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

Cao, Dao-Min

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 10, No. 6, 593-604 (1993).

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

Keywords:

semilinear elliptic equations; multiple solutions; concentration- compactness method; dual variational principle

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