The existence and multiplicity of positive solutions for equation(1.1)with the new critical exponent 4<p<2·2*shall be investigated in a high dimension.The conclusions extend the relative results recently at...The existence and multiplicity of positive solutions for equation(1.1)with the new critical exponent 4<p<2·2*shall be investigated in a high dimension.The conclusions extend the relative results recently attained in[1]for the one-dimensional case.More precisely,as the coefficient a(x)in the nonlinearity is sign-changing,the modified term 2(Δ(|u|2))u is still helpful for obtaining multiple positive solutions in a high dimension,even if a sign condition like∫_(R)N a(x)e_(1)^(p)dx<0(also named“a necessary condition”see[2,3])does not hold.展开更多
A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect ...A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.展开更多
基金Partially the NSFC(11671085)the Science Foundation of Fujian Province(2020J01160)。
文摘The existence and multiplicity of positive solutions for equation(1.1)with the new critical exponent 4<p<2·2*shall be investigated in a high dimension.The conclusions extend the relative results recently attained in[1]for the one-dimensional case.More precisely,as the coefficient a(x)in the nonlinearity is sign-changing,the modified term 2(Δ(|u|2))u is still helpful for obtaining multiple positive solutions in a high dimension,even if a sign condition like∫_(R)N a(x)e_(1)^(p)dx<0(also named“a necessary condition”see[2,3])does not hold.
基金Supported by NSFC Grant(11401100,10601021)the foundation of Fujian Education Department(JB14021)the innovation foundation of Fujian Normal University(IRTL1206)
文摘A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.
