Multiple symmetric solutions of semilinear elliptic equations in unbounded symmetric domains (Q1369332)

Let \(\Omega\subset \mathbb{R}^n\), \(n\geq 3\), be a suitable unbounded domain, which contains the \(x_n\)-axis and is axially symmetric with respect to this axis. The authors look for solutions \(u\) of the Dirichlet problem \[ -\Delta u+u=\lambda b(x)|u|^{p-1}u+ c(x)|u|^{q-1}u \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega, \] having the same kind of symmetry as the domain. Therefore, also the smooth nonnegative bounded coefficients \(b\) and \(c\) are assumed to be axially symmetric. Furthermore, \(b(x)\) and \(c(x)\not\equiv 0\) have to tend to a positive constant and to zero, resp. The eigenvalue parameter \(\lambda\) has to be positive and the growth of the nonlinear terms is assumed to be superlinear and subcritical: \(1<p,q<(n+2)/(n-2)\). Under these assumptions the authors show existence of a positive solution for \(\lambda\) close to zero. If \(c(x)>0\) a.e. and \(\lambda\searrow 0\) they find an arbitrarily increasing number of mutually distinct solutions, thereby generalizing a paper by \textit{D.-M. Cao} [Ann. Inst. H. Poincaré, Anal. Nonl. 10, No. 6, 593-604 (1993; Zbl 0797.35039)].