1,195 research outputs found
A note on the implicit function theorem for quasi-linear eigenvalue problems
We consider the quasi-linear eigenvalue problem
subject to Dirichlet boundary conditions on a bounded open set , where
is a locally Lipschitz continuous functions. Imposing no further conditions
on or we show that for small the problem has a bounded
solution which is unique in the class of all small solutions. Moreover, this
curve of solutions depends continuously on .Comment: 7 page
Existence and multiplicity of solutions to equations of Laplacian type with critical exponential growth in
In this paper, we deal with the existence and multiplicity of solutions to
the nonuniformly elliptic equation of the N-Lapalcian type with a potential and
a nonlinear term of critical exponential growth and satisfying the
Ambrosetti-Rabinowitz condition. In spite of a possible failure of the
Palais-Smale compactness condition, in this article we apply minimax method to
obtain the weak solution to such an equation. In particular, in the case of
Laplacian, using the minimization and the Ekeland variational principle, we
obtain multiplicity of weak solutions.
Finally, we will prove the above results when our nonlinearity doesn't
satisfy the well-known Ambrosetti-Rabinowitz condition and thus derive the
existence and multiplicity of solutions for a much wider class of nonlinear
terms .Comment: 30 pages. First draft in November, 201
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