Convergence rates in a weighted Fučik problem (Q2923146)

Let \(\Omega\subset \mathbb R^N\), \(N\geq 1\), be a bounded domain and \(\varepsilon\) be a real positive number. The author studies the asymptotic behaviour as \(\varepsilon\to 0\) of the spectrum of the asymmetric elliptic problem NEWLINE\[NEWLINE -\Delta_p u_\varepsilon = \alpha_\varepsilon m_\varepsilon(x)(u_\varepsilon^+)^{p-1}- \beta_\varepsilon n_\varepsilon(x)(u_\varepsilon^-)^{p-1}\qquad\text{in}\;\,\Omega, \eqno{(1.1)} NEWLINE\]NEWLINE either with homogeneous Dirichlet or with Neumann boundary conditions.NEWLINENEWLINEHere, \(\Delta_p\) denotes the \(p\)-Laplace differential operator with \(1<p<\infty\) and \(u^\pm:=\max\{\pm u,0\}\). The parameters \(\alpha_\varepsilon\) and \(\beta_\varepsilon\) are real numbers depending on \(\varepsilon>0\). Consider functions \(m_\varepsilon,n_\varepsilon\) such that for constants \(m_-\leq m_+\), \(n_-\leq n_+\) we have NEWLINE\[NEWLINE 0<m_-\leq m_\varepsilon(x)\leq m_+\leq +\infty\quad\text{and}\quad 0< n_-\leq n_\varepsilon(x)\leq n_+\leq +\infty\,. \eqno{(2.8)} NEWLINE\]NEWLINE Also, assume that there exist functions \(m_0(x)\) and \(n_0(x)\) satisfying (2.8) such that, as \(\varepsilon\to 0\), NEWLINE\[NEWLINE { m_\varepsilon(x)\rightharpoonup m_0(x)\qquad\text{weakly\(^*\) in}\;\, L^\infty(\Omega),}\atop{n_\varepsilon(x)\rightharpoonup n_0(x)\;\;\qquad\text{weakly\(^*\) in}\;\, L^\infty(\Omega).} \eqno(2.9) NEWLINE\]NEWLINE The main results are formulated for the Dirichlet problem. When \(\varepsilon\to 0\), the natural limit problem for (1.1) is NEWLINE\[NEWLINE \begin{cases} -\Delta_p u_0 = \alpha_0 m_0(x)(u_0^+)^{p-1}- \beta_0 n_0(x)(u_0^-)^{p-1}\quad & \text{in}\;\,\Omega,\\ u_0=0\;& \text{on}\;\,\partial\Omega,\end{cases} \tag{2.10} NEWLINE\]NEWLINE where \(m_0,n_0\) are given in (2.9). Then, the main result is the following.NEWLINENEWLINE{ Theorem 2.1} Let \(m_\varepsilon, n_\varepsilon\) satisfy (2.8), (2.9). Then, the first non-trivial curve of the Fučik spectrum of problem (1.1) NEWLINE\[NEWLINE C_\varepsilon := C_1(m_\varepsilon, n_\varepsilon)=\{\alpha_\varepsilon(s), \beta_\varepsilon(s),\, s\in \mathbb R^+\} NEWLINE\]NEWLINE converges to the first non-trivial curve of the limit problem (2.10) NEWLINE\[NEWLINE C_\varepsilon := C_1(m_0, n_0)=\{\alpha_0(s), \beta_0(s),\, s\in \mathbb R^+\} NEWLINE\]NEWLINE as \(\varepsilon\to 0\) in the sense that \(\alpha_\varepsilon(s)\to \alpha_0(s)\) and \(\beta_\varepsilon(s)\to \beta_0(s)\), for all \(s\in \mathbb R^+\).NEWLINENEWLINEThe author also obtained the convergence rates to the first non-trivial curve of the Fučik spectrum.NEWLINENEWLINE{ Theorem 2.3} Under the same hypothesis of Theorem 2.1, if the weights \(m_\varepsilon\) and \(n_\varepsilon\) are given in terms of \(Q\)-periodic functions \(m,n\) (\(Q\) being the unit cube in \(\mathbb R^N\)) in the form \(m_\varepsilon(x)=m(x/\varepsilon)\) and \(n_\varepsilon(x)=n(x/\varepsilon)\), for each \(s\in \mathbb R^+\), we have the estimates NEWLINE\[NEWLINE |\alpha_\varepsilon(s)-\alpha_0(s)|\leq c(1+s)\tau(s)\varepsilon\,,\quad |\beta\varepsilon(s)-\beta_0(s)|\leq c(1+s)\tau(s)\varepsilon,\eqno{(2.13)} NEWLINE\]NEWLINE where \(c\) is given explicitly by NEWLINE\[NEWLINE pc_1c_p^{p-1}\max\{\|m-\tilde{m}_{L^\infty(R^N)}\|, \|n-\tilde{n}_{L^\infty(R^N)}\}(\min\{m_-^{-1},n_-^{-1}\}\mu_2)^2, NEWLINE\]NEWLINE where \(c_1\), \(c_p\) are the Poincaré constants in \(L^1(Q)\) and \(L^p(\Omega)\), respectively, \(\mu_2\) is the second Dirichlet \(p\)-Laplacian eigenvalue in \(\Omega\), \(\tilde{m},\tilde{n}\) are real numbers given by the averages of \(m,n\) over \(Q\), respectively, and \(\tau\) is defined by NEWLINE\[NEWLINE \tau(s)=\left\{{1, \qquad \;\;\; s\geq 1,}\atop{s^{-2}, \qquad s<1.}\right. \eqno{(2.14)} NEWLINE\]NEWLINE The author also analyses the trivial curve of the Fučik spectrum and the Neumann problem. To prove these results, the author uses the variational characterization of the first eigenvalues of the corresponding problems.