## Finite Morse index solutions of supercritical problems.(English)Zbl 1158.35013

The author studies properties of bounded solutions to $-\Delta u = f(u)\qquad\text{on }\mathbb{R}^N$ that have finite Morse index, in the sense that the spectrum of $$-\Delta-f'(u)$$ in $$L^2(\mathbb{R}^N)$$ has only a finite number of negative points, each with finite algebraic multiplicity. Apparently, only $$N\geq2$$ is considered. Denote $$p^*:=\infty$$ if $$N=2$$, $$p^*:=(N+2)/(N-2)$$ if $$N\geq3$$, $$p^f:=\infty$$ if $$N\leq4$$, and $$p^f:=N/(N-4)$$ if $$N>4$$. The main interest lies in the case where $$f$$ grows supercritically at $$\infty$$, i. e., $$| f(t)| /| t| ^{p^*}\to\infty$$ as $$| t| \to\infty$$.
The first result states that for $$p^*<p\leq p^f$$ and $$f(t):=| t| ^{p-1}t$$ the only bounded finite Morse index solution is $$u\equiv0$$. Similar results are proved for equations with homogeneous Dirichlet boundary conditions on a half space or in exterior domains. In some cases also $$p=p^*$$ is allowed.
In the second result the existence of a bounded, non-constant finite Morse index solution $$u$$ is assumed, and conclusions are drawn concerning the shape of $$u$$ and properties of $$f$$. More specifically, it is assumed that $$f\in C^1$$, $$f\geq0$$ on $$[\inf u,\sup u]$$, and that $$f$$ behaves like a power function near its zeros. Then $$u$$ is radially symmetric with respect to some point and asymptotically constant at $$\infty$$. Moreover, some restrictions on the interaction of the zeros of $$f$$ and the values of $$u$$ are proved.
As one application the author proves a Bahri-Lions type result for supercritical problems: Boundedness of solutions is equivalent to boundedness of Morse indices, for a Dirichlet problem on a bounded domain. Other applications are the existence of infinitely many bifurcation points on the positive solution branch for a real analytic Dirichlet problem on a bounded, star shaped domain, and some results about domain variation in Dirichlet problems.

### MSC:

 35B35 Stability in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B45 A priori estimates in context of PDEs 35B25 Singular perturbations in context of PDEs 35J60 Nonlinear elliptic equations 47J30 Variational methods involving nonlinear operators 35J65 Nonlinear boundary value problems for linear elliptic equations

### Keywords:

Morse index; bounded solution; supercritical nonlinearity
Full Text:

### References:

  DOI: 10.1090/S0894-0347-00-00345-3 · Zbl 0968.35041  DOI: 10.1002/cpa.3160450908 · Zbl 0801.35026  Berestycki H., Ann. Scu. Norm. Sup. Pisa 25 pp 69– (1996)  DOI: 10.1080/03605309708821315 · Zbl 0910.35048  DOI: 10.1007/BF01243922 · Zbl 0755.35036  DOI: 10.1007/s002050000087 · Zbl 0962.76012  DOI: 10.1002/cpa.3160420304 · Zbl 0702.35085  Clement P., Ann. Scu. Norm. Sup. Pisa 14 pp 97– (1987)  DOI: 10.1090/S0002-9947-04-03543-3 · Zbl 1145.35369  DOI: 10.1512/iumj.2004.53.2354 · Zbl 1183.35125  Dancer E. N., Di{\currency}. Int. Eqns. 17 pp 961– (2004)  DOI: 10.1007/s00208-002-0352-2 · Zbl 1040.35033  DOI: 10.1016/0022-0396(88)90021-6 · Zbl 0662.34025  Dancer E. N., Topol. Meth. Nonlin. Anal. 7 pp 95– (1996)  DOI: 10.1007/BF02505896 · Zbl 1030.35073  DOI: 10.1007/PL00004666 · Zbl 0933.35068  Dancer E. N., Di{\currency}. Int. Eqns. 5 pp 903– (1992)  DOI: 10.1017/S0004972700012089 · Zbl 0777.35005  DOI: 10.1016/S0362-546X(98)00307-1 · Zbl 0960.35035  DOI: 10.1007/BF01209398 · Zbl 0608.35017  Farina A., CR Acad. Sci. Paris 341 pp 415– (2005)  DOI: 10.1002/cpa.3160340406 · Zbl 0465.35003  Joseph D., Arch. Rat. Mech. Anal. 49 pp 241– (1973)  Moschini L., Anal. Nonlin. 22 pp 11– (2005)  DOI: 10.1002/cpa.3160380105 · Zbl 0581.35021  DOI: 10.1006/jfan.1993.1064 · Zbl 0793.35039  DOI: 10.1080/03605307908820096 · Zbl 0462.35016  Schaaf R., Adv. Di{\currency}. Eqns. 5 pp 1201– (2000)  DOI: 10.1006/jdeq.1995.1105 · Zbl 0844.35028  DOI: 10.1512/iumj.1996.45.986 · Zbl 0864.35009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.