In this paper, we aim at establishing a new existence result for the quasilinear elliptic equation -div(a(x,u,∇u))+At(x,u,∇u)+V(x)|u|p-2u=g(x,u)inRN\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} - \textrm{div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) + V(x) {\vert u \vert }^{p-2} u= g(x,u) \quad \quad \hbox { in }{{\mathbb {R}}}^{N} \end{aligned}$$\end{document}with p>1\documentclass[12pt]{minimal}
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\begin{document}$$p>1$$\end{document}, N≥2\documentclass[12pt]{minimal}
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\begin{document}$$N\ge 2\ $$\end{document} and V:RN→R\documentclass[12pt]{minimal}
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\begin{document}$$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$\end{document} suitable measurable positive function. Here, we suppose A:RN×R×RN→R\documentclass[12pt]{minimal}
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\begin{document}$$A: {\mathbb {R}}^N \times {\mathbb {R}}\times {\mathbb {R}}^N \rightarrow {\mathbb {R}}$$\end{document} is a given C1\documentclass[12pt]{minimal}
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\begin{document}$${C}^{1}$$\end{document}-Carathéodory function which grows as |ξ|p\documentclass[12pt]{minimal}
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\begin{document}$$|\xi |^p$$\end{document}, with At(x,t,ξ)=∂A∂t(x,t,ξ)\documentclass[12pt]{minimal}
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\begin{document}$$A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )$$\end{document}, a(x,t,ξ)=∇ξA(x,t,ξ)\documentclass[12pt]{minimal}
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\begin{document}$$a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )$$\end{document}, V:RN→R\documentclass[12pt]{minimal}
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\begin{document}$$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$\end{document} is a suitable measurable function and g:RN×R→R\documentclass[12pt]{minimal}
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\begin{document}$$g:{\mathbb {R}}^N \times {\mathbb {R}}\rightarrow {\mathbb {R}}$$\end{document} is a given Carathéodory function which grows as |ξ|q\documentclass[12pt]{minimal}
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\begin{document}$$|\xi |^q$$\end{document} with 1<q<p\documentclass[12pt]{minimal}
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\begin{document}$$1<q<p$$\end{document}. Since the coefficient of the principal part depends on the solution itself, under suitable assumptions on A(x,t,ξ),V(x)\documentclass[12pt]{minimal}
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\begin{document}$$A(x,t,\xi ), V(x)$$\end{document} 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.
