JournalsrlmVol. 31, No. 1pp. 121–130

Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains

  • Riccardo Molle

    Università di Roma Tor Vergata, Italy
  • Donato Passaseo

    Università del Salento, Lecce, Italy
Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains cover
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Abstract

We deal with nonlinear elliptic Dirichlet problems of the form

div(Dup2Du)+f(u)=0\mboxinΩ,u=0 \mboxonΩ\mathrm {div}(|D u|^{p-2}D u )+f(u)=0\quad\mbox{ in }\Omega,\qquad u=0\ \mbox{ on }\partial\Omega

where Ω\Omega is a bounded domain in Rn\mathbb R^n, n2n\ge 2, p>1p > 1 and ff has supercritical growth from the viewpoint of Sobolev embedding. Our aim is to show that there exist bounded contractible non star-shaped domains Ω\Omega, arbitrarily close to domains with nontrivial topology, such that the problem does not have nontrivial solutions. For example, we prove that if n=2n=2, 1<p<21 < p < 2, f(u)=uq2uf(u)=|u|^{q-2}u with q>2p2pq > {2p\over 2-p} and Ω={(ρcosθ,ρsinθ) :θ<α, ρ1<s}\Omega=\{(\rho\cos\theta,\rho\sin\theta)\ :\|\theta|<\alpha,\ |\rho -1| < s\} with 0<α<π0 < \alpha < \pi and 0<s<10 < s < 1, then for all q>2p2pq > {2p\over 2-p} there exists sˉ>0\bar s > 0 such that the problem has only the trivial solution u0u\equiv 0 for all α(0,π)\alpha\in (0,\pi) and s(0,sˉ)s\in (0,\bar s).

Cite this article

Riccardo Molle, Donato Passaseo, Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 31 (2020), no. 1, pp. 121–130

DOI 10.4171/RLM/882