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A class of generalized quasilinear Schrödinger equations

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  • We establish the existence of nontrivial solutions for the following quasilinear Schrödinger equation with critical Sobolev exponent: \begin{eqnarray} -\Delta u+V(x) u-\Delta [l(u^2)]l'(u^2)u= \lambda u^{\alpha2^*-1}+h(u),\ \ x\in \mathbb{R}^N, \end{eqnarray} where $V(x):\mathbb{R}^N\rightarrow \mathbb{R}$ is a given potential and $l,h$ are real functions, $\lambda\geq 0$, $\alpha>1$, $2^*=2N/(N-2)$, $N\geq 3$. Our results cover two physical models $l(s)=s^{\frac{\alpha}{2}}$ and $l(s) = (1+s)^{\frac{\alpha}{2}}$ with $\alpha\geq 3/2$.
    Mathematics Subject Classification: Primary: 35J20, 35J62; Secondary: 35Q55.


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