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Radial solutions of elliptic equations with critical exponents in \({\mathbb{R}}^ N\). (English) Zbl 1042.35011

The authors study the existence of radial solutions to the problem \[ \Delta ^2 u +a u =\lambda u | u| ^{q-1} + u | u| ^{p-1} \quad \text{ in } {\mathbb R}^N \tag{P} \] where \(a\), \(\lambda \) are positive constants, \(1<q<p=(N+4)/(N-4)\) and \(N\geq 5\). They assume either \(N\geq 8\) and \(q>1\) or \(5\leq N\leq 7\) and \(p-2<q<p\) and prove in Theorem 1.1 that (P) has a nontrivial radial solution for any \(\lambda >0\). Moreover, if \(5\leq N\leq 7\) and \(1<q\leq p-2\) then (P) has a nontrivial radial solution for any \(\lambda \) large enough.
The authors use the Mountain Pass Theorem without (PS) condition, the embedding theorems, the technique of Brézis and Nirenberg and the fact that radial critical point of the corresponding functional is a weak solution of (P).
The analogous result for the problem \[ -\Delta u +a u =\lambda u | u| ^{q-1} + u | u| ^{2^* -2} \quad \text{ in } {\mathbb R}^N \tag{P'} \] is contained in Theorem 1.2: Let us assume either \(N\geq 4\) and \(q>1\) or \(N=3\) and \(3<q<5\). Then for any \(\lambda >0\) there is a nontrivial radial solution of (P’). Moreover, if \(N=3\) and \(1<q\leq 3\) then (P’) has a radial solution for any \(\lambda \) sufficiently large.
Reviewer: Jan Eisner (Praha)

MSC:

35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B33 Critical exponents in context of PDEs
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