On heterogeneous radially symmetric semilinear elliptic equations with critical nonlinearities in [formula omitted

Abstract

This is a continuation of our recent study of the effect of heterogeneity on existence and multiplicity results for elliptic PDEs with critical nonlinearities a la Trudinger–Moser in dimension N=2. Our results here are ultimately connected to the increase in the threshold of compactness that can be achieved in the Trudinger–Moser inequality under radial symmetry and in the presence of a rapidly vanishing radial weight function in H0,r1(BR). Such results are generalization to dimension N=2 of the 1981 seminal result of Ni [1] in dimension N≥3 (cf. other papers also cited in the References).In this work we prove and employ a simple weak continuity result in the subspace of radially symmetric H01(BR) functions for a class of functionals that include the classical Trudinger–Moser functional in the presence of a rapidly vanishing weight functions. As it is well known, such results are crucial in proving suitable compactness conditions for the corresponding energy functionals of the elliptic PDEs under consideration and enables us to prove the existence of a positive, a negative and a sign-changing solution for the equation (0.1)−Δu=a(r)h(u)eαu2inB(R1,R2)u=0on∂B(R1,R2)where B(R1,R2)≔{x∈R2|R1<|x|<R2},0≤R1<R2,without any growth restriction on the lower order terms of the nonlinearity. In addition we also prove existence of Infinitely Many Sign-Changing Solutions in the case of a ball under similar hypotheses. To the best of our knowledge, by highlighting the effect of the heterogeneity in relaxing the usual growth restrictions on the lower order terms of critical nonlinearities, our results complement and generalize a number of existing results in the literature of such problems.