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Quasilinear elliptic equations on noncompact Riemannian manifolds. (English) Zbl 1382.35110

This paper deals with the existence of solutions \(u\in W^{1,p}(M)\) of the following class of semilinear elliptic equations on an \(n\)-dimensional Riemannian manifold \(M\): \[ \int_M|\nabla u|^{p-2}\nabla u\cdot\nabla v\,d{\mathcal H}^n= \int_M f(u) vd{\mathcal H}^n\tag{1} \] for every \(v\in W^{1,p}(M)\), \(1<p<\infty\), where \(\nabla\) is the gradient operator on \(M\) and \({\mathcal H}^n\) is the volume measure on \(M\) induced by the corresponding metric. \(M\) is connected, without boundary and \({\mathcal H}^n(M)<\infty\). The real-valued scalar function \(f\) is continuous and satisfies suitable growth conditions of polynomial type. The authors define the so-called isocapacitary function \(\nu_{M,p}(s)\) of \(M\) and the class of manifolds \(C_p(\nu)= \{M:\nu_{M,p}(s)\geq\nu(s)\) for \(s\) near \(0\}\). Here \(\nu:\mathbb{R}^1_+\to\overline{\mathbb{R}}^1_+\) is a prescribed quasi-concave function (that implies \(\nu\in C^0(\mathbb{R}^1_+\)).
This is the main result of that paper.
Theorem 1.1. Let \(\nu\) be a quasi-concave function and \(M\in C_p(\nu)\), while \(f\) fulfills several appropriate growth conditions. Assume that either \(\nu(0^+)>0\) or \(\nu(0^+)=0\) and \(\lim_{t\to\infty} \nu^{-1}(t^{-p})+f(kt)= 0\), \(\forall k\in\mathbb{R}\), where \(\nu^{-1}\) is the inverse function of \(\nu\). Then (1) possesses a nontrivial (nonconstant) solution. Bvp with homogeneous Neumann conditions in possibly irregular domains are included as a special instance.

MSC:

35J62 Quasilinear elliptic equations
35R01 PDEs on manifolds
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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