# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. 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. 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. 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. [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 $\mathbbR^N$,, Rev. Mat. Iberoamericana, 6 (1990), 1. [6] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. 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. [8] K. Chen and C. Peng, Multiplicity and bifurcation of positive solutions for nonhomogeneous semilinear elliptic problems,, J. Differential Equations, 240 (2007), 58. 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. doi: 10.1007/BF00282325. [10] D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbbR^N$,, Proc. Roy. Soc. Edinburgh Sect., 126 (1996), 443. [11] Y. Deng, Existence of multiple positive solutions for a semilinear equation with critical exponent,, Proc. Roy. Soc. Edinburgh Sect., 122 (1992), 161. [12] Y. B. Deng, Q. Gao and D. D. Zhang, Nodal Solutions for Laplace Equations with Critical Sobolev and Hardy Exponents on $\mathbbR$,, Discrete and Continuous Dynamical Systems (DCDS-A), 19 (2007), 211. [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. 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. 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. 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. [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. [18] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125. [19] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order,", Springer-Verlag, (1983). [20] J. Graham-Eagle, Monotone method for semilinear elliptic equations in unbounded domains,, J. Math. Anal. Appl., 137 (1989), 122. 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. 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. [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. [24] P.-L. Lions, The concentration-compactness principle in the calculus of variations,, The limit case. I. Rev. Mat. Iberoamericana, 1 (1985), 145. [25] J. Yang, Positive solutions of semilinear elliptic problems in exterior domains,, J. Differential Equations, 106 (1993), 40. 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. 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. [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.

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##### 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. 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. 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. 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. [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 $\mathbbR^N$,, Rev. Mat. Iberoamericana, 6 (1990), 1. [6] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. 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. [8] K. Chen and C. Peng, Multiplicity and bifurcation of positive solutions for nonhomogeneous semilinear elliptic problems,, J. Differential Equations, 240 (2007), 58. 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. doi: 10.1007/BF00282325. [10] D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbbR^N$,, Proc. Roy. Soc. Edinburgh Sect., 126 (1996), 443. [11] Y. Deng, Existence of multiple positive solutions for a semilinear equation with critical exponent,, Proc. Roy. Soc. Edinburgh Sect., 122 (1992), 161. [12] Y. B. Deng, Q. Gao and D. D. Zhang, Nodal Solutions for Laplace Equations with Critical Sobolev and Hardy Exponents on $\mathbbR$,, Discrete and Continuous Dynamical Systems (DCDS-A), 19 (2007), 211. [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. 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. 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. 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. [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. [18] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125. [19] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order,", Springer-Verlag, (1983). [20] J. Graham-Eagle, Monotone method for semilinear elliptic equations in unbounded domains,, J. Math. Anal. Appl., 137 (1989), 122. 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. 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. [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. [24] P.-L. Lions, The concentration-compactness principle in the calculus of variations,, The limit case. I. Rev. Mat. Iberoamericana, 1 (1985), 145. [25] J. Yang, Positive solutions of semilinear elliptic problems in exterior domains,, J. Differential Equations, 106 (1993), 40. 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. 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. [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.
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