Boundary value problems for semilinear Schrödinger equations with singular potentials and measure data
The authors consider the boundary value problem with measure data in a bounded \(C^2\)-smooth domain \(\Omega\) in \(\mathbb{R}^N\), \(N\ge 3\) for the semilinear equation\N\[\N\begin{aligned}\N-\Delta u-Vu +f(u)&=\tau\quad \text{in }\Omega\\\N\operatorname{tr}_V u&=\nu \quad\text{on }\partial\Omega\,.\N\end{aligned}\tag{1}\N\]\NHere it is assumed that \(V\in C^1(\Omega)\) satisfies \(|V(x)| \leq \bar a \operatorname{dist}(x,\partial\Omega)^{-2}\) in \(\Omega\) for some \(\bar{a}>0\) and \(\gamma_-<1<\gamma_+\), where \(\gamma_-:=\inf E_0\), \(\gamma_+:=\sup E_0\) and \(E_0:=\{\gamma: \exists u_\gamma>0\text{ such that }L_{\gamma V} (u_\gamma)=0\}\).\NThen \(L_{V}:=\Delta + V\) has a (minimal) ground state \(\Phi_V\) in \(\Omega\) and in addition, \(-L_V\) admits a (minimal) Green function \(G_V\) and Martin kernel \(K_V\).\NThe function \(f\in C(R)\) is assumed monotonely increasing with \(f(0)=0\) and \(\operatorname{tr}_V u\) denotes the measure boundary trace of \(u\) associated with \(L_V\). Furthermore, \(\nu\in\mathfrak{M}(\partial \Omega)\) and \(\tau\in\mathfrak{M}(\Omega;\Phi_V)\).\N\NHere \(\mathfrak{M}(\partial \Omega)\) denotes the space of finite Borel measures on \(\partial \Omega\) and \(\mathfrak{M}(\Omega;\Phi_V)\) the space of real Borel measures \(\tau\) on \(\Omega\) such that \(\int_\Omega \Phi_V d |\tau|<\infty\).\NBy convention \(\mathfrak{M}_+(\partial\Omega)\) and \(\mathfrak{M}_+(\Omega;\Phi_V)\) denote the positive cones of these spaces respectively.\N\NThe authors prove the existence for the boundary value problem (1) in the sense of reduced measures for couples \((\tau,\nu)\) of positive measures. Then, such result is extended to the case of signed measures \(\tau\) and \(\nu\), which may require more complex tools and techniques.\NThe idea of `reduced measure' was introduced for \(V=0\) in [\textit{H. Brezis} et al., Ann. Math. Stud. 163, 55--109 (2007; Zbl 1151.35034)].\NThe approach is to provide a reduction process that converges to the `good' part of \(\tau\) and \(\nu\) when (1) has no solution.\NIn the setting of signed measures, some theorems in the article present new results, even in the absence of a potential, than [\textit{H. Brezis} and \textit{A. C. Ponce}, J. Funct. Anal. 229, No. 1, 95--120 (2005; Zbl 1081.31007)].
scientific article; zbMATH DE number 7932311
scientific article