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Infinitely many positive solutions for a nonlinear field equation with super-critical growth. (English) Zbl 1342.35114

In the paper under review, the authors consider the following nonlinear field equation with super-critical growth: \[ \begin{aligned} -\Delta u+\lambda u=Q(y)u^{(N+2)/(N-2)}, \quad u>0 \quad & \text{in } \mathbb{R}^{N+m}, \\ u(y)\to 0 \quad & \text{as } |y| \to +\infty, \end{aligned} \] where \(m \geq 1\), \(\lambda \geq 0\), and \(Q\) is a bounded positive function which satisfies some symmetry conditions. The exponent \((N+2)/(N-2)\) is super-critical in \(\mathbb{R}^{N+m}\).
By constructing solutions that concentrate at a large number of \(m\)-dimensional manifolds, the authors prove the existence of infinitely many positive solutions for the considered problem.

MSC:

35J61 Semilinear elliptic equations
35B33 Critical exponents in context of PDEs
35B25 Singular perturbations in context of PDEs
35B09 Positive solutions to PDEs
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