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Many solutions for elliptic equations with critical exponents

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Institution:

(1) Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, P. R. China

Abstract:

Let Ω be an open bounded domain in ℝ^N(N ≥ 3) and $2^* = \frac{{2N}}{{N - 2}}$ . We are concerned with two kinds of critical elliptic problems. The first one is

$- \Delta u - \mu \frac{u}{{\left| x \right|^2 }} = \lambda u + \left| u \right|^{m - 2} u + \theta \left| u \right|^{2^* - 2} u u \in H_0^1 (\Omega ),$

(*)

where 0 ∈ Ω, $0 < \mu < (\frac{{N - 2}}{2})^2$ , 2 < m < 2* and λ > 0. By using the fountain theorem and concentration estimates, if N ≥ 7 and θ > 0, we establish the existence of infinitely many solutions for the following regularization of (*) with small number ϵ > 0

$- \Delta u - \mu \frac{u}{{\left| x \right|^2 + \varepsilon }} = \lambda u + \left| u \right|^{m - 2} u + \theta \left| u \right|^{2^* - 2} u u \in H_0^1 (\Omega ).$

Then if θ > 0 is suitably small, we obtain many solutions for problem (*) by taking the process of approximation. The second problem is

$- \Delta u = \left| u \right|^{2^* - 2} u + t\left| u \right|^{q - 1} u u \in H_0^1 (\Omega ),$

where q ∈ (0, 1), t > 0. By using similar methods as in (*), we prove that if N ≥ 7, $\frac{4}{{N - 2}} < q < 1$ and t > 0, there exist infinitely many solutions with positive energy. In particular, we give a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami 1].

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