Abstract
We prove existence results in all of {mathbb {R}}^N for an elliptic problem of (p, q)-Laplacian type involving a critical term, nonnegative weights and a positive parameter lambda . In particular, under suitable conditions on the exponents of the nonlinearity, we prove existence of infinitely many weak solutions with negative energy when lambda belongs to a certain interval. Our proofs use variational methods and the concentration compactness principle. Towards this aim we give a detailed proof of tight convergence of a suitable sequence.
Highlights
In this paper we are interested in nontrivial weak solutions in D1,p(RN ) ∩ D1,q (RN ) of the following nonlinear elliptic problem of ( p, q)-Laplacian type involving a critical term
As discussed before, in order to recover compactness, in the spirit of the celebrated first concentration compactness principle by Lions [27,28,29,30], we have to deal with tight convergence
Let Cb(Y ) be the space of real bounded functions defined in Y and we report the definition of tight convergence of measures in the same setting as above
Summary
In this paper we are interested in nontrivial weak solutions in D1,p(RN ) ∩ D1,q (RN ) of the following nonlinear elliptic problem of ( p, q)-Laplacian type involving a critical term. We apply Theorem 1.9 in [39] in which the Krasnoselskii genus is involved with its properties and the standard and crucial compactness condition (PS)c is required to be satisfied by Eλ, for c < 0 This is a delicate point, for critical problems in all of RN this compactness condition is often loss, for this reason some of the papers treating problems on unbounded domains use special function spaces where the compactness is preserved, such as spaces of radially symmetric functions or weighted Sobolev spaces. We point out that the unbounded case is sensibly more complicated than the bounded case since, only in the latter case, tight
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