Regularizing Effect of Two Hypotheses on the Interplay Between Coefficients in Some Hamilton–Jacobi Equations

Abstract

Abstract 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 ∫ Ω M ⁢ ( x , u ) ⁢ ∇ ⁡ u ⋅ ∇ ⁡ φ + ∫ Ω a ⁢ ( x ) ⁢ u ⁢ φ = ∫ Ω b ⁢ ( x ) ⁢ | ∇ ⁡ u | q ⁢ φ + ∫ Ω f ⁢ ( x ) ⁢ φ   for all  ⁢ φ ∈ W 0 1 , 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 ℝ N {\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 × ( ℝ N ∖ { 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 ) ∈ L 2 2 - q ⁢ ( Ω ) {b(x)\in L^{\frac{2}{2-q}}(\Omega)} , 1 < q < 2 {1<q<2} and 0 ≤ a ⁢ ( x ) ∈ L 1 ⁢ ( Ω ) {0\leq a(x)\in L^{1}(\Omega)} . We prove the existence of a weak solution u belonging to W 0 1 , 2 ⁢ ( Ω ) {W_{0}^{1,2}(\Omega)} and to L ∞ ⁢ ( Ω ) {L^{\infty}(\Omega)} when either b ∈ L 2 ⁢ m 2 - q ⁢ ( Ω ) ⁢  for some  ⁢ m > N 2 ⁢  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 ∈ L m ⁢ ( Ω ) ⁢  for some  ⁢ m > N 2 ⁢  and \displaystyle f\in L^{m}(\Omega)\text{ for some }m>\frac{N}{2}\text{ and} (0.2) ∃ R > 0 ⁢  such that  ⁢ | b ⁢ ( x ) | 2 2 - 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 ∈ L 1 ⁢ ( Ω ) {f\kern-1.0pt\in\kern-1.0ptL^{1}(\Omega)} and b ⁢ ( x ) ∈ L 2 2 - 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.