一类p-Laplacian方程解的存在性及多重性
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摘要
文中首先考虑如下带有Dirichlet边界条件的p-Laplacian方程：(1)其中△_pu为p-Laplacian算子：△_pu=div(|▽u|~(p-2)▽u)，f∈C((?)×R，R)，λ>0为参数．假设p>1，Ω为R~N(N≥1)中的带有光滑边界(?)Ω的有界区域．运用山路引理得到方程(1)解及多解的存在性．
主要结果如下：
定理1假设在方程(1)中f满足以下条件：(f_1)f∈C((?)×R，R)，当t∈R，x∈(?)时f(x，t)≥0；(f_2)(?)f(x,t)/t~(p-1)=0关于x∈(?)几乎处处一致成立；(f_3)当N≤p时存在q∈(0，+∞)，当N>p时存在q∈(p，pN/N-p-1)使得(?)f(x,t)/t~q=0关于x∈(?)几乎处处一致成立；(f_4)存在θ>p，r>0使得当t≥r时对任意(x，t)∈(?)×R~+有0<θF(x，t)≤tf(x，t)，其中F(x，t)=∫_0~tf(x，s)ds.那么对所有λ>0方程(1)至少有一个正解．
定理2假设在方程(1)中f满足(f_1~*)f∈C((?)×R，R)，当t∈R，x∈(?)时f(x，t)t≥0；(f_2~*)(?)f(x,t)/|t|~(p-2)t=0关于x∈(?)几乎处处一致成立；(f_3~*)当N≤p时存在q∈(0，+∞)，当N>p时存在q∈(p，pN/N-p-1)使得(?)f(x,t)/|t|~q=0关于x∈(?)几乎处处一致成立；(f_4~*)存在θ>p，r>0使得当|t|≥r时对所有(x，t)∈(?)×R有0<θF(x，t)≤tf(x，t)．那么对所有λ>0方程(1)至少有一正一负两解．接下来讨论R~N上的p-Laplacian方程-△_pu+|u|~(p-2)u=f(u)，u∈W~(1,p)(R~N) (2)最小能量解的存在性，其中1 主要结果如下：
定理3假设在方程(2)中f∈C(R，R)满足(f_5)(?)f(t)/|t|→0|t|~(p-2)t=0；(f_6)存在pN/p(q-p)和常数a>0使得当t∈R＼{0}时f(t)t-pF(t)≥a|t|~μ>0．那么方程(2)至少有一个最小能量解．最后讨论了R~N上的p-Laplacian方程-△_pu+V(x)|u|~(p-2)u=f(u)，u∈W~(1,p)(R~N) (3)正解及最小能量解的存在性，其中1 主要结果如下：
定理4设在方程(3)中f和V满足以下条件(f_(10))当t>0时，f(t)≥0；当t≤0时，f(t)≡0；(f_(11))当N=p时存在q<∞，当N>p时存在q 定理5在定理4的假设下，方程(3)有一个最小能量解．即是说，存在一个解w∈W~(1,p)(R~N)使得I(w)=m，其中m=inf{I(u)|u≠0，I′(u)=0}．
In this paper,firstly,we consider the following p-Laplacain equations with Dirichletboundary value condition:(1)where△_pu=div(|▽u|~(p-2)▽u)，is the p-Laplacian operator with 10 is a real parameter f:(?)×R→'R is a continuous function.We will proof the existence of positive solution and ground statesolution of (1)by using the Mountain Pass Theorem.
The main result is the following theorem
Theorem 1 Suppose that the function f satisfies the following assumptions(f_1)f∈C((?)×R，R)，f(x,t≥0 for all t∈R，x∈(?)；(f_2)f(x,t)=o(t~(p-1) as t→0~+ uniformly for a.e.x∈(?);(f_3)there exists q>0 if N≤p,pp such that(?)f(x,t)/t~q=0,uniformly for a.e x∈(?);(f_4)there existθ>p,r>0 such that0<θF(x,t)≤tf(x,t),(?)(x,t)∈(?)×R~+,t≥r,where F(x,t)=∫_0~tf(x，s)ds.Then problem (1)has a positive solution for everyλ>0.
