Existence and multiplicity of solutions for Dirichlet problems involving nonlinearities with arbitrary growth

• Main Authors: Giovanni Anello, Francesco Tulone
• Format: Article
• Language: English
• Published: Texas State University 2014-09-01
• Series: Electronic Journal of Differential Equations
• Subjects: Existence and multiplicity of solutions; Dirichlet problem; growth condition; critical point theorem
• Online Access: http://ejde.math.txstate.edu/Volumes/2014/200/abstr.html

In this article we study the existence and multiplicity of solutions for the Dirichlet problem $$\displaylines{ -\Delta_p u=\lambda f(x,u)+ \mu g(x,u)\quad\hbox{in }\Omega,\cr u=0\quad\hbox{on } \partial \Omega }$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $f,g:\Omega \times \mathbb{R}\to \mathbb{R}$ are Caratheodory functions, and $\lambda,\mu$ are nonnegative parameters. We impose no growth condition at $\infty$ on the nonlinearities f,g. A corollary to our main result improves an existence result recently obtained by Bonanno via a critical point theorem for $C^1$ functionals which do not satisfy the usual sequential weak lower semicontinuity property.