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On the nonhomogeneous Neumann problem with weight and with critical nonlinearity in the boundary. (English) Zbl 1133.35050

For a given \(N\geq 3\), a bounded domain \(\Omega\subseteq\mathbb{R}^N\) with smooth boundary, \(p\in H^1(\Omega)\cap C(\overline{\Omega})\), \(Q\in C(\partial\Omega)\), \(p,Q\) being positive functions, \(f\in (H^{-1}(\Omega)\cap C(\overline{\Omega})) \backslash\{0\}\), and a real parameter \(\lambda\), the author considers weak solutions to the problem
\[ \begin{cases}-\text{div}(p(x)\nabla u)= \lambda u+f(x),&\text{in }\Omega,\\ \frac{\partial u}{\partial\nu}=Q(x)|u|^{q-2}u,&\text{on }\partial\Omega, \end{cases}\tag{1} \]
where \(\nu\) is the outer unit normal to \(\partial\Omega\). The exponent \(q=\frac{2(N-1)}{N-2}\) is critical for the trace embedding of \(H^1(\Omega)\) into \(L^q(\partial\Omega)\). The general assumptions for the existence results are the following: to exclude solutions that vanish identically on \(\partial\Omega\) it is always assumed that the problem
\[ \begin{cases}-\text{div}(p(x)\nabla u)= \lambda u+f(x),&\text{in }\Omega,\\ \frac{\partial u}{\partial\nu}=u=0,&\text{on }\partial\Omega, \end{cases}\tag{2} \]
has no solution. Moreover, it is assumed that there is a point \(x_0\in\partial\Omega\) such that \(\Omega\) locally lies on one side of the tangent hyperspace of \(\partial\Omega\) at \(x_0\), that the mean curvature with respect to \(\nu\) is positive in \(x_0\); the function \(p/Q^{N-2}\) achieves its minimum over \(\partial\Omega\) at \(x_0\), that \(p\) and \(Q\) are differentiable at \(x_0\) with derivative equals to \(0\).
Theorem 1: Suppose that \(\lambda<0\) and that \(\|f\|_{H^{-1}(\Omega)}\) is sufficiently small (but not \(0\)). Under additional regularity assumptions on \(p\), \(Q\), and \(f\) Eq. (1) has at least two weak solutions. Denote by \(0=\lambda_1<\lambda_2<\lambda_3<\dots\) the distinct eigenvalues of \(L:=\operatorname{div}(p\nabla\,\cdot)\) with respect to Neumann boundary condition. Fix \(k\geq 2\) and denote by \(E_{k}^-\) the generalized eigenspace of \(L\) corresponding to \(\{\lambda_1,\lambda_2,\dots,\lambda_{k-1}\}\). Let \(E_k^+\) be its orthogonal complement in \(H^1(\Omega)\) (the author does not specify the used scalar product; most likely, it is the \(H^1\)-product with \(p\)-weight in the gradient term).
Theorem 2: Suppose that \(\lambda\in(\lambda_{k-1},\lambda_k)\) and \(f\in E_k^+\). If \(N=3,4\), in addition suppose that \(f(x_0)\neq0\). Then (1) has a weak solution. The author mentions that (2) has no solutions for a function \(f\) with constant sign on \(\Omega\), but his argument is unclear. Moreover, the latter property never holds under the conditions of Theorem 2. Hence, no information is given about how large the set of data is that satisfies the conditions of Theorem 2.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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