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On the number of blowing-up solutions to a nonlinear elliptic equation with critical growth. (English) Zbl 1163.35009

The paper deals with the problem: \[ -\Delta w+V(x)w=n(n-2)w^{\frac{n+2}{n-2}-\varepsilon}\quad \text{in}\,\,\mathbb R^n,\, n\geqslant 3,\,\varepsilon>0, \]
\[ w(x)>0\quad \forall\,x\in\mathbb R^n,\, w\in D^{1,2}(\mathbb R^n), \] where positive potential \(V:\mathbb R^n\to\mathbb R^1\) satisfies suitable conditions, \(D^{1,2}(\mathbb R^n)\) is the completion of \(C^\infty_0(\mathbb R^n)\) with respect to the norm \(\| u\|_{1,2}=(\int_{\mathbb R^n}|\nabla u|^2 dx)^{1/2}\). There are proved estimates from below of the number of solutions which blow up at a suitable critical point of the potential \(V\) as the parameter \(\varepsilon\) goes to zero. Additionally, there are obtained sufficient conditions on the structure of the potential \(V\) in the neighborhood of its critical points \(y_0\), which guarantee absence of such a blowing up and concentrating at \(y_0\) solutions.

MSC:

35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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References:

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