We study of the regularizing effect of the interaction between the coefficient of the zero-order term and the lower-order term in quasilinear Dirichlet problems whose model is ∫ΩM⁢(x,u)⁢∇⁡u⋅∇⁡φ+∫Ωa⁢(x)⁢u⁢φ=∫Ωb⁢(x)⁢|∇⁡u|q⁢φ+∫Ωf⁢(x)⁢φ for all ⁢φ∈W01,2⁢(Ω)∩L∞⁢(Ω),\int_{\Omega}M(x,u)\nabla u\cdot\nabla\varphi+\int_{\Omega}a(x)u\varphi=\int_{% \Omega}b(x)|\nabla u|^{q}\varphi+\int_{\Omega}f(x)\varphi\quad\text{for all }% \varphi\in W_{0}^{1,2}(\Omega)\cap L^{\infty}(\Omega), where Ω is a bounded open set of ℝN{\mathbb{R}^{N}}, M⁢(x,s){M(x,s)} is a Carathéodory matrix on Ω×ℝ{\Omega\times\mathbb{R}} which is elliptic (that is, M⁢(x,s)⁢ξ⋅ξ≥α⁢|ξ|2>0{M(x,s)\xi\cdot\xi\geq\alpha|\xi|^{2}>0} for every (x,s,ξ)∈Ω×ℝ×(ℝN∖{0}){(x,s,\xi)\in\Omega\times\mathbb{R}\times(\mathbb{R}^{N}\setminus\{0\})}) and bounded (that is, |M⁢(x,s)|≤β{|M(x,s)|\leq\beta} for every (x,s)∈Ω×ℝ{(x,s)\in\Omega\times\mathbb{R}}), b⁢(x)∈L22-q⁢(Ω){b(x)\in L^{\frac{2}{2-q}}(\Omega)}, 1<q<2{1<q<2} and 0≤a⁢(x)∈L1⁢(Ω){0\leq a(x)\in L^{1}(\Omega)}. We prove the existence of a weak solution u belonging to W01,2⁢(Ω){W_{0}^{1,2}(\Omega)} and to L∞⁢(Ω){L^{\infty}(\Omega)} when either b∈L2⁢m2-q⁢(Ω)⁢ for some ⁢m>N2⁢ and\displaystyle b\in L^{\frac{2m}{2-q}}(\Omega)\text{ for some }m>\frac{N}{2}% \text{ and}(0.1)∃Q>0⁢ such that ⁢|f⁢(x)|≤Q⁢a⁢(x)\displaystyle\exists\,Q>0\text{ such that }|f(x)|\leq Qa(x) or f∈Lm⁢(Ω)⁢ for some ⁢m>N2⁢ and\displaystyle f\in L^{m}(\Omega)\text{ for some }m>\frac{N}{2}\text{ and}(0.2)∃R>0⁢ such that ⁢|b⁢(x)|22-q≤R⁢a⁢(x).\displaystyle\exists\,R>0\text{ such that }|b(x)|^{\frac{2}{2-q}}\leq Ra(x). In addition, we also prove the existence for every f∈L1⁢(Ω){f\kern-1.0pt\in\kern-1.0ptL^{1}(\Omega)} and b⁢(x)∈L22-q⁢(Ω){b(x)\kern-1.0pt\in\kern-1.0ptL^{\frac{2}{2-q}}(\Omega)} satisfying both conditions (0.1) and (0.2) jointly.

中文翻译：

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两个假设对某些Hamilton–Jacobi方程系数之间相互作用的正则化影响

s）| \ leq \ beta}对于每个（x，s）∈Ω×ℝ{（（x，s）\ in \ Omega \ times \ mathbb {R}}），b⁢（x）∈L22-q⁢（ Ω）{b（x）\ in L ^ {\ frac {2} {2-q}}（\ Omega）}，1 <q <2 {1 <q <2}和0≤a⁢（x）∈ L1⁢（Ω）{0 \ leq a（x）\ in L ^ {1}（\ Omega）}。我们证明存在一个弱解u，它属于W01,2⁢（Ω）{W_ {0} ^ {1,2}（\ Omega）}和L∞⁢（Ω）{L ^ {\ infty}（ \ omega）}，当b∈L2⁢m2-q⁢（Ω）⁢等于⁢m>N2⁢和\ displaystyle b \ in L ^ {\ frac {2m} {2-q}}（\ Omega）\ text {代表} m> \ frac {N} {2}％\ text {和}（0.1）∃Q>0⁢，使得⁢|f⁢（x）|≤Q⁢a⁢（x）\ displaystyle \存在\，Q> 0 \ text {使得} | f（x）| \ leq Qa（x）或f∈Lm⁢（Ω）⁢对于某些⁢m>N2⁢和\ displaystyle f \在L ^ {m }（\ Omega）\ text {对于某些} m> \ frac {N} {2} \ text {和}（0.2）∃R>0⁢，使得⁢|b⁢（x）|22-q≤R⁢ a⁢（x）。\ displaystyle \ exists \，R> 0 \ text {这样} | b（x）| ^ {\ frac {2} {2-q}} \ leq Ra（x）。此外，