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Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents. (English) Zbl 0675.35036
The paper is concerned with the boundary value problem: \[ -\nabla \cdot (f| \nabla u|^{m-2}\nabla u)-b\cup^ p=\lambda hu^{m-1}\quad in\quad \Omega;\quad u>0\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega \] where \(1<m<n\), \(\Omega \subset R^ n\) smooth, bounded, open and connected and p the critical Sobolev exponent \((p=mn/(n-m))\). The results proved regard the existence and non-existence of solutions \(u\in H_ 0^{1,m}(\Omega).\)
Thus it is proved that there exists \(\lambda^*\) such that for \(\lambda^*<\lambda <\lambda_ 0\) there exist solutions and for \(\lambda >\lambda_ 0\) there are no solutions of the problem where \(\lambda_ 0\) is the first eigenvalue of: \[ -\nabla \{f| \nabla u|^{m-2} \nabla u)=\lambda hu^{m-1}\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega \] under certain conditions on the coefficients and on the domain for \(n\geq m^ 2\) sharp existence and non- existence results are obtained for which a Pohozaev identity is used.
Reviewer: B.Vernescu

35J65 Nonlinear boundary value problems for linear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
Full Text: DOI
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