In this paper, we deal with the following problem：By variational method, we prove the existenceof a nontrivial weak solution whenand the existence of a cylindricalweak solution when
We study the following Schr?dinger equation with variable exponent−Δu+u=u^(p+∈a)(x),u>0 in R^(N),where∈>0,1<p<N+2/N−2,a(x)∈C^(1)(R^(N))∩L^(∞)(R^(N)),N≥3.Under certain assumptions on a vector field r...We study the following Schr?dinger equation with variable exponent−Δu+u=u^(p+∈a)(x),u>0 in R^(N),where∈>0,1<p<N+2/N−2,a(x)∈C^(1)(R^(N))∩L^(∞)(R^(N)),N≥3.Under certain assumptions on a vector field related to a(x),we use the Lyapunov–Schmidt reduction to show the existence of single peak solutions to the above problem.We also obtain local uniqueness and exact multiplicity results for this problem by the Pohozaev type identity.展开更多
We developed a simple polynomial taper equation for poplars growing on former farmland in Sweden and also evaluated the performance of some well-known taper equations. In Sweden there is an increasing interest in the ...We developed a simple polynomial taper equation for poplars growing on former farmland in Sweden and also evaluated the performance of some well-known taper equations. In Sweden there is an increasing interest in the use of poplar. Effective management of poplar plantations for high yield production would be facilitated by taper equations providing better predictions of stem volume than currently available equations. In the study a polynomial stem taper equation with five parameters was established for individual poplar trees growing on former farmland. The outputs of the polynomial taper equation were compared with five published equations. Data for fitting the equations were collected from 69 poplar trees growing at 37 stands in central and southern Sweden （lat. 55–60° N）. The mean age of the stands was 21 years （range 14–43）, the mean density 984 stems·ha？1 （198–3,493）, and the mean diameter at breast height （outside bark） 25 cm （range 12–40）. To verify the tested equations, performance of accuracy and precision diameter predictions at seven points along the stem was closely analyzed. Statistics used for evaluation of the equations indicated that the variable exponent taper equation presented by Kozak （1988） performed best and can be recommended. The stem taper equation by Kozak （1988） recommended in the study is likely to be beneficial for optimising the efficiency and profitability of poplar plantation management. The constructed polynomial equation and the segmented equation presented by Max ＆ Burkhart （1976） were second and third ranked. Due to the statistical complexity of Kozak’s equation, the constructed polynomial equation is alternatively recommended when a simple model is requested and larger bias is accepted.展开更多
This paper deals with the following doubly nonlinear parabolic equations(u + |u|r(x)-2u)t-div(|?u|m(x)-2?u) = |u|p(x)-2u, where the exponents of nonlinearity r(x), m(x) and p(x) are given functions. Under some appropr...This paper deals with the following doubly nonlinear parabolic equations(u + |u|r(x)-2u)t-div(|?u|m(x)-2?u) = |u|p(x)-2u, where the exponents of nonlinearity r(x), m(x) and p(x) are given functions. Under some appropriate assumptions on the exponents of nonlinearity, and with certain initial data, a blow-up result is established with positive initial energy.展开更多
We study the Dirichlet problem associated to strongly nonlinear parabolic equations involving p（x） structure in W;L;（Q）. We prove the existence of weak solutions by applying Galerkin’s approximation method.
Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical ...Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients：{-div（｜△u｜p-2△u）=｜x｜s｜u｜p＊（s）-2u＋λ｜x｜t｜u｜p-2u, x∈B1, u｜σB1 =0, where t, s〉-p, 2≤p〈N, p＊（s）= （N＋s）pN-p andλ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N 〉p（p-1）t＋p（p2-p＋1） andλ∈（0,λ1,t）, whereλ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ （ps＋p） min{1, p＋t/p＋s}＋p2p-（p-1） min{1, p＋tp＋s} andλ〉0 is small.展开更多
This paper is devoted to the study of a nonlinear anisotropic elliptic e-quation with degenerate coercivity,lower order term and L1 datum in appropriate anisotropic variable exponents Sobolev spaced We obtain the exis...This paper is devoted to the study of a nonlinear anisotropic elliptic e-quation with degenerate coercivity,lower order term and L1 datum in appropriate anisotropic variable exponents Sobolev spaced We obtain the existence of distributional solutions.展开更多
We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.
