Multiplicity of solutions for a fourth order problem with exponential nonlinearity
Consider for \(\lambda \geq 0,\) the boundary value problem \[ \Delta^2 u = \lambda e^u\text{ in the unit ball }B, \] \[ u=0, \partial u /\partial n =0\text{ on }\partial B \] (\(n\) is the exterior unit normal vector). First, it is proved that if \(N \geq 5\), there exists a unique \(\lambda_S>0\) such that the given problem with \(\lambda=\lambda_S\) admits a radial singular solution; moreover, this solution is unique in the class of radial solutions. In order to state the main result, one has to consider the largest value of \(\lambda \geq 0\), denoted by \(\lambda^*,\) such that the given problem has a radial classical solution. It is then proved that if \(5 \leq N \leq 12\) then \(\lambda_S < \lambda^*\) and with \(\lambda = \lambda^*\) there exist infinitely many regular radial solutions. For \(\lambda \neq \lambda_S\) there exist a finite number of regular radial solutions, and this number goes to infinity as \(\lambda \to \lambda_S\). Finally, in the case \(N\geq 13\) it is proved that \(\lambda_S = \lambda^*\) and for all \(0 < \lambda < \lambda^*\) there exists a unique regular radial solution. For \(\lambda = \lambda^*\) there exists a unique singular radial solution. Variants of the above results are performed in case of Navier boundary conditions. For the proofs of the multiplicity results, the ODE arising from the given problem when searching radial solutions is trasformed into a suitable first-order 4-dimensional autonomous system; then phase space analysis methods are used.
scientific article; zbMATH DE number 5637048
scientific article