### Multiple nonnegative solutions for elliptic boundary value problems involving the $p$-Laplacian

#### Abstract

In this paper we present a result concerning the existence of two

nonzero nonnegative solutions for the following Dirichlet problem involving

the $p$-Laplacian

$$

\cases

-\Delta_p u=\lambda f(x,u) &\text{\rm in\ } \Omega,\\

u=0 &\text{\rm on\ } \partial \Omega,

\endcases

$$

using variational methods. In particular, we will determine an

explicit real interval $\Lambda$ for which these solutions exist

for every $\lambda\in \Lambda$. We also point out that our result

improves and extends to higher dimension a recent multiplicity

result for ordinary differential equations.

nonzero nonnegative solutions for the following Dirichlet problem involving

the $p$-Laplacian

$$

\cases

-\Delta_p u=\lambda f(x,u) &\text{\rm in\ } \Omega,\\

u=0 &\text{\rm on\ } \partial \Omega,

\endcases

$$

using variational methods. In particular, we will determine an

explicit real interval $\Lambda$ for which these solutions exist

for every $\lambda\in \Lambda$. We also point out that our result

improves and extends to higher dimension a recent multiplicity

result for ordinary differential equations.

#### Keywords

Variational methods; weak solutions; nonnegative solutions; p-Laplacian; Dirichlet problem

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