TY - JOUR
T1 - Critical points in open sublevels and multiple solutions for parameter-depending quasilinear elliptic equations
AU - Livrea, Roberto
AU - Carl, null
AU - Livrea, Roberto
AU - Candito, Pasquale
PY - 2014
Y1 - 2014
N2 - We investigate the existence of multiple nontrivial solutions of a quasilinear elliptic Dirichlet problem depending on a parameter $\lambda> 0$ of the form $-\Delta_p u = \lambda f(u)$ in $\Omega$, $u = 0$ on $\partial\Omega$, where $\Omega$ is a bounded domain, $\Delta_p$, $1 < p < +\infty$, is the p-Laplacian, and $f: R\to R$ is a continuous function satisfying a sub-critical growth condition. More precisely, we establish a variational approach that when combined with differential inequality techniques, allows us to explicitly describe intervals for the parameter $\lambda$ for which the problem under consideration admits nontrivial constant-sign as well as nodal (sign-changing) solutions. In our approach, a crucial role plays an abstract critical point result for functionals whose critical points are attained in certain open level sets. To the best of our knowledge, the novelty of this paper is twofold. First, neither an asymptotic condition for f at zero nor at infinity is required to ensure multiple constant-sign solutions. Second, only by imposing some lim inf and lim sup condition of f at zero, the existence of at least three nontrivial solutions including one nodal solution can be proved.
AB - We investigate the existence of multiple nontrivial solutions of a quasilinear elliptic Dirichlet problem depending on a parameter $\lambda> 0$ of the form $-\Delta_p u = \lambda f(u)$ in $\Omega$, $u = 0$ on $\partial\Omega$, where $\Omega$ is a bounded domain, $\Delta_p$, $1 < p < +\infty$, is the p-Laplacian, and $f: R\to R$ is a continuous function satisfying a sub-critical growth condition. More precisely, we establish a variational approach that when combined with differential inequality techniques, allows us to explicitly describe intervals for the parameter $\lambda$ for which the problem under consideration admits nontrivial constant-sign as well as nodal (sign-changing) solutions. In our approach, a crucial role plays an abstract critical point result for functionals whose critical points are attained in certain open level sets. To the best of our knowledge, the novelty of this paper is twofold. First, neither an asymptotic condition for f at zero nor at infinity is required to ensure multiple constant-sign solutions. Second, only by imposing some lim inf and lim sup condition of f at zero, the existence of at least three nontrivial solutions including one nodal solution can be proved.
KW - Dirichlet problem
KW - p-Laplacian
KW - Dirichlet problem
KW - p-Laplacian
UR - http://hdl.handle.net/10447/258593
UR - http://projecteuclid.org/download/pdf_1/euclid.ade/1408367285
M3 - Article
VL - 19
SP - 1021
EP - 1042
JO - Advances in Differential Equations
JF - Advances in Differential Equations
SN - 1079-9389
ER -