Global boundedness for Generalized Schrödinger-Type Double Phase Problems in $\mathbb{R}^N$ and Applications to Supercritical Double Phase Problems
Hoang Hai Ha, Ky Ho, Bui The Quan, Inbo Sim
TL;DR
This work develops a robust regularity-to-existence framework for generalized Schrödinger-type double phase problems with variable exponents on $\mathbb{R}^N$. By working in Musielak–Orlicz–Sobolev spaces and employing a localized De Giorgi iteration, the authors establish global $L^{\infty}$ bounds and decay for subcritical and critical growth, respectively. They then leverage these a priori estimates to treat supercritical double phase problems via a Rabinowitz truncation approach, proving the existence of nontrivial nonnegative weak solutions both on $\mathbb{R}^N$ and on bounded domains. The results extend the boundedness and regularity theory to unbounded domains and variable exponent settings, with novelty even for constant exponents.
Abstract
We establish two global boundedness results for weak solutions to generalized Schrödinger-type double phase problems with variable exponents in $\mathbb{R}^N$ under new critical growth conditions optimally introduced in [26, 32]. More precisely, for the case of subcritical growth, we employ the De Giorgi iteration with a suitable localization method in $\mathbb{R}^N$ to obtain a-priori bounds. As a byproduct, we derive the decay property of weak solutions. For the case of critical growth, using the De Giorgi iteration with a localization adapted to the critical growth, we prove the global boundedness. As an interesting application of these results, the existence of weak solutions for supercritical double phase problems is shown. These results are new even for problems with constant exponents in $\mathbb{R}^N$.