## Solutions with multiple peaks for nonlinear elliptic equations.(English)Zbl 0928.35048

The nonlinear elliptic problem: $-\triangle u+\lambda ^{2}u=Q(x)| u| ^{p-2}u \quad \text{ in }\mathbb{R}^{N},\quad u\in H^{1}(\mathbb{R}^{N}),$ is considered, where $$N\geq 3$$, $$\lambda \neq 0$$ and $$2<p<2N/(N-2)$$. Here, $$Q(x)\geq 0$$ is of class $$C^{3}(\mathbb{R}^{N})$$ and $$D^{m}Q$$ are bounded on $$\mathbb{R}^{N},$$ $$m=0,1,2,3$$. It is supposed that $$Q(x)$$ has $$k$$ non-degenerate critical points, that is to say, $$Q(x)$$ has $$k$$ points $$a^{j}\in \mathbb{R}^{N}$$, $$j=1,2,\dots, k$$ satisfying $$Q(a^{j})>0$$, $$DQ(a^{j})=0$$ and $$\det (D^{2}Q(a^{j}))\neq 0$$. It is shown that the preceding essential hypothesis on the leading term $$Q(x)$$ influence the existence of at least $$2^{k}-1$$ multipeak positive solutions with peaks ranging in numbers from $$1$$ to $$k$$. It is remarked that these solutions constructed here, concentrating as $$\lambda \rightarrow \infty$$ at the points $$a^{j}\in \mathbb{R}^{N}$$, $$j=1,2,\dots, k,$$ are not least-energy solutions. The proofs of the main results are presented in detail. For some related results see for instance D. Cao and E. S. Noussair [Ann. Inst. Henri Poincaré Anal. Non Linéaire 13, No. 5, 576-588 (1996; Zbl 0859.35032)], and W. M. Ni and J. Wei [Commun. Pure Appl. Math. 48, No. 7, 731-768 (1995; Zbl 0838.35009)].

### MSC:

 35J60 Nonlinear elliptic equations 35A15 Variational methods applied to PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

### Citations:

Zbl 0859.35032; Zbl 0838.35009
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### References:

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