2< p< 2^*\). Using the Palais-Smale theory the author presents the concentration and dynamic system of solutions. Moreover, the author proves that the equation (1) in axially symmetric bounded domain has three positive solution.

MSC:

35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J60 Nonlinear elliptic equations
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