Existence of Bounded Solutions for a Weighted Quasilinear Elliptic Equation in R^N

Abstract

In this paper we establish a new existence result for the quasilinear elliptic equation

$$\displaystyle - \mathrm {div} (A(x, u)\vert \nabla u\vert ^{p-2} \nabla u) + \frac {1}{p}A_t (x, u)\vert \nabla u\vert ^{p} + V(x) {\vert u \vert }^{p-2} u= \xi (x) {\vert u \vert }^{q-2} u \quad \quad \mbox{ in }{\mathbb {R}}^{N} $$

with N ≥ 2 and 1 < q < p. Here, we suppose $A:\mathbb {R}^N\times \mathbb {R}\rightarrow \mathbb {R}$ is a C¹-Carathéodory function such that $A_t(x, t) = \frac {\partial A}{\partial t} (x, t)$ and $V:\mathbb {R}^N\to \mathbb {R}$, $\ \xi :\mathbb {R}^N\rightarrow \mathbb {R}$ are suitable measurable functions. Since the coefficient of the principal part depends on the solution itself, the study of the interaction of two different norms in a suitable Banach space is needed.

Thus, a variational approach and approximation arguments on bounded sets can be used to state the existence of a nontrivial weak bounded solution.

Dedicated to Francesco Altomare, with great esteem and gratitude.