We establish some results on the existence of multiple nontrivial solutions for a class of p(x)-Lap-lacian elliptic equations without assumptions that the domain is bounded. The main tools used in the proof are the va...We establish some results on the existence of multiple nontrivial solutions for a class of p(x)-Lap-lacian elliptic equations without assumptions that the domain is bounded. The main tools used in the proof are the variable exponent theory of generalized Lebesgue-Sobolev spaces, variational methods and a variant of the Mountain Pass Lemma.展开更多
This paper deals with the extinction of weak solutions of the initial and boundary value problem for ut = div（（｜u｜σ ＋ d0）｜ u｜^p（x）-2 u）. When the exponent belongs to different intervals, the solution has ...This paper deals with the extinction of weak solutions of the initial and boundary value problem for ut = div（（｜u｜σ ＋ d0）｜ u｜^p（x）-2 u）. When the exponent belongs to different intervals, the solution has different singularity （vanishing in finite time）.展开更多
We show the existence and multiplicity of solutions to degenerate p(x)-Laplace equations with Leray-Lions type operators using direct methods and critical point theories in Calculus of Variations and prove the uniquen...We show the existence and multiplicity of solutions to degenerate p(x)-Laplace equations with Leray-Lions type operators using direct methods and critical point theories in Calculus of Variations and prove the uniqueness and nonnegativeness of solutions when the principal operator is monotone and the nonlinearity is nonincreasing. Our operator is of the most general form containing all previous ones and we also weaken assumptions on the operator and the nonlinearity to get the above results. Moreover, we do not impose the restricted condition on p(x) and the uniform monotonicity of the operator to show the existence of three distinct solutions.展开更多
We study a degenerate elliptic system with variable exponents. Using the variational approach and some recent theory on weighted Lebesgue and Sobolev spaces with variable exponents, we prove the existence of at least ...We study a degenerate elliptic system with variable exponents. Using the variational approach and some recent theory on weighted Lebesgue and Sobolev spaces with variable exponents, we prove the existence of at least two distinct nontrivial weak solutions of the system. Several consequences of the main theorem are derived;in particular, the existence of at lease two distinct nontrivial non-negative solutions is established for a scalar degenerate problem. One example is provided to show the applicability of our results.展开更多
We are concerned with the following quasilinear wave equation involving variable sources and supercritical damping:■Generally speaking,when one tries to use the classical multiplier method to analyze tRhe asymptotic ...We are concerned with the following quasilinear wave equation involving variable sources and supercritical damping:■Generally speaking,when one tries to use the classical multiplier method to analyze tRhe asymptotic behavior of solutions,an inevitable step is to deal with the integralΩ|ut|^(m−2)utudx.A usual technique is to apply Young’s inequality and Sobolev embedding inequality to use the energy function and its derivative to control this integral for the subcritical or critical damping.However,for the supercritical case,the failure of the Sobolev embedding inequality makes the classical method be impossible.To do this,our strategy is to prove the rate of the integral RΩ|u|^(m)dx grows polynomially as a positive power of time variable t and apply the modified multiplier method to obtain the energy functional decays logarithmically.These results improve and extend our previous work[12].Finally,some numerical examples are also given to authenticate our results.展开更多
文摘We establish some results on the existence of multiple nontrivial solutions for a class of p(x)-Lap-lacian elliptic equations without assumptions that the domain is bounded. The main tools used in the proof are the variable exponent theory of generalized Lebesgue-Sobolev spaces, variational methods and a variant of the Mountain Pass Lemma.
基金Partially supported by the NSF(11271154)of China the 985 program of Jilin University
文摘This paper deals with the extinction of weak solutions of the initial and boundary value problem for ut = div（（｜u｜σ ＋ d0）｜ u｜^p（x）-2 u）. When the exponent belongs to different intervals, the solution has different singularity （vanishing in finite time）.
基金supported by the National Research Foundation of Korea Grant Funded by the Korea Government (Grant No. NRF-2015R1D1A3A01019789)
文摘We show the existence and multiplicity of solutions to degenerate p(x)-Laplace equations with Leray-Lions type operators using direct methods and critical point theories in Calculus of Variations and prove the uniqueness and nonnegativeness of solutions when the principal operator is monotone and the nonlinearity is nonincreasing. Our operator is of the most general form containing all previous ones and we also weaken assumptions on the operator and the nonlinearity to get the above results. Moreover, we do not impose the restricted condition on p(x) and the uniform monotonicity of the operator to show the existence of three distinct solutions.
基金supported in part by a University of Tennessee at Chattanooga SimCenter-Center of Excellence in Applied Computational Science and Engineering (CEACSE) grant
文摘We study a degenerate elliptic system with variable exponents. Using the variational approach and some recent theory on weighted Lebesgue and Sobolev spaces with variable exponents, we prove the existence of at least two distinct nontrivial weak solutions of the system. Several consequences of the main theorem are derived;in particular, the existence of at lease two distinct nontrivial non-negative solutions is established for a scalar degenerate problem. One example is provided to show the applicability of our results.
基金supported by the Scientific and Technological Project of jilin Province's Education Department in Thirteenth Five-Year(JKH20180111KI)supported by NSFC(11301211).
文摘We are concerned with the following quasilinear wave equation involving variable sources and supercritical damping:■Generally speaking,when one tries to use the classical multiplier method to analyze tRhe asymptotic behavior of solutions,an inevitable step is to deal with the integralΩ|ut|^(m−2)utudx.A usual technique is to apply Young’s inequality and Sobolev embedding inequality to use the energy function and its derivative to control this integral for the subcritical or critical damping.However,for the supercritical case,the failure of the Sobolev embedding inequality makes the classical method be impossible.To do this,our strategy is to prove the rate of the integral RΩ|u|^(m)dx grows polynomially as a positive power of time variable t and apply the modified multiplier method to obtain the energy functional decays logarithmically.These results improve and extend our previous work[12].Finally,some numerical examples are also given to authenticate our results.
