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<p>We establish two global boundedness results for weak solutions to generalized Schrödinger-type double phase problems with variable exponents in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({{\mathbb {R}}}^N\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> under new critical growth conditions optimally introduced in Ha and Ho (J Math Anal Appl 541:128748, 2025) and Ho and Winkert (Calc Var Partial Differ Equ 62(8):227, 2023). More precisely, for the case of subcritical growth, we employ the De Giorgi iteration with a suitable localization method in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\({{\mathbb {R}}}^N\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> to obtain <i>a priori</i> 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\({{\mathbb {R}}}^N\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>.</p>

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Global boundedness for generalized Schrödinger-type double phase problems in \({{\mathbb {R}}}^N\) and applications to supercritical double phase problems

  • 期刊论文

  • 发表日期：

    2026年5月14日

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  • Hoang Hai Ha,
  • Ky Ho,
  • Bui The Quan,
  • Inbo Sim

摘要

We establish two global boundedness results for weak solutions to generalized Schrödinger-type double phase problems with variable exponents in \({{\mathbb {R}}}^N\) R N under new critical growth conditions optimally introduced in Ha and Ho (J Math Anal Appl 541:128748, 2025) and Ho and Winkert (Calc Var Partial Differ Equ 62(8):227, 2023). More precisely, for the case of subcritical growth, we employ the De Giorgi iteration with a suitable localization method in \({{\mathbb {R}}}^N\) 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 \({{\mathbb {R}}}^N\) R N .