The behaviour of the \(p(x)\)-Laplacian eigenvalue problem as \(p(x)\rightarrow \infty \). (English) Zbl 1182.35176
Summary: We study the behaviour of the solutions to the eigenvalue problem corresponding to the \(p(x)\)-Laplacian operator
\[ \begin{cases} -\text{div}(|\nabla u|^{p(x)-2}\nabla u)= \Lambda_{p(x)}|u|^{p(x)-2}u &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega, \end{cases} \]
as \(p(x)\to\infty\). We consider a sequence of functions \(p_n(x)\) that goes to infinity uniformly in \(\overline{\Omega}\). Under adequate hypotheses on the sequence \(p_n\), namely that the limits
\[ \nabla\ln p_n(x)\to\xi(x) \quad\text{and}\quad \frac{p_n}{n}(x)\to q(x) \]
exist, we prove that the corresponding eigenvalues \(\Lambda_{p_n}\) and eigenfunctions \(u_{p_n}\) verify that
\[ (\Lambda_{p_n})^{1/n}\to \Lambda_\infty, \quad u_{p_n}\to u_\infty \quad\text{uniformly in }\overline{\Omega}, \]
where \(\Lambda_\infty\), \(u_\infty\) is a nontrivial viscosity solution of the following problem
\[ \begin{cases} \min\big\{-\Delta_\infty u_\infty-|\nabla u_\infty|^2 \log(|\nabla u_\infty|) \langle\xi\nabla u_\infty\rangle,\;|\nabla u_\infty|^q-\Lambda_\infty u_\infty^q\big\}=0 &\text{in }\Omega,\\ u_\infty=0 &\text{on }\partial\Omega.\end{cases} \]
\[ \begin{cases} -\text{div}(|\nabla u|^{p(x)-2}\nabla u)= \Lambda_{p(x)}|u|^{p(x)-2}u &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega, \end{cases} \]
as \(p(x)\to\infty\). We consider a sequence of functions \(p_n(x)\) that goes to infinity uniformly in \(\overline{\Omega}\). Under adequate hypotheses on the sequence \(p_n\), namely that the limits
\[ \nabla\ln p_n(x)\to\xi(x) \quad\text{and}\quad \frac{p_n}{n}(x)\to q(x) \]
exist, we prove that the corresponding eigenvalues \(\Lambda_{p_n}\) and eigenfunctions \(u_{p_n}\) verify that
\[ (\Lambda_{p_n})^{1/n}\to \Lambda_\infty, \quad u_{p_n}\to u_\infty \quad\text{uniformly in }\overline{\Omega}, \]
where \(\Lambda_\infty\), \(u_\infty\) is a nontrivial viscosity solution of the following problem
\[ \begin{cases} \min\big\{-\Delta_\infty u_\infty-|\nabla u_\infty|^2 \log(|\nabla u_\infty|) \langle\xi\nabla u_\infty\rangle,\;|\nabla u_\infty|^q-\Lambda_\infty u_\infty^q\big\}=0 &\text{in }\Omega,\\ u_\infty=0 &\text{on }\partial\Omega.\end{cases} \]
MSC:
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35B40 | Asymptotic behavior of solutions to PDEs |
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