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Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents

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  • We study the existence, concentration and multiplicity of weak solutions to the quasilinear Schrödinger equation with critical Sobolev growth \begin{equation*} \left\{ \begin{gathered} - {\varepsilon ^2}\Delta u + V(x)u - {\varepsilon ^2}\Delta (u^2)u = W(x){u^{q - 1}} + {u^{2\cdot{2^*} - 1}} {\text{ in }}{\mathbb{R}^N},\\ u > 0{\text{ in }}{\mathbb{R}^N},\\ \end{gathered} \right. \end{equation*} where $\varepsilon $ is a small positive parameter, $N \ge 3$, ${2^ * } = \frac{{2N}} {{N - 2}}$, $4 < q < 2 \cdot {2^ * }$, $\min V > 0$ and $\inf W > 0$. Under proper assumptions, we obtain the existence and concentration phenomena of soliton solutions of the above problem. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple soliton solutions by employing the topology of the set where the potentials $V(x)$ attains its minimum and $W(x)$ attains its maximum.
    Mathematics Subject Classification: Primary: 35J20, 35J60, 35J92.

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