Theorem 2 Assume that f satisfies the following conditions (f_1~*)f∈C((?)×R，R)，f(x，t)t≥0 for all t∈R，x∈(?);(f_2~*)f(x,t)o(|t|~(p-2)t)as|t|→0 uniformly for a.e.x∈(?);(f_3~*)there exists q>0 if N≤p,pp such that(?)f(x,t)/|t|~q=0,uniformly for a.e x∈(?);(f_4~*)there existθ>p,r>0 such that 0<θF(x,t)≤tf(x,t),(?)(x,t)∈(?)×R,|t|≥r.Then problem (1)has at least two nontrivial solutions for everyλ>0 in which one ispositive and the other is negative.Next we consider the existence of ground state solution for the following p-Laplacianequation in R~N.-△_pu+|u|~(p-2)u=f(u)，u∈W~(1,p)(R~N) (2)where 1 The main result is the following theorem
Theorem 3 Suppose that f∈C(R,R)satisfies the following assumptions(f_5)f(t)=o(|t|~(p-2)t)as|t|→0;(f_6)there exists pN/p(q-p)and a constant a>0 such that f(t)t-pF(t)≥a|t|~μ>0 for all t∈R\{0}.Then problem (2)has a ground state solution.At last,we consider the following p-Laplacian equation in R~N-△_pu+V(x)|u|~(p-2)u=f(u)，u∈W~(1,p)(R~N) (3)where 1 The main results are the following:
Theorem 4 Assume that f and V satisfy the following conditions (f_(10))f(t)≥0 for all t>0 and f(t)≡0 for all t≤0;(f_(11)) there is q<∞if N=p,qp such that(?)f(t)/t~q=0(f_(12))(?)f(t)/t~(p-1)=a,where a≥0;(f_(13))(?)f(t)/t(p-1)=+∞;(v_1)a<α_0，whereα_0=(?)∫R~N(|▽u|~p+V(x)|u|~p)dx/∫R~N|u|~pdx;(v_2)0＜(?)V(x)≤V(x)≤(?)V(x)=V(∞)＜+∞;(v_3) there exists a function (?)∈L~2(R~N)∩W~(1,∞)(R~N) such that|x||▽V(x)|≤(?)(x)~2,(?)_x∈R~N.Then the problem (3) has a non trivial positive solution.
Theorem 5 Under the assumption of Theorem 4, the problem (3) has a ground statesolution. Namely, there exists a solution w∈W~(1,p)(R~N) such that I(w)=m, where m= inf{I(u)|u≠0,I′(u)=0}.
主要结果如下：
定理1假设在方程(1)中f满足以下条件：(f_1)f∈C((?)×R，R)，当t∈R，x∈(?)时f(x，t)≥0；(f_2)(?)f(x,t)/t~(p-1)=0关于x∈(?)几乎处处一致成立；(f_3)当N≤p时存在q∈(0，+∞)，当N>p时存在q∈(p，pN/N-p-1)使得(?)f(x,t)/t~q=0关于x∈(?)几乎处处一致成立；(f_4)存在θ>p，r>0使得当t≥r时对任意(x，t)∈(?)×R~+有0<θF(x，t)≤tf(x，t)，其中F(x，t)=∫_0~tf(x，s)ds.那么对所有λ>0方程(1)至少有一个正解．
定理2假设在方程(1)中f满足(f_1~*)f∈C((?)×R，R)，当t∈R，x∈(?)时f(x，t)t≥0；(f_2~*)(?)f(x,t)/|t|~(p-2)t=0关于x∈(?)几乎处处一致成立；(f_3~*)当N≤p时存在q∈(0，+∞)，当N>p时存在q∈(p，pN/N-p-1)使得(?)f(x,t)/|t|~q=0关于x∈(?)几乎处处一致成立；(f_4~*)存在θ>p，r>0使得当|t|≥r时对所有(x，t)∈(?)×R有0<θF(x，t)≤tf(x，t)．那么对所有λ>0方程(1)至少有一正一负两解．接下来讨论R~N上的p-Laplacian方程-△_pu+|u|~(p-2)u=f(u)，u∈W~(1,p)(R~N) (2)最小能量解的存在性，其中1
定理3假设在方程(2)中f∈C(R，R)满足(f_5)(?)f(t)/|t|→0|t|~(p-2)t=0；(f_6)存在p
定理4设在方程(3)中f和V满足以下条件(f_(10))当t>0时，f(t)≥0；当t≤0时，f(t)≡0；(f_(11))当N=p时存在q<∞，当N>p时存在q
In this paper,firstly,we consider the following p-Laplacain equations with Dirichletboundary value condition:(1)where△_pu=div(|▽u|~(p-2)▽u)，is the p-Laplacian operator with 1
The main result is the following theorem
Theorem 1 Suppose that the function f satisfies the following assumptions(f_1)f∈C((?)×R，R)，f(x,t≥0 for all t∈R，x∈(?)；(f_2)f(x,t)=o(t~(p-1) as t→0~+ uniformly for a.e.x∈(?);(f_3)there exists q>0 if N≤p,p
