arXiv subjects
Publications and source records attributed to K. O. Silva.
It is established existence, uniqueness and multiplicity of solutions for a quasilinear elliptic problem problems driven by $Φ$-Laplacian operator. Here we consider the reflexive and nonreflexive cases using an auxiliary problem. In order to prove our main results we employ variational methods, regularity results and truncation techniques.
In this paper we prove that the equation $ -( r^αϕ(|u'(r)|)u'(r))' = λr^γf(u(r)), ~0 0$ is a real parameter and $f:{\bf{R}}\to{\bf{R}}$ is continuous, admits an infinite sequence of sign-changing solutions satisfying $u'(0) =u(R) =0$. The function $f$ is required to satisfy $tf(t)>0$ for $ t\neq 0$. Our technique explores fixed point arguments applied to suitable integral equations and shooting arguments. Our main result extends earlier ones in the case $ϕ$ is in the form $ϕ(t) = |t|^β$ for an appropriate constant $γ$.
In 2009 Loc and Schmitt established a result on sufficient conditions for multiplicity of solutions of a class of nonlinear eignvalue problems for the p-Laplace operator under Dirichlet boundary conditions, extending an earlier result of 1981 by Peter Hess for the Laplacian. Results on necessary conditions for existence were also established. In the present paper the authors extend the main results by Loc and Schmitt to the $Φ$-Laplacian. To overcome the difficulties with this much more general operator it was necessary to employ regularity results by Lieberman, a strong maximum principle by Pucci and Serrin and a general result on lower and upper solutions by Le.