Relative category and multiplicity of positive solutions for the equation \(\Delta u+u^{2^*-1}=0\). (English) Zbl 0929.35054
Summary: Let us consider the problem
\[
\begin{cases} \Delta u+ u^{2^*-1}= 0\quad & \text{in }\Omega,\\ u>0\quad & \text{in }\Omega,\\ u= 0\quad & \text{on }\partial\Omega,\end{cases}\tag{1}
\]
where \(\Omega\) is a smooth bounded domain of \(\mathbb{R}^n\) with \(n\geq 3\) and \(2^*= {2n\over n-2}\) (the critical Sobolev exponent).
It is well known that the existence of solutions of problem (1) is strictly related to the shape of \(\Omega\). In this paper, we show that every perturbation of a given domain, which modifies its topological properties and is obtained removing a subset of small capacity, gives rise to solutions in the perturbed domain; moreover, these solutions converge weakly to zero and concentrate near a point as the capacity of the perturbation goes to zero.
The method we use is the following: we analyze the sublevels of the corresponding functional, we relate their topological properties to the shape of the domain, and then we use the notion of relative category to evaluate the number of solutions.
It is well known that the existence of solutions of problem (1) is strictly related to the shape of \(\Omega\). In this paper, we show that every perturbation of a given domain, which modifies its topological properties and is obtained removing a subset of small capacity, gives rise to solutions in the perturbed domain; moreover, these solutions converge weakly to zero and concentrate near a point as the capacity of the perturbation goes to zero.
The method we use is the following: we analyze the sublevels of the corresponding functional, we relate their topological properties to the shape of the domain, and then we use the notion of relative category to evaluate the number of solutions.
MSC:
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 35B20 | Perturbations in context of PDEs |
| 35B33 | Critical exponents in context of PDEs |
Keywords:
nonlinear elliptic problems; critical Sobolev exponent; positive solutions; capacity; relative categoryReferences:
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