On a class of nonhomogeneous elliptic equation on compact Riemannian manifold without boundary (Q723806)

This paper deals with the quasilinear elliptic equation \[ -\text{div}_g(a(x,\nabla_g u))+ u|u|^{n-2}= f(x,u)+\lambda h(x)\quad\text{in }M,\tag{1} \] where \((M,g)\) is \(n\)-dimensional compact Riemannian manifold without boundary (\(n\geq 2\)). The function \(u\) is a weak solution of (1) and belongs to the classical Sobolev space \(W^{1,n}(M)\), while \(h\not\equiv 0\) belongs to its dual space \(W^{-1,n'}(M)\). It is assumed that the continuous function \(a(x,\xi)\) satisfies the conditions (i) \(a(x,\xi)=\nabla_gA(x,\xi)\) for some continuous function \(A(x,\xi)\), and (ii) \(a(x,\xi)\) satisfies Leray-Lions conditions (the latter being a combination of growth, coercivity and monotonicity conditions). The nonlinearity \(f(x,u)\) is measurable in an appropriate sense, \(f(x,s)\) is increasing in \(s\) and enjoys exponential critical growth in \(s\). Then it is proved in Theorem 1.1 that there exists \(\lambda>0\) such that for all \(0<\lambda\leq\lambda_0\) the equation (1) possesses a nontrivial weak solution.