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Some results about the existence of a second positive solution in a quasilinear critical problem. (English) Zbl 0822.35048

We consider the problem \[ -\Delta_ p u\equiv- \text{div} (| \nabla u|^{p-2} \nabla u)= \lambda| u|^{q-2} u+ | u|^{p^*-2} u \text{ in }\Omega,\;\;u|_{\delta \Omega}=0, \tag{P}\(_ \lambda\) \] where \(\Omega \subset \mathbb{R}^ N\) is a smooth bounded domain, \(1<q<p <N\), \(\lambda>0\), \(p^*= Np/ (N-p)\). The goal of this work is to prove the existence of \(\lambda_ 0\) such that for \(0<\lambda< \lambda_ 0\) the problem \((\text{P}_ \lambda)\) has at least two positive solutions if either \(2N/ (N+2)<p <3\) and \(1<q<p\), or \(p\geq 3\) and \(p>q> p^*- 2/(p-1)\). The results in the paper, depending on the values of the parameters, are the following:
Theorem 1. Let \(2\leq p<3\) and \(1<q<p\). Then there exists a constant \(\lambda_ 0> 0\) such that if \(0< \lambda< \lambda_ 0\) then \((\text{P}_ \lambda)\) has at least two positive solutions.
Theorem 2. The same conclusion as in Theorem 1 holds in the case \(p\geq 3\), if \(p>q> q_ 0= p^*- 2/(p-1)\).
Theorem 3. The same conclusion as in Theorem 1 holds in the case \(2N/ (N+2)< p<2\) and \(1<q <p\).
The three main steps of the proof are the following: Step 1. The local Palais-Smale condition. Step 2. The Mountain Pass Lemma. Step 3. The energy estimate. In fact the geometric properties of the energy functional allows to use the Mountain Pass Lemma (for \(\lambda\) small enough). On the other hand, one can prove a local Palais-Smale condition, when the energy is bounded from above by \(c_ 0+ {1\over N} S^{N/p}\), where \(c_ 0\) is the energy of \(u_ 0\) (local minimum of the energy functional) and \(S\) is the best Sobolev constant. The goal of the third step is showing that the two previous results are compatible, by some delicate energy estimates.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations
35J20 Variational methods for second-order elliptic equations
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