一类半线性椭圆型方程组正解的存在性
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摘要
主要研究一类半线性椭圆型方程组正解的存在性.利用变分法将椭圆型方程组解的问题转化为相应能量泛函的临界点问题,进一步证明了方程组能量泛函临界点的存在性.
This paper mainly studies the existence of positive solutions for a class of semilinear elliptic systems. By using the variational method,the problem of solving elliptic equations is transformed into the critical point problem of the corresponding energy functional,and the existence of the critical point of the energy functional of the equations is further proved.
This paper mainly studies the existence of positive solutions for a class of semilinear elliptic systems. By using the variational method,the problem of solving elliptic equations is transformed into the critical point problem of the corresponding energy functional,and the existence of the critical point of the energy functional of the equations is further proved.
引文
[1]LIU Z X,HAN P.Infinitely many solutions for elliptic systems with critical exponents[J].Journal of Mathematical Analysis and Applications,2009,353(2):544-552.
[2]CAO D,HAN P.High energy positive solutions of Neumann problem for an elliptic system of equations with critical nonlinearities[J].Calculus of Variations and Partial Differential Equations,2006,25(2):161-185.
[3]WU T F.The Nehari mainfold for a semilinear elliptic system involving sign-changing weight function[J].Nonlinear Analysis,2008,68(6):1733-1745.
[4]ALVES C O,MORAIS FILHO D C D,SOUTO M A S.On systems of elliptic equations involving subcritical or critical sobolev exponents[J].Nonlinear Analysis,2000,42(5):771-787.
[5]BEOWN K J,WU T F.A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function[J].Journal of Mathematical Analysis and Applications,2008,337(2):1326-1336.
[6]ZHANG Y J.Multiple solutions of an inhomogeneous Neumann problem for an elliptic system with critical Sobolev exponent[J].Nonlinear Analysis Theory Methods and Applications,2012,75(4):2047-2059.
[7]LIU Z X.Infinitely many solutions for some nonlinear scalar system of two elliptic equations[J].Journal of Mathematical Analysis and Applications,2011,382(2):731-747.
[8]CAO D M,ZHOU H S.Multiple positive solutions of nonhomgeneous semilinear elliptic equations in RN[J].Proceedings of the Royal Society of Edinburgh,1996,126(2):443-463.
[9]BREZIS H,NIRENBERG L.Positive solution of nonlinear elliptic equations involving critical Sobolev exponents[J].Communications on Pure and Applied Mathematics,2010,36(4):437-477.
[10]DING W Y,NI W M.On the existence of entire solution of a semilinear elliptic equation[J].Archive for Rational Mechanics and Analysis,1986,91(4):283-308.
[11]LIONS P L.The concentration-compactness principle in the calculus of variations:The locally compact case,PartⅠandⅡ[J].Annales De Linstitut Henri Poincare Non Linear Analysis,1984,1(2):109-145.
[12]LIONS P L.On positive solutions of semilinear elliptic equations in unbounded domains[J].1988,13:85-122.
[13]STRAUSS W A.Existence of solitary waves in higher dimensions[J].Communications in Mathematical Physics,1977,55(2):149-162.
[14]WILLEM M.Minimax Theorems[M].Springer Science and Business Media,1997.
[15]王振华,张为元,李艳艳.一类非线性次椭圆拉普拉斯方程解的对程性[J].西南民族大学报(自然科学版),2017,43(4):408-413.
[16]GRAHAM-EAGLE J.Monotone method for semilinear elliptic equations in unbounded domains[J].Journal of Mathematical Analysis and Applications,1989,137(1):122-131.
[17]TARANTELLO G.On nonhomogeneous elliptic equations involving critical Sobolev exponent[J].Annales De Linstitut Henri Poincare Non Linear Analysis,2016,9(3):281-304.
[18]GILBARG D,TRUDINGER N.S.Elliptic Partial Differential Equations of Second Order.Springer,Berlin[M].2001.
[19]AUBIN J P,EKELAND I.Applied Nonlinear Analysis[J].Communications on Pure and Applied Mathematics,1984.
[2]CAO D,HAN P.High energy positive solutions of Neumann problem for an elliptic system of equations with critical nonlinearities[J].Calculus of Variations and Partial Differential Equations,2006,25(2):161-185.
[3]WU T F.The Nehari mainfold for a semilinear elliptic system involving sign-changing weight function[J].Nonlinear Analysis,2008,68(6):1733-1745.
[4]ALVES C O,MORAIS FILHO D C D,SOUTO M A S.On systems of elliptic equations involving subcritical or critical sobolev exponents[J].Nonlinear Analysis,2000,42(5):771-787.
[5]BEOWN K J,WU T F.A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function[J].Journal of Mathematical Analysis and Applications,2008,337(2):1326-1336.
[6]ZHANG Y J.Multiple solutions of an inhomogeneous Neumann problem for an elliptic system with critical Sobolev exponent[J].Nonlinear Analysis Theory Methods and Applications,2012,75(4):2047-2059.
[7]LIU Z X.Infinitely many solutions for some nonlinear scalar system of two elliptic equations[J].Journal of Mathematical Analysis and Applications,2011,382(2):731-747.
[8]CAO D M,ZHOU H S.Multiple positive solutions of nonhomgeneous semilinear elliptic equations in RN[J].Proceedings of the Royal Society of Edinburgh,1996,126(2):443-463.
[9]BREZIS H,NIRENBERG L.Positive solution of nonlinear elliptic equations involving critical Sobolev exponents[J].Communications on Pure and Applied Mathematics,2010,36(4):437-477.
[10]DING W Y,NI W M.On the existence of entire solution of a semilinear elliptic equation[J].Archive for Rational Mechanics and Analysis,1986,91(4):283-308.
[11]LIONS P L.The concentration-compactness principle in the calculus of variations:The locally compact case,PartⅠandⅡ[J].Annales De Linstitut Henri Poincare Non Linear Analysis,1984,1(2):109-145.
[12]LIONS P L.On positive solutions of semilinear elliptic equations in unbounded domains[J].1988,13:85-122.
[13]STRAUSS W A.Existence of solitary waves in higher dimensions[J].Communications in Mathematical Physics,1977,55(2):149-162.
[14]WILLEM M.Minimax Theorems[M].Springer Science and Business Media,1997.
[15]王振华,张为元,李艳艳.一类非线性次椭圆拉普拉斯方程解的对程性[J].西南民族大学报(自然科学版),2017,43(4):408-413.
[16]GRAHAM-EAGLE J.Monotone method for semilinear elliptic equations in unbounded domains[J].Journal of Mathematical Analysis and Applications,1989,137(1):122-131.
[17]TARANTELLO G.On nonhomogeneous elliptic equations involving critical Sobolev exponent[J].Annales De Linstitut Henri Poincare Non Linear Analysis,2016,9(3):281-304.
[18]GILBARG D,TRUDINGER N.S.Elliptic Partial Differential Equations of Second Order.Springer,Berlin[M].2001.
[19]AUBIN J P,EKELAND I.Applied Nonlinear Analysis[J].Communications on Pure and Applied Mathematics,1984.