Electron. J. Diff. Equ., Vol. 2014 (2014), No. 200, pp. 1-7.

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

Giovanni Anello, Francesco Tulone

In this article we study the existence and multiplicity of solutions for the Dirichlet problem
 -\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.

Submitted May 20, 2014. Published September 26, 2014.
Math Subject Classifications: 35J20, 35J25.
Key Words: Existence and multiplicity of solutions; Dirichlet problem; growth condition; critical point theorem.

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  Giovanni Anello
Department of Mathematics and Computer Science
Messina University
Viale F. Stagno D'Alcontres 31, 98166, Messina, Italy
email: ganello@unime.it
Francesco Tulone
Department of Mathematics and Computer Science
Palermo University
Via Archirafi 34, 90123, Palermo, Italy
email: francesco.tulone@unipa.it

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