Theorem 2 Assume that f satisfies the following conditions (f_1~*)f∈C((?)×R，R)，f(x，t)t≥0 for all t∈R，x∈(?);(f_2~*)f(x,t)o(|t|~(p-2)t)as|t|→0 uniformly for a.e.x∈(?);(f_3~*)there exists q>0 if N≤p,p
Theorem 3 Suppose that f∈C(R,R)satisfies the following assumptions(f_5)f(t)=o(|t|~(p-2)t)as|t|→0;(f_6)there exists p
Theorem 4 Assume that f and V satisfy the following conditions (f_(10))f(t)≥0 for all t>0 and f(t)≡0 for all t≤0;(f_(11)) there is q<∞if N=p,q
Theorem 5 Under the assumption of Theorem 4, the problem (3) has a ground statesolution. Namely, there exists a solution w∈W~(1,p)(R~N) such that I(w)=m, where m= inf{I(u)|u≠0,I′(u)=0}.
引文
[1] 张恭庆．临界点理论及其应用，科学技术出版社，1986．
[2] 钟承奎，范先令，陈文山原．非线性分析理论[M]，兰州大学出版社，1998．
[3] El Amrouss A R.Moussaoui M.Minimax principles for critical-point theory in applications to quasilinear boundary-value problems [J].Electron.J.Differential Equations 2000,no.18,1-9.
[4] Z.H.Chen,Y.T.Shen.On the existence of nontrivial solutions of quasi-asymptotically linear problem for the p-Laplacian [J].Acta Math.Appl.Sin.Engl.Ser.18 (2002),no.4,599-606.
[5] Y.S.Huang,H.S.Zhou.Positive solution for-△_pu=f(x,u)with f(x,u)growing as u~(p-1) at infinity [J].Appl.Math.Lett.17 (2004),no.8,881-887.
[6] G.A.Afrouzi.On positive mountain pass solutions for a semilinear elliptic boundary value problem [J].Appl.Math.Comput.167 (2005),no.1,76-80.
[7] P.H.Rabinowitz.Minimax Methods in Critical Point Theory with Appliction to DifferentialEquation [J].CBMS Reg.Conf.Ser.in Math,American Mathematical Society,Province,RI,1986,vol.65.
[8] J.L.Vazquez.A strong maximum principle for some quasilinear elliptic equations [J].Appl.Math.Optim.12 (1984),no.3,191-202.
[9] Z.H.Chen,Y.T.Shen,Y.X.Yao.Some existence results of solutions for p-Laplacian.Acta Math.Sci.Ser.B Engl.Ed.23 (2003),no.4,487-496.
[10] D.G.Costa,O.H.Miyagaki.Nontrivial solutions for perturbations of the p-Laplacian on unbounded domains.J.Math.Anal.Appl.193 (1995),no.3,737-755.
[11] G.M.Figueiredo,M.F.Furtado.Multiplicity of positive solutions for a class of elliptic equations in divergence form.J.Math.Anal.Appl.321 (2006),no.2,705-721.
[12] L.Jeanjean,K.Tanaka.A remark on least energy solutions in R~N.Proc.Amer.Math.Soc.131 (2003),no.8,2399-2408.
[13] Q.S.Jiu,J.B.Su.Existence and multiplicity results for Dirichlet problems with p-Laplacian.J.Math.Anal.Appl.281 (2003),no.2,587-601.
[14] P.L.Lions.The concentration-compactness principle in the calculus of variations.The locally compact case.II.Ann.Inst.H.Poincar é Anal.Non Lin é aire 1(1984),no.4,223-283.
[15] M.Musso,D.Passaseo.Some nonlinear elliptic equations in R~N.Nonlinear Anal.39(2000),no.7,Ser.A:Theory Methods,837-860.
[16] E. A. Silva, S. H. M. Soares. Quasilinear Dirichlet problems in R~N with critical growth. Nonlinear Anal. 43 (2001), no. 1, 1-20.
