Existence of a nontrivial weak solution to quasilinear elliptic equations with singular weights and multiple critical exponents. (English) Zbl 1185.35102
Summary: We consider the existence of a non-trivial weak solution to a quasilinear elliptic equation with singular weights and multiple critical exponents in the whole space. Firstly, we get the existence of a local Palais-Smale sequence by verifying the geometric conditions of the Mountain Pass Lemma. Secondly, we study the concentration properties of the Palais-Smale sequence of a zero weak limit. Thirdly, we deduce by contradiction the elimination of the possibility of a zero weak limit case. Lastly, applying a monotonic inequality, we prove that the nontrivial weak limit of the Palais-Smale sequence is indeed a weak solution.
MSC:
| 35J61 | Semilinear elliptic equations |
| 35D30 | Weak solutions to PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
Keywords:
quasilinear elliptic equation; singular weights; multiple critical exponents; mountain pass LemmaReferences:
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