Generalized Quasilinear Elliptic Equations in $${\mathbb {R}}^N$$

Abstract

In this paper, we aim at establishing a new existence result for the quasilinear elliptic equation $$\begin{aligned} - \textrm{div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) + V(x) {\vert u \vert }^{p-2} u= g(x,u) \quad \quad \hbox { in }{{\mathbb {R}}}^{N} \end{aligned}$$ - div ( a ( x , u , ∇ u ) ) + A t ( x , u , ∇ u ) + V ( x ) | u | p - 2 u = g ( x , u ) in R N with $$p>1$$ p > 1 , $$N\ge 2\ $$ N ≥ 2 and $$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$ V : R N → R suitable measurable positive function. Here, we suppose $$A: {\mathbb {R}}^N \times {\mathbb {R}}\times {\mathbb {R}}^N \rightarrow {\mathbb {R}}$$ A : R N × R × R N → R is a given $${C}^{1}$$ C 1 -Carathéodory function which grows as $$|\xi |^p$$ | ξ | p , with $$A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )$$ A t ( x , t , ξ ) = ∂ A ∂ t ( x , t , ξ ) , $$a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )$$ a ( x , t , ξ ) = ∇ ξ A ( x , t , ξ ) , $$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$ V : R N → R is a suitable measurable function and $$g:{\mathbb {R}}^N \times {\mathbb {R}}\rightarrow {\mathbb {R}}$$ g : R N × R → R is a given Carathéodory function which grows as $$|\xi |^q$$ | ξ | q with $$1<q<p$$ 1 < q < p . Since the coefficient of the principal part depends on the solution itself, under suitable assumptions on $$A(x,t,\xi ), V(x)$$ A ( x , t , ξ ) , V ( x ) and g(x, t), we study the interaction of two different norms in a suitable Banach space with the aim of obtaining a good variational approach. Thus, a minimization argument on bounded sets can be used to state the existence of a nontrivial weak bounded solution on an arbitrary bounded domain. Then, one nontrivial bounded solution of the given equation can be found by passing to the limit on a sequence of solutions on bounded domains. Finally, under slightly stronger hypotheses, we can able to find a positive solution of the problem.