Nonlinear equations with \(p\)-Laplacian and exponential function on compact Riemannian manifolds
The authors commence their article by a nice presentation of certain direct methods of Calculus of Variation looking to the following elliptic equation whose principal part is the \(p\)-Laplacian. Let \((M, g)\) be a compact \(n\)-dimensional Riemannian manifold. Consider the equation \[ \triangle_p\varphi + a =f(x)\text{ e}^\varphi\tag{1} \] where \(a\) is a real number, \(f(x)\) is a smooth function, and \(\triangle_p\) is the \(p\)-Laplacian \((p > 1)\) given by \[ \triangle_p\varphi =-\nabla^i(|\nabla\varphi|^{p-2}\nabla_i\varphi). \] In fact, to solve their problems, they give a full lecture how to choose a suitable set of admissile functions, and how to apply some classical and recent sharp estimates, together with regularity results. Firstly, they prove the following theorem: Let \((M, g)\) be a compact Riemannian manifold and \(f\in C^0.\) Then the critical equation \((1)\) with \(p=n\) admits a \(C^{1, \alpha}\) solution, for certain \(0 <\alpha <1\) if one of the following conditions holds: (a) \(a <0\), \(f\) is strictly negative; (b) \(a =0\), \(\int f dV<0\) and \(\sup f >0\); (c) \(0 < a < 1/nV\mu_n,\) and \(\sup f >0,\) where \(\mu_n =(n -1)^{n -1}n^{1 -2n}\omega_{n-1}^{-1}\) (\(\omega_{n-1}\) is the volume of \(S_{n-1}\)). Secondly, on the \(n\)-dimensional sphere, they are concerned with the problem of best constant in certain Sobolev type sharp estimates, related to the supercritical exponent. They consider equation \((1)\), where \(f\) is invariant w.r.t. the group \(G= O(m)\times O(k)\subset O(n +1),\) with \(m+k=n+1\) and \(k\geqslant m\geqslant 2.\) Just the case \(a >0\) needs some additional argument which requires a new inequality that the authors provide. They prove an analogous theorem to above for \(p=k,\) under certain conditions on \(a\) and \(f.\) They also consider the following equation on the sphere \(\mathbb{S}^n,\;(n >2)\) for \(q > p\): \[ \triangle_p u + a(x) u^{p -1} =f(x) u^{q -1},\qquad u>0\tag{2} \] where \(a(x)\) and \(f(x)\) are continuous functions and \(G\)-invariant, \(G=O(m)\times O(k)\subset O(n +1),\) with \( m+k =n+1\) and \(k\geqslant m\geqslant 2.\) This equation was studied in a bounded domain of \(\mathbb{R}^n,\) by \textit{M. Guedda} and \textit{L. Veron} [Nonlinear Anal., Theory Methods Appl. 13, No.~8, 879-902 (1989; Zbl 0714.35032)]. Here , \(p^\ast=pk/(k -p) > pn/(n -p)\) is supercritical (\( k < n\)). If \(\sup f >0\), if the functional related to equation \((2)\) with \(q =p^\ast\) satisfies a certain coercivity condition and the \(\inf\) of the functional satisfies a certain inequality, then this problem has a positive \(C^{1, \alpha}\) \((0 <\alpha<1)\) solution.
scientific article; zbMATH DE number 1436200