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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
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References:

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