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

Abstract:
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.

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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