Multiplicity results for (p, q)-Laplacian equations with critical exponent in $${\mathbb {R}}^N$$ and negative energy
Laura Baldelli, Ylenia Brizi, Roberta Filippucci
Abstract
Abstract We prove existence results in all of $${\mathbb {R}}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> for an elliptic problem of ( p , q )-Laplacian type involving a critical term, nonnegative weights and a positive parameter $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> . In particular, under suitable conditions on the exponents of the nonlinearity, we prove existence of infinitely many weak solutions with negative energy when $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> 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.