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Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term. (English) Zbl 0729.35051
This work is a continuation of the work of the authors [Commun. Partial Differ. Equations 12, 1389-1430 (1987; Zbl 0637.35069)]. The authors study the existence of nontrivial solutions for the nonlinear degenerate elliptic Dirichlet problems in a bounded domain \(\Omega\) in \({\mathbb{R}}^ n\) \[ -div(| \nabla u|^{p-2}\nabla u)=| u|^{p^*- 2}u+\lambda | u|^{q-2}u\text{ in } \Omega \text{ and } u|_{\partial \Omega}=0, \] where \(\lambda >0\) and \(p^*\) is the critical Sobolev exponent of imbedding \(W_ 0^{1,p}\subset L^{p^*}\). The existence of solutions is reduced to the existence of critical points of the functional \[ F(u)=\frac{1}{p}\int_{\Omega}| \nabla u|^ p-\frac{\lambda}{q}\int_{\Omega}| u|^ q- \frac{1}{p^*}\int_{\Omega}| u|^{p^*} \] on \(W_ 0^{1,p}(\Omega)\). The main results are the following:
(1) If \(p<q<p^*\), there exists \(\lambda_ 0>0\) such that for all \(\lambda >\lambda_ 0\) there exists a nontrivial solution.
(2) If \(\max (p,p^*-p/(p-1))<q<p^*\), there exists a nontrivial solution for all \(\lambda >0.\)
(3) If \(1<q<p\), there exists \(\lambda_ 1\) such that there exist infinitely many solutions for all \(0<\lambda <\lambda_ 1.\)
With the help of the concentration compactness principle of P. L. Lions, the authors obtain a local Palais-Smale condition for F(u). To prove (1) and (2), by using a mountain pass lemma they get a critical point with positive critical value. To prove (3) they introduce some truncated functional and they get infinitely many critical points with negative critical values by using the classical critical point theory. They also obtain a multiplicity result in a noncritical problem for p-Laplacian when the associated functional is not symmetric by using the method of P. Rabinowitz [Trans. Am. Math. Soc. 272, 753-769 (1982; Zbl 0589.35004)].

35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations
35J20 Variational methods for second-order elliptic equations
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
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