The author first studies the Lipschitz properties of the monotone and relative rearrangement mappings in variable exponent Lebesgue spaces completing the result given in [9]. This paper is ended by establishing the Li...The author first studies the Lipschitz properties of the monotone and relative rearrangement mappings in variable exponent Lebesgue spaces completing the result given in [9]. This paper is ended by establishing the Lipschitz properties for quasilinear problems with variable exponent when the right-hand side is in some dual spaces of a suitable Sobolev space associated to variable exponent.展开更多
The existence of bounded weak solutions,to a class of nonlinear elliptic equations with variable exponents,is investigated in this article.A uniform a priori L^(∞) estimate is obtained by the De Giorgi iterative tech...The existence of bounded weak solutions,to a class of nonlinear elliptic equations with variable exponents,is investigated in this article.A uniform a priori L^(∞) estimate is obtained by the De Giorgi iterative tech-nique.Thanks to the weak convergence method and Minty's trick,the existence result is proved through limit process.展开更多
基金Supported by the National Science Foundation of China(11071245 and 11101418)
文摘In this paper, we deal with the following problem：By variational method, we prove the existenceof a nontrivial weak solution whenand the existence of a cylindricalweak solution when
基金National Natural Science Foundation of China(Grant Nos.11971147 and12371111)The second author is partially supported by National Natural Science Foundation of China(Grant No.11831009)+1 种基金the Fundamental Research Funds for the Central Universities(Grant Nos.KJ02072020-0319,CCNU22LJ002)The third author is supported by National Natural Science Foundation of China(Grant No.12201232)。
文摘We study the following Schr?dinger equation with variable exponent−Δu+u=u^(p+∈a)(x),u>0 in R^(N),where∈>0,1<p<N+2/N−2,a(x)∈C^(1)(R^(N))∩L^(∞)(R^(N)),N≥3.Under certain assumptions on a vector field related to a(x),we use the Lyapunov–Schmidt reduction to show the existence of single peak solutions to the above problem.We also obtain local uniqueness and exact multiplicity results for this problem by the Pohozaev type identity.
基金financially supported by Skogssll-skapet foundation
文摘We developed a simple polynomial taper equation for poplars growing on former farmland in Sweden and also evaluated the performance of some well-known taper equations. In Sweden there is an increasing interest in the use of poplar. Effective management of poplar plantations for high yield production would be facilitated by taper equations providing better predictions of stem volume than currently available equations. In the study a polynomial stem taper equation with five parameters was established for individual poplar trees growing on former farmland. The outputs of the polynomial taper equation were compared with five published equations. Data for fitting the equations were collected from 69 poplar trees growing at 37 stands in central and southern Sweden （lat. 55–60° N）. The mean age of the stands was 21 years （range 14–43）, the mean density 984 stems·ha？1 （198–3,493）, and the mean diameter at breast height （outside bark） 25 cm （range 12–40）. To verify the tested equations, performance of accuracy and precision diameter predictions at seven points along the stem was closely analyzed. Statistics used for evaluation of the equations indicated that the variable exponent taper equation presented by Kozak （1988） performed best and can be recommended. The stem taper equation by Kozak （1988） recommended in the study is likely to be beneficial for optimising the efficiency and profitability of poplar plantation management. The constructed polynomial equation and the segmented equation presented by Max ＆ Burkhart （1976） were second and third ranked. Due to the statistical complexity of Kozak’s equation, the constructed polynomial equation is alternatively recommended when a simple model is requested and larger bias is accepted.
基金supported by the National Natural Science Foundation of China(No.11801145)
文摘This paper deals with the following doubly nonlinear parabolic equations(u + |u|r(x)-2u)t-div(|?u|m(x)-2?u) = |u|p(x)-2u, where the exponents of nonlinearity r(x), m(x) and p(x) are given functions. Under some appropriate assumptions on the exponents of nonlinearity, and with certain initial data, a blow-up result is established with positive initial energy.
文摘We study the Dirichlet problem associated to strongly nonlinear parabolic equations involving p（x） structure in W;L;（Q）. We prove the existence of weak solutions by applying Galerkin’s approximation method.
基金supported by the National Natural Science Foundation of China(11326139,11326145)Tian Yuan Foundation(KJLD12067)Hubei Provincial Department of Education(Q20122504)
文摘Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients：{-div（｜△u｜p-2△u）=｜x｜s｜u｜p＊（s）-2u＋λ｜x｜t｜u｜p-2u, x∈B1, u｜σB1 =0, where t, s〉-p, 2≤p〈N, p＊（s）= （N＋s）pN-p andλ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N 〉p（p-1）t＋p（p2-p＋1） andλ∈（0,λ1,t）, whereλ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ （ps＋p） min{1, p＋t/p＋s}＋p2p-（p-1） min{1, p＋tp＋s} andλ〉0 is small.
文摘This paper is devoted to the study of a nonlinear anisotropic elliptic e-quation with degenerate coercivity,lower order term and L1 datum in appropriate anisotropic variable exponents Sobolev spaced We obtain the existence of distributional solutions.
文摘We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.
文摘The author first studies the Lipschitz properties of the monotone and relative rearrangement mappings in variable exponent Lebesgue spaces completing the result given in [9]. This paper is ended by establishing the Lipschitz properties for quasilinear problems with variable exponent when the right-hand side is in some dual spaces of a suitable Sobolev space associated to variable exponent.
基金supported in part by the National Natural Science Foundation of China(Grant No.11901131)Youth science and technology talent growth project in Guizhou Province for ordinary institutions of higher learning(KY[2021]142).
文摘The existence of bounded weak solutions,to a class of nonlinear elliptic equations with variable exponents,is investigated in this article.A uniform a priori L^(∞) estimate is obtained by the De Giorgi iterative tech-nique.Thanks to the weak convergence method and Minty's trick,the existence result is proved through limit process.
