Multi-peak solutions for the Hénon equation with slightly subcritical growth. (English) Zbl 1134.35047

Pistoia, Angela; Serra, Enrico

Math. Z. 256, No. 1, 75-97 (2007).

Summary: We study the Dirichlet problem for the Hénon equation
\[ -\Delta u=|x|^\alpha u^{\frac{N+2}{N-2}-\varepsilon}\quad\text{in } \Omega,\qquad u > 0 \quad\text{in } \Omega,\qquad u=0 \quad\text{on } \partial\Omega, \]
where \(\Omega\) is the unit ball in \(\mathbb{R}^N\), with \(N \geq 3\), the power \(\alpha\) is positive and \(\varepsilon\) is a small positive parameter. We prove that for every integer \(k \geq 1\) the above problem has a solution which blows up at \(k\) different points of \(\partial\Omega\) as \(\varepsilon\) goes to zero. We also show that the ground state solution (which blows up at one point) is unique.

MSC:

35J60	Nonlinear elliptic equations
35J20	Variational methods for second-order elliptic equations
35J25	Boundary value problems for second-order elliptic equations

Keywords:

Hénon equation; blowing up solutions; critical exponent; ground state solution

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