# Solutions of perturbed Schrödinger equations with critical nonlinearity

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## Abstract

We consider the perturbed Schrödinger equation

$$\left\{\begin{array}{ll}{- \varepsilon ^2 \Delta u + V(x)u = P(x)|u|^{p - 2} u + k(x)|u|^{2* - 2} u} & {\text{for}}\, x \in {\mathbb{R}}^N\\ \qquad \qquad \quad {u(x) \rightarrow 0} & \text{as}\, {|x| \rightarrow \infty} \end{array} \right.$$

where $$N\geq 3, \ 2^*=2N/(N-2)$$ is the Sobolev critical exponent, $$p\in (2, 2^*)$$ , P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that $$\varepsilon\leq{\mathcal{E}}$$ ; for any $$m\in{\mathbb{N}}$$ , it has m pairs of solutions if $$\varepsilon\leq{\mathcal{E}}_{m}$$ ; and suppose there exists an orthogonal involution $$\tau:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$$ such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that $$\varepsilon\leq{\mathcal{E}}$$ , where $${\mathcal{E}}$$ and $${\mathcal{E}}_{m}$$ are sufficiently small positive numbers. Moreover, these solutions $$u_\varepsilon\to 0$$ in $$H^1({\mathbb{R}}^N)$$ as $$\varepsilon\to 0$$ .

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Correspondence to Fanghua Lin.

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Ding, Y., Lin, F. Solutions of perturbed Schrödinger equations with critical nonlinearity. Calc. Var. 30, 231–249 (2007). https://doi.org/10.1007/s00526-007-0091-z