Existence of positive solutions for a class of quasilinear elliptic problems with exponential growth via the Nehari manifold method

Abstract

In this paper we will be concerned with the problem $$\begin{aligned} -\,\text{ div }(a(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u)= f(u) \ \text{ in } \ \Omega , \ \ u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}$$ where $$\Omega \subset \mathbb {R}^{N}$$ is bounded, $$1{<}p{<}N$$ , $$f:\mathbb {R}\rightarrow \mathbb {R}$$ is a superlinear continuous function with exponential subcritical or exponential critical growth and the function a is $$C^{1}$$ . We use as a main tool the Nehari manifold method and our results include a large class of problems.