Anoop, Thazhe Veetil; Biswas, Nirjan On global bifurcation for the nonlinear Steklov problems. (English) Zbl 1495.35020 Topol. Methods Nonlinear Anal. 58, No. 2, 731-763 (2021). The authors consider the following nonlinear Steklov bifurcation problem: \[ \begin{cases} -\Delta_p\phi =0 & \mbox{in}\;\; \Omega, \\ |\nabla \phi |^{p-2} \displaystyle\frac{\partial \phi }{\partial \nu} = \lambda (g|\phi |^{p-2}\phi + fr(\phi) ) & \mbox{on}\;\; \partial \Omega, \end{cases}\tag{1.1} \] where \(\Omega\) is an open bounded Lipschitz domain in \(\mathbb{R}^N\) (\(N \ge 2\)) with the boundary \(\partial \Omega\), \(p\in (1,\infty)\), \(-\Delta_p(\phi )=\operatorname{div}(|\nabla \phi |^{p-2}\nabla\phi )\), \(f, g \in L^1(\partial \Omega)\) are indefinite weights functions and \(r\in C(\mathbb{R})\) satisfying \(r(0) = 0\). A function \(\phi \in W^{1,p}(\Omega)\) is said to be a solution to (1.1) if \[ \int_\Omega |\nabla \phi |^{p-2}\nabla \phi \cdot \nabla v\, dx = \lambda \int_{\partial\Omega} (g|\phi |^{p-2}\phi v + fr(\phi )v)\, d\sigma\,,\tag{1.2} \] for all \(v\in W^{1,p}(\Omega)\). They say a real number \(\lambda\) is a bifurcation point of (1.1) if there exists a sequence \(\{(\lambda_n, \phi_n)\}\) of nontrivial weak solutions to (1.1) such that \(\lambda_n \to \lambda\) and \(\phi_n \to 0\) in \(W^{1,p}(\Omega)\) as \(n\to \infty\).To study the bifurcation problem (1.1), the authors consider the nonlinear eigenvalue problem: \[ \begin{cases} -\Delta_p\phi =0 & \mbox{in}\;\; \Omega, \\ |\nabla \phi |^{p-2} \displaystyle\frac{\partial \phi }{\partial \nu} = \lambda g|\phi |^{p-2}\phi & \mbox{on}\;\; \partial \Omega.\end{cases} \tag{1.5} \] They say an eigenvalue \( \lambda\) is principal if there exists an eigenfunction of (1.5) corresponding to \( \lambda\) that does not change its sign in \(\overline\Omega\). For \(1 \le d < \infty\), the authors denote:● \(\mathcal{F}_d :=\) closure of \(C^1(\partial \Omega)\) in the Lorentz space \(L^{d,\infty}(\partial \Omega)\),● \(\mathcal{G}_d :=\) closure of \(C^1(\partial \Omega)\) in the Lorentz-Zygmund space \(L^{d,\infty;}N(\partial \Omega)\) and prove{Theorem 1.1.} Let \(p\in (1,\infty)\) and \(N \ge p\). Let \(g^+ \not\equiv 0\), \(\int_{\partial\Omega} g < 0\) and \[ g\in \begin{cases} \mathcal{F}_{(N-1)/(p-1)}& \mbox{for}\;\; N > p\,,\\ \mathcal{G}_1 & \mbox{for}\;\; N = p\,. \end{cases} \] Then \[ \lambda_1 = \inf\left\{\int_\Omega |\nabla \phi |^p : \phi \in W^{1,p}(\Omega), \int_{\partial \Omega} g|\phi |^p = 1\right\} \] is the unique positive principal eigenvalue of (1.5). Furthermore, \(\lambda_1\) is simple and isolated.Further the authors establish the existence of a continuum that bifurcates from (\(\lambda_1, 0)\). They consider the following set \[ \mathcal{S}= \{(\lambda,\phi)\in \mathbb{R}\times W^{1,p}(\Omega) :(\lambda,\phi) \;\mbox{is a solution to (1.1) and}\; \phi \not\equiv 0\} \] and say \(\mathcal{C}\subset \mathcal{S}\) is a continuum of nontrivial solutions to (1.1) if it is connected in \(\mathbb{R}\times W^{1,p}(\Omega)\).For \(p\in (1,\infty)\) and \(g\) as in Theorem 1.1, depending on the dimension the authors assume:H1(a) \(\lim\limits_{|s|\to 0} |r(s)|/|s|^{p-1}=0\) and \[ |r(s)| \le C|s|^{\gamma-1}\;\;\mbox{for some}\;\;\gamma\in\left( 1, \frac{p(N - 1)}{N - p}\right). \]H1(b) \(g\in \mathcal{F}_{(N-1)/(p-1)}\) and \[ f \in \begin{cases} \mathcal{F}_{\tilde{p}} & \mbox{if}\;\; \gamma \ge p,\;\;\mbox{where}\;\; \frac{1}{\tilde p} + \frac{\gamma (N - p)}{p(N - 1)}= 1, \\ \mathcal{F}_{(N-1)/(p-1)} & \mbox{if}\;\; \gamma < p. \end{cases} \]H2(a) \(\lim\limits_{|s|\to 0} |r(s)|/|s|^{N-1} = 0\) and \(|r(s)| \le C|s|^{\gamma-1}\) for some \(\gamma\in (1,\infty)\).H2(b) \(g\in \mathcal{G}_1\), \(f\in \mathcal{G}_d\) with \(d > 1\).Then the authors prove the following theorems.{Theorem 1.2}. Let \(p\in (1,\infty)\). Assume that \(r, g\) and \(f\) satisfy (H1) for \(N > p\) and satisfy (H2) for \(N = p\). Then \(\lambda_1\) is a bifurcation point of (1.1). Moreover, there exists a continuum of nontrivial solutions \(\mathcal{C}\) of (1.1) such that \((\lambda_1, 0)\in \overline{\mathcal{C}}\) and either (a) \(\mathcal{C}\) is unbounded, or(b) \(\mathcal{C}\) contains the point \((\lambda, 0)\), where \(\lambda\) is an eigenvalue of (1.5) and \(\lambda \not= \lambda_1\). Reviewer: Petr Tomiczek (Plzeň) MSC: 35B32 Bifurcations in context of PDEs 35J25 Boundary value problems for second-order elliptic equations 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 35J50 Variational methods for elliptic systems 35J66 Nonlinear boundary value problems for nonlinear elliptic equations Keywords:bifurcation; Steklov eigenvalue problem; weighted trace inequalities; Lorentz spaces; Lorentz-Zygmund spaces × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] D.R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. (2) 128 (1988), no. 2, 385-398. · Zbl 0672.31008 [2] W. Allegretto and Y.X. Huang, A Picone’s identity for the \(p\)-Laplacian and applications, Nonlinear Anal. 32 (1998), no. 7, 819-830. · Zbl 0930.35053 [3] A. Ambrosetti and A. 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