March  2012, 32(3): 795-826. doi: 10.3934/dcds.2012.32.795

Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent

1. 

Department of Mathematics, Huazhong Normal University, Wuhan 430079, China, China

2. 

Department of Mathematics, Central China Normal University, Wuhan 430079, China

Received  March 2010 Revised  August 2011 Published  October 2011

In this paper, we consider the following problem $$ \left\{ \begin{array}{ll} -\Delta u+u=u^{2^{*}-1}+\lambda(f(x,u)+h(x))\ \ \hbox{in}\ \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),\ \ u>0 \ \hbox{in}\ \mathbb{R}^{N}, \end{array} \right. (\star) $$ where $\lambda>0$ is a parameter, $2^* =\frac {2N}{N-2}$ is the critical Sobolev exponent and $N>4$, $f(x,t)$ and $h(x)$ are some given functions. We prove that there exists $0<\lambda^{*}<+\infty$ such that $(\star)$ has exactly two positive solutions for $\lambda\in(0,\lambda^{*})$ by Barrier method and Mountain Pass Lemma and no positive solutions for $\lambda >\lambda^*$. Moreover, if $\lambda=\lambda^*$, $(\star)$ has a unique solution $(\lambda^{*}, u_{\lambda^{*}})$, which means that $(\lambda^{*}, u_{\lambda^{*}})$ is a turning point in $H^{1}(\mathbb{R}^{N})$ for problem $(\star)$.
Citation: Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795
References:
[1]

A. Ambrosetti and M. Struwe, A note on the problem $-\Delta u=\lambda u+u|u| ^{2^\mathbf{star}-2}$, Manuscripta Math., 54 (1986), 373-379. doi: 10.1007/BF01168482.

[2]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300. doi: 10.1007/BF00282048.

[3]

A. Bahri and P.-L. Lions, Morse index of some min-max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math., 41 (1988), 1027-1037. doi: 10.1002/cpa.3160410803.

[4]

A. Bahri and P.-L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 365-413.

[5]

A. Bahri and Y. Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb{R}^N2$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.

[6]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[7]

G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 341-350.

[8]

K. Chen and C. Peng, Multiplicity and bifurcation of positive solutions for nonhomogeneous semilinear elliptic problems, J. Differential Equations, 240 (2007), 58-91. doi: 10.1016/j.jde.2007.05.023.

[9]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325.

[10]

D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb{R}^N2$, Proc. Roy. Soc. Edinburgh Sect., A 126 (1996), 443-463.

[11]

Y. Deng, Existence of multiple positive solutions for a semilinear equation with critical exponent, Proc. Roy. Soc. Edinburgh Sect., A 122 (1992), 161-175.

[12]

Y. B. Deng, Q. Gao and D. D. Zhang, Nodal Solutions for Laplace Equations with Critical Sobolev and Hardy Exponents on $\mathbb{R}$, Discrete and Continuous Dynamical Systems (DCDS-A), 19 (2007), 211-233.

[13]

Y. Deng and Y. Li, Existence and bifurcation of the positive solutions for a semilinear equation with critical exponent, J. Differential Equations, 130 (1996), 179-200. doi: 10.1006/jdeq.1996.0138.

[14]

Y. Deng, Z. Guo and G. Wang, Nodal solutions for $p$-Laplace equations with critical growth, Nonlinear Anal. TMA., 54 (2003), 1121-1151. doi: 10.1016/S0362-546X(03)00129-9.

[15]

Y. Deng, Y. Ma and X. Zhao, Existence and properties of multiple positive solutions for semi-linear equations with critical exponents, Rocky Mountain J. Math., 35 (2005), 1479-1512. doi: 10.1216/rmjm/1181069647.

[16]

Y. Deng, L. Jin and S. Peng, Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, Commun. Math. Sci,. 9 (2011), 859-878.

[17]

G. Cerami and R. Molle, On some Schrodinger equations with non regular potential at infinity, Discrete and Continuous Dynamical Systems (DCDS-A), 28 (2010), 827-844.

[18]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.

[19]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order," Springer-Verlag, Berlin, 1983.

[20]

J. Graham-Eagle, Monotone method for semilinear elliptic equations in unbounded domains, J. Math. Anal. Appl., 137 (1989), 122-131. doi: 10.1016/0022-247X(89)90276-X.

[21]

N. Hirano, Existence of entire positive solutions for nonhomogeneous elliptic equations, Nonlinear Anal., 29 (1997), 889-901. doi: 10.1016/S0362-546X(96)00176-9.

[22]

L. Jeanjean, Two positive solutions for a class of nonhomogeneous elliptic equations, Differential Integral Equations, 10 (1997), 609-624.

[23]

C. Mercuri and M. Willem, A global compactness result for the p-Laplacian involving critical nonlinearities, Discrete and Continuous Dynamical Systems (DCDS-A), 28 (2010), 469-493.

