## 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

### Keywords:

Palais-Smale condition; Mountain Pass Lemma
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