EXISTENCE AND NO EXISTENCE OF POSITIVE SOLUTIONS FOR ROBIN (P,Q)-LAPLACE EQUATION WITH TWO PARAMETERS

Abstract

The purpose of this paper is to study the (p,q)-Laplace equation under the
Robin boundary condition in a bounded region of ℝ???? through the following
problem

The principal results of this article concerns the existence and not existence of
positive solution. For this we will discuss all the possible cases depending on the
parameters ????, ???? and the first eigenvalue of p-Laplacian (respectively q-
Laplacian). Using various amount of methods mostly the Min-Max theory wich
include the Palais-Smale condition, the global and local minimizer and the Col
theorem, We will also use the notion of super solution. Adding to that we willgive a definition for the curve witch separates between the regions of existence
and not existence of positive solution in (????, ????) plane and we will make sur that
this curve wich we will be noted ???? is continuous and locally finite. Moreover we
are going to explor the existence of positive solution in the region where the curve
???? is included. We have chosen to put our attention on the study of (p,q)-Laplace
equation under the Robin boundary condition because of its importance for
modeling nonlinear phenomena in physics, biology, ecology, and other fields. We
can take as an example in ecology it can be used to modeling the phenomena of
population dynamics or in physics to modeling the nonlinear heat conduction.