Weak perturbations of the \(p\)-Laplacian (Q2349166)

The asymptotic expansion of the lowest eigenvalue of the \(p\)-Laplacian in \(\mathbb{R}^{d}\) perturbed by a weakly coupled potential \(V\in L^{1}(\mathbb{R}^{d})\) with \(\int_{\mathbb{R}^{d}} {{V(x)}\text{d}x} >0\) is studied using variational methods. More precisely, let \(Q_{V}[u] = \int_{\mathbb{R}^{d}}{( | \nabla u | ^{p} - V | u | ^{p})} \text{ d}x\) be the functional associated to the \(p\)-Laplacian and \( \lambda(V)= \inf_{u \in W^{1,p}(\mathbb{R}^{d})} \frac{Q_{V}[u]}{\int_{\mathbb{R}^{d}} {| u | ^{p}} \text{ d}x}\). The leading term of the asymptotic expansion of \(\lambda(\alpha V)\) when \(\alpha \rightarrow 0\), \(\alpha >0\) is computed. Two cases are considered. If \(p>d \geq 1\), the leading term depends (besides \(\alpha, p\) and \(d\)) on the potential \(V\) and on the optimal constant in the Sobolev interpolation inequality. If \(p=d\), then one also has to assume that \(V\in L^{q}(\mathbb{R}^{d})\) for some \(q>1\). In this case, \(\lambda(\alpha V)\) vanishes exponentially fast when \(\alpha \rightarrow 0\), so the leading term for the asymptotic expansion of \(\log | \lambda(\alpha V) |\) is obtained.