[24]

P.-L. Lions, The concentration-compactness principle in the calculus of variations, The limit case. I. Rev. Mat. Iberoamericana, 1 (1985), 145-201.

[25]

J. Yang, Positive solutions of semilinear elliptic problems in exterior domains, J. Differential Equations, 106 (1993), 40-69. doi: 10.1006/jdeq.1993.1098.

[26]

X. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Differential Equations, 92 (1991), 163-178. doi: 10.1016/0022-0396(91)90045-B.

[27]

X. Zhu and D. Cao, The concentration-compactness principle in nonlinear elliptic equations, Acta Math. Sci., 9 (1989), 307-328.

[28]

X. Zhu and H. Zhou, Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 301-318.

show all references

References:
[1]

A. Ambrosetti and M. Struwe, A note on the problem $-\Delta u=\lambda u+u|u| ^{2^\mathbf{star}-2}$, Manuscripta Math., 54 (1986), 373-379. doi: 10.1007/BF01168482.

[2]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300. doi: 10.1007/BF00282048.

[3]

A. Bahri and P.-L. Lions, Morse index of some min-max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math., 41 (1988), 1027-1037. doi: 10.1002/cpa.3160410803.

[4]

A. Bahri and P.-L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 365-413.

[5]

A. Bahri and Y. Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb{R}^N2$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.

[6]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[7]

G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 341-350.

[8]

K. Chen and C. Peng, Multiplicity and bifurcation of positive solutions for nonhomogeneous semilinear elliptic problems, J. Differential Equations, 240 (2007), 58-91. doi: 10.1016/j.jde.2007.05.023.

[9]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325.

[10]

D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb{R}^N2$, Proc. Roy. Soc. Edinburgh Sect., A 126 (1996), 443-463.

[11]

Y. Deng, Existence of multiple positive solutions for a semilinear equation with critical exponent, Proc. Roy. Soc. Edinburgh Sect., A 122 (1992), 161-175.

[12]

Y. B. Deng, Q. Gao and D. D. Zhang, Nodal Solutions for Laplace Equations with Critical Sobolev and Hardy Exponents on $\mathbb{R}$, Discrete and Continuous Dynamical Systems (DCDS-A), 19 (2007), 211-233.

[13]

Y. Deng and Y. Li, Existence and bifurcation of the positive solutions for a semilinear equation with critical exponent, J. Differential Equations, 130 (1996), 179-200. doi: 10.1006/jdeq.1996.0138.

[14]

Y. Deng, Z. Guo and G. Wang, Nodal solutions for $p$-Laplace equations with critical growth, Nonlinear Anal. TMA., 54 (2003), 1121-1151. doi: 10.1016/S0362-546X(03)00129-9.

[15]

Y. Deng, Y. Ma and X. Zhao, Existence and properties of multiple positive solutions for semi-linear equations with critical exponents, Rocky Mountain J. Math., 35 (2005), 1479-1512. doi: 10.1216/rmjm/1181069647.

[16]

Y. Deng, L. Jin and S. Peng, Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, Commun. Math. Sci,. 9 (2011), 859-878.

[17]

G. Cerami and R. Molle, On some Schrodinger equations with non regular potential at infinity, Discrete and Continuous Dynamical Systems (DCDS-A), 28 (2010), 827-844.

[18]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.

[19]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order," Springer-Verlag, Berlin, 1983.

[20]

J. Graham-Eagle, Monotone method for semilinear elliptic equations in unbounded domains, J. Math. Anal. Appl., 137 (1989), 122-131. doi: 10.1016/0022-247X(89)90276-X.

[21]

N. Hirano, Existence of entire positive solutions for nonhomogeneous elliptic equations, Nonlinear Anal., 29 (1997), 889-901. doi: 10.1016/S0362-546X(96)00176-9.

[22]

L. Jeanjean, Two positive solutions for a class of nonhomogeneous elliptic equations, Differential Integral Equations, 10 (1997), 609-624.

[23]

C. Mercuri and M. Willem, A global compactness result for the p-Laplacian involving critical nonlinearities, Discrete and Continuous Dynamical Systems (DCDS-A), 28 (2010), 469-493.

[24]

P.-L. Lions, The concentration-compactness principle in the calculus of variations, The limit case. I. Rev. Mat. Iberoamericana, 1 (1985), 145-201.

[25]

J. Yang, Positive solutions of semilinear elliptic problems in exterior domains, J. Differential Equations, 106 (1993), 40-69. doi: 10.1006/jdeq.1993.1098.

[26]

X. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Differential Equations, 92 (1991), 163-178. doi: 10.1016/0022-0396(91)90045-B.

[27]

X. Zhu and D. Cao, The concentration-compactness principle in nonlinear elliptic equations, Acta Math. Sci., 9 (1989), 307-328.

[28]

X. Zhu and H. Zhou, Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 301-318.

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