Continuous spectrum for a two phase eigenvalue problem with an indefinite and unbounded potential
The authors investigate the following two phase eigenvalue Robin problem \[ \begin{cases} -\Delta_p u -\Delta_q u +\xi(z)|u|^{p-2}u= \lambda_p|u|^{p-2}u \quad \mbox{in}\; \Omega, \cr \displaystyle\frac{\partial u}{\partial n_{pq}}+\beta(z)|u|^{p-2}u = 0 \quad \mbox{on}\; \partial\Omega, \quad \lambda\in \mathbb{R}^n,\;1<q<p<+\infty. \end{cases} \tag{\(P_\lambda\)} \] where \(\Omega\subseteq \mathbb{R}^n\) is a bounded domain with a \(C^2\)-boundary \(\partial \Omega\), and, for every \(r\in(1,\infty)\), \(\Delta_r\) is the \(r\)-Laplace operator, \(\Delta_r u := \mbox{div} (|\nabla u|^{r-2}\nabla u) \). In the boundary condition, \(\frac{\partial u}{\partial n_{pq}}\) denotes the conormal derivative, if \(u\in C^1(\overline{\Omega})\), then \[ \frac{\partial u}{\partial n_{pq}}= (|\nabla u|^{p-2}\nabla u + |\nabla u|^{q-2}\nabla u, n)_{\mathbb{R}^N} , \] with \(n(\cdot)\) being the outward unit normal on \(\partial \Omega\). The authors say that \(\lambda \in \mathbb{R}\) is an ``eigenvalue'' of \((P_\lambda)\), if the problem admits a nontrivial weak solution. They assume that \(H_0\): \(\xi \in L^s(\Omega )\) with \(s >N/p\) if \(1 < p \le N\), \(s = 1\) if \(N <p\), \(\beta \in W^{1,\infty} (\partial \Omega )\), \(\beta (z) \ge 0\) for all \(z \in \partial \Omega\), \( \beta \not\equiv 0\) and \[ \int_\Omega \xi (z)\,dz + \int_{\partial\Omega} \beta (z)\,d\sigma >0. \] Here \(\sigma (\cdot)\) denotes the \((N -1)\)-dimensional Hausdorff measure on \(\partial \Omega\), and \(| \cdot |_N\) denotes the Lebesgue measure on \(\mathbb{R}^N\). The authors set \[ \lambda_0 = \frac{1}{|\Omega|_N}\left[\int_\Omega \xi (z)\, dz +\int_{\partial\Omega}\beta (z)\,d\sigma \right] > 0. \] Further let \(\gamma_p : W^{1,p}(\Omega ) \to\mathbb{R}\) be the \(C^1\)-functional defined by \[ \gamma_p(u)=\| \nabla u\|^p_p +\int_\Omega \xi (z)|u|^p\, dz + \int_{\partial \Omega} \beta (z)|u|^p\,d\sigma \quad\mbox{for all}\; u \in W^{1,p}(\Omega ) \] and \[ \widehat{\lambda}_1 = \inf \left[\frac{\gamma_p(u)}{\|u\|^p_p} : u \in W^{1,p}(\Omega ), u \not= 0\right]. \] The main result of this article is the following theorem. Theorem 1. If hypothesis \(H_0\) holds, then every \(\lambda\in (\widehat\lambda_1, \lambda_0)\) is an eigenvalue of problem \((P_\lambda)\) with corresponding eigenfunction \(u_\lambda \in W^{1,p}(\Omega) \setminus\{0\}\), \(u_\lambda\ge 0\). To prove this theorem the authors modify the Nehari manifold method. They consider the energy functional \(\varphi_\lambda :W^{1,p}(\Omega ) \to\mathbb{R}\) for problem \((P_\lambda)\) defined by \[ \varphi_\lambda(u) = \frac{1}{p} \gamma_p(u)+ \frac{1}{q} \|\nabla u\|^q_q -\lambda_p \|u\|^p_p \quad\mbox{for all}\quad u \in W^{1,p}(\Omega ) \] and the corresponding Nehari manifold for this functional defined by \[ N_\lambda =\left\{u \in W^{1,p}(\Omega ) : \langle\varphi'_\lambda(u), u\rangle = 0 \right\}. \] Since \(W^{1,p}(\Omega )\) contains the constant functions the authors work with the following subset of \(N_\lambda\) \[ \widehat{N}_\lambda =\left\{u \in N_\lambda : \int_\Omega \xi (z)|u|^{p-2}u\,dz +\int_{\partial \Omega} \beta (z)|u|^{p-2}u\,d\sigma = \lambda \int_\Omega |u|^{p-2}u\,dz \right\}. \]
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