Nontrivial weak solutions for a (p,q)-Laplacian equation involving discontinuities with critical exponent

Abstract

We review the existence of nontrivial weak solutions for the following problem{−Δpu−Δqu=f(u)+λH(u−β)uq⁎−1inΩ,u=0 on ∂Ω where Ω⊂RN is a smooth bounded domain, q⁎ is the critical Sobolev exponent, f:R⟶R is a discontinuous function that does not satisfy necessarily the Ambrosetti-Rabinowitz condition and H is the Heaviside function. Applying convenable critical point theory for nondifferentiable functionals, we prove that such problem has at least one nontrivial solution for any parameters λ>0,β>0. We also establish other results if the nonlinearity satisfies the Ambrosetti-Rabinowitz condition.