On quasilinear elliptic problems with finite or infinite potential wells

Abstract

We consider quasilinear elliptic problems of the form \[ -\operatorname{div}\big(ϕ(|\nabla u|)\nabla u\big)+V(x)ϕ(|u|)u=f(u)\qquad u\in W^{1,Φ}(\mathbb{R}^{N}), \] where $\varphi$ and $f$ satisfy suitable conditions. The positive potential $V\in C({\mathbb{R}}^N)$ exhibits a finite or infinite potential well in the sense that $V(x)$ tends to its supremum $V_{\infty }\le +\infty$ as $|x|\to \infty$. Nontrivial solutions are obtained by variational methods. When $V_{\infty }=+\infty$, a compact embedding from a suitable subspace of $W^{1,\Phi }({\mathbb{R}}^N)$ into $L^{\Phi }({\mathbb{R}}^N)$ is established, which enables us to get infinitely many solutions for the case that $f$ is odd. For the case that $V(x)=\lambda a(x)+1$ exhibits a steep potential well controlled by a positive parameter $\lambda$, we get nontrivial solutions for large $\lambda$.

21 pages