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Sign-changing solutions and multiplicity results for some quasi-linear elliptic Dirichlet problems. (English) Zbl 1199.35104
The paper deals with boundary value problem for quasilinear elliptic equation \[ \text{div}(A(x,u)\nabla u)=f(\lambda ,x,y) \tag \(P_\lambda \) \] on smooth \(\Omega \subset {\mathbb R}^N\), \(u| _{\partial \Omega }=0\). The role of \(\lambda \) is unclear. The problem is converted to a critical value problem for a functional \(J_\lambda (u)\). The main result asserts that for some \(\lambda \) the problem \(P_\lambda\) admits at least six nontrivial solutions, at least two positive, two negative and two sign-changing solutions. The proof using functional analysis methods is based on mountain pass theorem in ordered intervals.
35J62 Quasilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
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