Discontinuous perturbations of nonhomogeneous strongly-singular Kirchhoff problems. (English) Zbl 1480.35228
The authors study the quasilinear Kirchhoff problem
\begin{align*}
\begin{cases} -M \left(\displaystyle \int_\Omega \Phi(|\nabla u|)\,\text{d}x\right) \Delta_\Phi u=\mu b(x)u^{-\delta} +\lambda f(x,u) &\text{in }\Omega,\\
u>0&\text{in } \Omega,\\
u=0 &\text{on }\partial\Omega, \end{cases}
\end{align*}
where \(M\colon [0,\infty)\to [0,\infty)\) is a continuous function, \(f\colon \Omega \times (0,\infty)\to (0,\infty)\) is of Heaviside type, \(0<b\in L^1(\Omega)\), \(\delta>1\), \(\lambda,\mu>0\) are real parameters and \( \Delta_\Phi \) denotes the \(\Phi\)-Laplacian. The authors present equivalent conditions for the existence of three solutions to the problem above.
Reviewer: Patrick Winkert (Berlin)
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J75 | Singular elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
Keywords:
Kirchhoff equation; strongly-singular nonlinearity; \(\Phi\)-Laplace operator; discontinuous perturbationReferences:
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