Exact multiplicity results for variational inequalities involving pointwise constraints on the derivatives
Let \(\Omega\) be a bounded domain in \(\mathbb R^ n\), let \(\psi \in H^1_0(\Omega)\) and let \(K_\psi\) denote the set of all \(u\in H^1_0(\Omega)\) such that \(\Delta u \leq \Delta\psi\) in the weak sense. The authors examine the problem of finding an element \(u\in K_\psi\) such that \[ \int_{\Omega} [Du\cdot D(u-v)-g(x,u)(v-u)+h(u-v)]\,dx \geq 0 \] for all \(v \in K_\psi\) for given functions \(g\) and \(h\). If \(K_\psi\) is the set of \(u \in H^1_0\) such that \(u \geq \psi\) in \(\Omega\), then this problem is the familiar obstacle problem, but the constraint on the Laplacian of the solution leads to rather different behavior. In particular, the current problem behaves like the corresponding elliptic boundary value problem \[ \Delta u+g(u)=h \text{ in } \Omega, \quad u=0 \text{ on } \partial\Omega. \] If \(\lambda_j\) denotes the \(j\)th eigenvalue of the Dirichlet Laplacian on \(\Omega\), then, for the boundary value problem, the number of solutions depends on the behavior of \(g(x,t)/t\) as \(t\to\pm\infty\). Here, the behavior as \(t\to-\infty\) is replaced by the assumption that the solution lies in \(K_\psi\). Specifically, if \(\lim_{t\to\infty} g(x,t)/t >\lambda_1\) and if \(e_1\) is the normalized eigenfunction of the Dirichlet Laplacian with eigenvalue \(\lambda_1\), then, for any \(\tilde h \in L^2\), there is a number \(\tau_0\) such that the original problem with \(h=\tilde h +\tau e_1\) has no solution if \(\tau<\tau_0\), at least one solution if \(\tau=\tau_0\), and at least two solutions if \(\tau>\tau_0\).
scientific article; zbMATH DE number 1880106
scientific article