## Multiple solutions of a semilinear elliptic equation in $$\mathbb{R}^ N$$.(English)Zbl 0797.35039

Summary: We are concerned with the existence of multiple solutions of $-\Delta u + u = \lambda b(x) | u |^{p-1} u + c(x) | u |^{q-1} u$ where $$1<p$$, $$q<(N+2)/(N-2)$$ if $$N \geq 3$$, $$1<p$$, $$q<+\infty$$ if $$N=2$$, $$\lambda>0$$. We obtain the existence of multiple solutions by using concentrations-compactness method and dual variational principle to establish the corresponding existence of critical points.

### MSC:

 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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### References:

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