Multiple solutions for prescribed boundary value problem (Q698857)

This paper deals with the semilinear elliptic boundary value problem \[ \begin{gathered} -\Delta u= f(x,u)\quad\text{in }\Omega,\\ u= g\quad\text{on }\partial\Omega\end{gathered} \] and the corresponding Bolza problem \(\ddot x+\nabla V(t,x)= 0\), \(0\leq t\leq T\), \(x(0)= x_0\), \(x(T)= x_1\), where \(\Omega\) is a bounded subset in \(\mathbb R^N\) with \(C^2\) boundary and \(g\) is a given continuous function on the boundary of \(\Omega\). Note that the most results in the literature are stated for the cases \(f(x,u)= |u|^{p-1}u+ h(x)\) or the time-independent systems \(V(t,x)= \widetilde V(x)+ h(t)\cdot x\), so the main goal of the author is to generalize the above-mentioned cases to the \(x\)-variable dependent variable nonlinearity \(f(x,u)\) and the fully time-dependent systems.