We study a quasilinear elliptic problem depending on a parameter $\lambda$ of the form $-\Delta_p u=\lambda f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$. We present a novel variational approach that allows us to obtain multiplicity, regularity and a priori estimate of solutions by assuming certain growth and sign conditions on f prescribed only near zero. More precisely, we describe an interval of parameters$\lambda$ for which the problem under consideration admits at least three nontrivial solutions: two extremal constant-sign solutions and one sign-changing solution. Our approach is based on an abstract localization principle of critical points of functionals of the form $\Phi-\lambda\Psi$ on open sublevels $\Phi^{-1}(]-\infty,r[)$, combined with comparison principles and the sub-supersolution method. Moreover, variational and topological arguments, such as the mountain pass theorem, in conjunction with truncation techniques are the main tools for the proof of sign-changing solutions. © 2012 Elsevier Ltd.

Candito, P., Carl, S., Livrea, R. (2012). Multiple solutions for quasilinear elliptic problems via critical points in open sublevels and truncation principles. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 395(1), 156-163 [10.1016/j.jmaa.2012.05.003].

### Multiple solutions for quasilinear elliptic problems via critical points in open sublevels and truncation principles

#### Abstract

We study a quasilinear elliptic problem depending on a parameter $\lambda$ of the form $-\Delta_p u=\lambda f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$. We present a novel variational approach that allows us to obtain multiplicity, regularity and a priori estimate of solutions by assuming certain growth and sign conditions on f prescribed only near zero. More precisely, we describe an interval of parameters$\lambda$ for which the problem under consideration admits at least three nontrivial solutions: two extremal constant-sign solutions and one sign-changing solution. Our approach is based on an abstract localization principle of critical points of functionals of the form $\Phi-\lambda\Psi$ on open sublevels $\Phi^{-1}(]-\infty,r[)$, combined with comparison principles and the sub-supersolution method. Moreover, variational and topological arguments, such as the mountain pass theorem, in conjunction with truncation techniques are the main tools for the proof of sign-changing solutions. © 2012 Elsevier Ltd.
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Candito, P., Carl, S., Livrea, R. (2012). Multiple solutions for quasilinear elliptic problems via critical points in open sublevels and truncation principles. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 395(1), 156-163 [10.1016/j.jmaa.2012.05.003].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/258527