Existence and local boundedness of solutions of a -Laplacian problem

ABSTRACT

In this paper, we consider the $\mathit{\varphi }$-Laplacian equation $$-\mathrm{div}\left(\mathit{\varphi }(|\mathrm{\nabla }u|)\mathrm{\nabla }u/|\mathrm{\nabla }u|\right)=\mathit{\lambda }g(·)\mathit{\varphi }(u)\text{in}{\mathbb{R}}^N$$

where $\mathit{\varphi }$ is an odd increasing homeomorphism from $\mathbb{R}$ onto $\mathbb{R}$ which is not necessarily differentiable. The term $\mathit{\lambda }$ is a suitable parameter and g is measurable. We prove that nontrivial and nonnegative weak solutions of this equation exist. The problem on the local boundedness of the solutions is also treated. Mild restrictions are imposed on g and $\mathit{\varphi }$.

KEYWORDS:

• Orlicz-Sobolev space
• eigenvalue problem
• $\mathit{\varphi }$-Laplacian
• local boundedness

AMS SUBJECT CLASSIFICATIONS:

• 35P30
• 35J20
• 46E30

Notes

No potential conflict of interest was reported by the authors.

1 Hypothesis $({\mathcal{H}}_2)$ generalizes the natural condition $g\in L^{N/p}({\mathbb{R}}^N)\cap L^{\infty }({\mathbb{R}}^N)$ imposed on the right-hand side of the eigenvalue differential problem considered in [10]. Indeed, in that reference the authors treat the classical case of the p-Laplacian ${\mathrm{\Delta }}_{p}u={\text{div}(|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u)$ for which $p_{\mathrm{\Phi }}=q_{\mathrm{\Phi }}=p$ and the quantity $q_{\mathrm{\Phi }}^{\ast }/(q_{\mathrm{\Phi }}^{\ast }-p_{\mathrm{\Phi }})=N/p$.