Elliptic equations with cylindrical potential and multiple critical exponents
Xiaomei Sun Yimin Zhang
Communications on Pure & Applied Analysis 2013, 12(5): 1943-1957 doi: 10.3934/cpaa.2013.12.1943
In this paper, we deal with the following problem: \begin{eqnarray*} -\Delta u-\lambda |y|^{-2}u=|y|^{-s}u^{2^{*}(s)-1}+u^{2^{*}-1}\ \ \ in \ \ R^N , y\neq 0\\ u\geq 0 \end{eqnarray*} where $u(x)=u(y,z): R^m\times R^{N-m}\longrightarrow R$, $N\geq 4$, $2 < m < N$, $\lambda < (\frac{m-2}{2})^2$ and $0 < s < 2$, $2^*(s)=\frac{2(N-s)}{N-2}$, $2^*=\frac{2N}{N-2}$. Using the Variational method, we proved the existence of a ground state solution for the case $0 < \lambda < (\frac{m-2}{2})^2$ and the existence of a cylindrical weak solution under the case $\lambda<0$.
keywords: Variational method cylindrical weight multiple critical exponents.
Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents
Yimin Zhang Youjun Wang Yaotian Shen
Communications on Pure & Applied Analysis 2011, 10(4): 1037-1054 doi: 10.3934/cpaa.2011.10.1037
By using a change of variable, the quasilinear Schrödinger equation is reduced to semilinear elliptic equation. Then, Mountain Pass theorem without $(PS)_c$ condition in a suitable Orlicz space is employed to prove the existence of positive standing wave solutions for a class of quasilinear Schrödinger equations involving critical Sobolev-Hardy exponents.
keywords: quasilinear Schrödinger equations Orlicz space. Critical Sobolev-Hardy exponents
Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity
Minbo Yang Jianjun Zhang Yimin Zhang
Communications on Pure & Applied Analysis 2017, 16(2): 493-512 doi: 10.3934/cpaa.2017025

In this paper, we study a class of nonlinear Choquard type equations involving a general nonlinearity. By using the method of penalization argument, we show that there exists a family of solutions having multiple concentration regions which concentrate at the minimum points of the potential V. Moreover, the monotonicity of f(s)=s and the so-called Ambrosetti-Rabinowitz condition are not required.

keywords: Multi-peak solutions Choquard equation semiclassical states penalization arguments Berestycki-Lions conditions
Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential
Helin Guo Yimin Zhang Huansong Zhou
Communications on Pure & Applied Analysis 2018, 17(5): 1875-1897 doi: 10.3934/cpaa.2018089

We study a Kirchhoff type elliptic equation with trapping potential. The existence and blow-up behavior of solutions with normalized $L^{2}$-norm for this equation are discussed.

keywords: Constrained variational method Kirchhoff equation elliptic equation Schwarz symmetrization Energy estimates

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