[17] Z. W. Tang. Existence and asymptotic behavior of the solutions for a nonlinear elliptic equation arising in astrophysics. ActaMath. Sci. Ser. B Engl. Ed. 26 (2006), no. 2, 229-245.
[18] M. Willem. Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, 24. Birkha|¨user Boston, Inc., Boston, MA, 1996.
[19] Y. Lin, C. L. Tang. Existence of nontrivial solutions for a class of p-Laplacian equations.《西南大学学报(自然科学版)》(核心期刊),30(2008),no.2,1-4.
[20] C. O. Alves, P. C. Carriao, O. H. Miyagaki. Existence and multiplicity results for a class of resonant quasilinear elliptic problems on R~N. Nonlinear Anal. 39 (2000), no. 1, Ser. A: Theory Methods, 99-110.
[21] C. O. Alves. Existence of positive solutions for a problem with lack of compactness involving the p-Laplacian. Nonlinear Anal. 51 (2002), no. 7, 1187-1206.
[22] D. G. Costa, C. A. Magalhaes. Existence results for perturbations of the p-Laplacian. Nonlinear Anal. 24 (1995), no. 3, 409-418.
[23] G. M. Figueiredo. Multiplicity of solutions for a quasilinear problem with supercritical growth. Electron. J. Differential Equations 2006, No. 31, 7 pp.
[24] K. Perera. Nontrivial solutions of p-superlinear p-Laplacian problems. Appl. Anal. 82 (2003), no. 9, 883-888.
[25] J. D. Zhu, X. L. Fan. Multiplicity of solutions for a quasilinear elliptic equation with conflicting nonlinearities on R~N. J. Lanzhou Univ. Nat. Sci. 37 (2001), no. 6, 1-5.
[26] J. M. do 6, E. S. Medeiros. Remarks on least energy solutions for quasilinear elliptic problems in R~N. Electron. J. Differential Equations 2003, No. 83, 14 pp.
[27] L. Jeanjean, K. Tanaka. A positive solution for a nonlinear Schrodinger equation on R~N. Indiana Univ. Math. J. 54 (2005), no. 2, 443-464.
[28] L. Jeanjean, J. F. Toland. Bounded Palais-Smale mountain-pass sequences. C. R. Acad. Sci. Paris S e r. I Math. 327 (1998), no. 1, 23-28.
[29] M. Badiale, G. Citti. Concentration compactness principle and quasilinear elliptic equations in R~n, Comm. Partial Differential Equations 16 (1991), no. 11, 1795-1818.
[30] M. F. Furtado. Multiple minimal nodal solutions for a quasilinear Schrodinger equation with symmetric potential. (English summary) J. Math. Anal. Appl. 304 (2005), no. 1, 170-188.
[31] Y. Q. Li, Z. Q. Wang, J. Zeng. Ground states of nonlinear Schrodinger equations with potentials. Ann. Inst. H. Poincar e Anal. Non Lin e aire 23 (2006), no. 6, 829-837.
[2] 钟承奎，范先令，陈文山原．非线性分析理论[M]，兰州大学出版社，1998．
[3] El Amrouss A R.Moussaoui M.Minimax principles for critical-point theory in applications to quasilinear boundary-value problems [J].Electron.J.Differential Equations 2000,no.18,1-9.
[4] Z.H.Chen,Y.T.Shen.On the existence of nontrivial solutions of quasi-asymptotically linear problem for the p-Laplacian [J].Acta Math.Appl.Sin.Engl.Ser.18 (2002),no.4,599-606.
[5] Y.S.Huang,H.S.Zhou.Positive solution for-△_pu=f(x,u)with f(x,u)growing as u~(p-1) at infinity [J].Appl.Math.Lett.17 (2004),no.8,881-887.
[6] G.A.Afrouzi.On positive mountain pass solutions for a semilinear elliptic boundary value problem [J].Appl.Math.Comput.167 (2005),no.1,76-80.
[7] P.H.Rabinowitz.Minimax Methods in Critical Point Theory with Appliction to DifferentialEquation [J].CBMS Reg.Conf.Ser.in Math,American Mathematical Society,Province,RI,1986,vol.65.
[8] J.L.Vazquez.A strong maximum principle for some quasilinear elliptic equations [J].Appl.Math.Optim.12 (1984),no.3,191-202.
[9] Z.H.Chen,Y.T.Shen,Y.X.Yao.Some existence results of solutions for p-Laplacian.Acta Math.Sci.Ser.B Engl.Ed.23 (2003),no.4,487-496.
