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 -