<p>We prove a Liouville type theorem for nonnegative solutions of the problem <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ -\Delta _p u + |\nabla u|^\gamma = u^q $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> <mi>u</mi> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mi>γ</mi> </msup> <mo>=</mo> <msup> <mi>u</mi> <mi>q</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ \mathbb {R}^N_+ $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> <mi>N</mi> </msubsup> </math></EquationSource> </InlineEquation> with zero Dirichlet boundary condition, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ p>2 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ q>\gamma > p-1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>></mo> <mi>γ</mi> <mo>></mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Our proof combines a recent monotonicity result with a new Liouville type theorem for nonnegative stable solutions in dimension <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ N<N^\sharp $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo><</mo> <msup> <mi>N</mi> <mo>♯</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ N^\sharp $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>N</mi> <mo>♯</mo> </msup> </math></EquationSource> </InlineEquation> is explicitly computed.</p>

Some Liouville Results for Quasilinear Hamilton-Jacobi Type Problems

期刊论文
发表日期：

2025年6月2日

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Phuoc Vinh Dinh,
Kim Anh T. Le,
Phuong Le

摘要

We prove a Liouville type theorem for nonnegative solutions of the problem $ -\Delta _p u + |\nabla u|^\gamma = u^q $ $- Δ_{p} u + {| \nabla u |}^{γ} = u^{q}$ in $ \mathbb {R}^N_+ $ $R_{+}^{N}$ with zero Dirichlet boundary condition, where $ p>2 $ $p > 2$ and $ q>\gamma > p-1 $ $q > γ > p - 1$ . Our proof combines a recent monotonicity result with a new Liouville type theorem for nonnegative stable solutions in dimension $ N<N^\sharp $ $N < N^{♯}$ , where $ N^\sharp $ $N^{♯}$ is explicitly computed.