Multiplicity results for (p, q)-Laplacian equations with critical exponent in $${\mathbb {R}}^N$$ and negative energy

Abstract

We prove existence results in all of $${\mathbb {R}}^N$$ R N for an elliptic problem of (p, q)-Laplacian type involving a critical term, nonnegative weights and a positive parameter $$\lambda $$ λ . In particular, under suitable conditions on the exponents of the nonlinearity, we prove existence of infinitely many weak solutions with negative energy when $$\lambda $$ λ 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.

On sufficient "local" conditions for existence results to generalized $p(\cdot)$-Laplace equations involving critical growth

“…It is worth pointing out that [1, Theorem 1] actually asserted the existence of a sequence of solutions provided the L ∞ norm of K(x) := θb(x) is sufficiently small. However, we found that their argument can only produce a finite number of solutions in the same way as the statement of Theorem 1.3 and discussed it with the authors of [1] about this issue, please see Remark 4.4 for more details.…”

“…where 1 < q < s < p and the weight m is possibly sign-changing. We obtained the existence of nontrivial nonnegative solutions to [1] for a fixed λ and some range of parameter θ (depending on λ). In [1], Baldelli et al obtained the multiplicity of solutions for problem (1.2) when Ω = R N in a reverse way, that is, for θ fixed and small, they find some range of λ.…”

“…We obtained the existence of nontrivial nonnegative solutions to [1] for a fixed λ and some range of parameter θ (depending on λ). In [1], Baldelli et al obtained the multiplicity of solutions for problem (1.2) when Ω = R N in a reverse way, that is, for θ fixed and small, they find some range of λ. The subcritical problem of sandwich-type growth involving variable exponent was studied by Mihailescu-Rǎdulescu in [26].…”

“…Multiplicity of solutions. In this subsection we borrow the idea from [1] to prove Theorem 1.3, which asserts the existence of k pairs of solutions to problem (1.10) for an arbitrarily given number k. To seek such a k pairs of solutions, we determine the critical points of the functional Ψ defined in (4.2). Throughout this subsection, in addition to (S), (W) and (L1) we always assume that m(x) > 0 a.e.…”

Multiplicity results for generalized quasilinear critical Schrödinger equations in R^N

“…A key role in the proof of our results is the concentration compactness principles by Lions. For a detailed discussion on them, we refer to [5] and [6]. In particular, we are interested in the second concentration compactness principle, which regards a possible concentration only at finite points and where two different types (since we are in unbounded domains) of convergences are considered: the tight convergence of measures, whose symbol is * , and the "weak" convergence, denoted with .…”