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 \(\varepsilon\to 0\) .