    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 & Continuous Dynamical Systems - A, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795
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show all references

##### References:
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Math., 35 (2005), 1479.  doi: 10.1216/rmjm/1181069647.  Google Scholar  Y. Deng, L. Jin and S. Peng, Solutions of Schrödinger equations with inverse square potential and critical nonlinearity,, Commun. Math. Sci, 9 (2011), 859.   Google Scholar  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. Google Scholar  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.  Google Scholar  D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order,", Springer-Verlag, (1983). Google Scholar  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.  Google Scholar  N. 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