On quasilinear elliptic problems with finite or infinite potential wells

Abstract

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