[10] D.G.Costa,O.H.Miyagaki.Nontrivial solutions for perturbations of the p-Laplacian on unbounded domains.J.Math.Anal.Appl.193 (1995),no.3,737-755.
[11] G.M.Figueiredo,M.F.Furtado.Multiplicity of positive solutions for a class of elliptic equations in divergence form.J.Math.Anal.Appl.321 (2006),no.2,705-721.
[12] L.Jeanjean,K.Tanaka.A remark on least energy solutions in R~N.Proc.Amer.Math.Soc.131 (2003),no.8,2399-2408.
[13] Q.S.Jiu,J.B.Su.Existence and multiplicity results for Dirichlet problems with p-Laplacian.J.Math.Anal.Appl.281 (2003),no.2,587-601.
[14] P.L.Lions.The concentration-compactness principle in the calculus of variations.The locally compact case.II.Ann.Inst.H.Poincar é Anal.Non Lin é aire 1(1984),no.4,223-283.
[15] M.Musso,D.Passaseo.Some nonlinear elliptic equations in R~N.Nonlinear Anal.39(2000),no.7,Ser.A:Theory Methods,837-860.
[16] E. A. Silva, S. H. M. Soares. Quasilinear Dirichlet problems in R~N with critical growth. Nonlinear Anal. 43 (2001), no. 1, 1-20.
[17] Z. W. Tang. Existence and asymptotic behavior of the solutions for a nonlinear elliptic equation arising in astrophysics. ActaMath. Sci. Ser. B Engl. Ed. 26 (2006), no. 2, 229-245.
[18] M. Willem. Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, 24. Birkha|¨user Boston, Inc., Boston, MA, 1996.
[19] Y. Lin, C. L. Tang. Existence of nontrivial solutions for a class of p-Laplacian equations.《西南大学学报(自然科学版)》(核心期刊),30(2008),no.2,1-4.
[20] C. O. Alves, P. C. Carriao, O. H. Miyagaki. Existence and multiplicity results for a class of resonant quasilinear elliptic problems on R~N. Nonlinear Anal. 39 (2000), no. 1, Ser. A: Theory Methods, 99-110.
[21] C. O. Alves. Existence of positive solutions for a problem with lack of compactness involving the p-Laplacian. Nonlinear Anal. 51 (2002), no. 7, 1187-1206.
[22] D. G. Costa, C. A. Magalhaes. Existence results for perturbations of the p-Laplacian. Nonlinear Anal. 24 (1995), no. 3, 409-418.
[23] G. M. Figueiredo. Multiplicity of solutions for a quasilinear problem with supercritical growth. Electron. J. Differential Equations 2006, No. 31, 7 pp.
[24] K. Perera. Nontrivial solutions of p-superlinear p-Laplacian problems. Appl. Anal. 82 (2003), no. 9, 883-888.
[25] J. D. Zhu, X. L. Fan. Multiplicity of solutions for a quasilinear elliptic equation with conflicting nonlinearities on R~N. J. Lanzhou Univ. Nat. Sci. 37 (2001), no. 6, 1-5.
[26] J. M. do 6, E. S. Medeiros. Remarks on least energy solutions for quasilinear elliptic problems in R~N. Electron. J. Differential Equations 2003, No. 83, 14 pp.
[27] L. Jeanjean, K. Tanaka. A positive solution for a nonlinear Schrodinger equation on R~N. Indiana Univ. Math. J. 54 (2005), no. 2, 443-464.
[28] L. Jeanjean, J. F. Toland. Bounded Palais-Smale mountain-pass sequences. C. R. Acad. Sci. Paris S e r. I Math. 327 (1998), no. 1, 23-28.
[29] M. Badiale, G. Citti. Concentration compactness principle and quasilinear elliptic equations in R~n, Comm. Partial Differential Equations 16 (1991), no. 11, 1795-1818.
[30] M. F. Furtado. Multiple minimal nodal solutions for a quasilinear Schrodinger equation with symmetric potential. (English summary) J. Math. Anal. Appl. 304 (2005), no. 1, 170-188.
[31] Y. Q. Li, Z. Q. Wang, J. Zeng. Ground states of nonlinear Schrodinger equations with potentials. Ann. Inst. H. Poincar e Anal. Non Lin e aire 23 (2006), no. 6, 829-837.