Eigenfunctions of the nonlinear equation Δu+νf(x,u) = 0 inR²

Two dimensional elliptic equation with critical nonlinear growth

“…Existence of a family of solutions {(u m , λ m )}, is assured by the result of Adimurthi [1] based on the Nehari critical point theory [14]. Besides this formulation, there are several existence results using the variational method (M. Flucher [7], M.-C. Shaw [21]). We shall discuss on their relations between those variational solutions as well as their regularity.…”

“…For example, by Adimurthi's method we can construct a weak solution of (1.1) by Nehari's critical point theory ( [14]). On the other hand, there is a result by M.-C. Shaw [21] in which a different kind of variational formulation is used. All those solutions satisfy the equation (1.1) in a weak sense.…”

“…Lions' semi-compactness argument ( [11]). M.-C. Shaw [21] considered a different kind of variational problem. Let…”

“…On the other hand, M.-C. Shaw [21] and Adimurthi [1] made use of variational methods to obtain solutions in bounded domains. Among other things, Adimurthi constructed positive solutions for any λ with 0 < λ < λ 1 , where λ 1 denotes the first eigenvalue of −∆ in Ω with homogeneous Dirichlet boundary condition.…”

Elliptic Equations and Systems with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition

“…In the case N = 2, motivated by the Moser-Trudinger inequality (see Lemma 3), existence of nontrivial solutions to problems of the above type when f has exponential growth has been studied by many authors; see, for example, Carleson-Chang [9], Atkinson-Peletier [6], Shaw [40], Adimurthi et al [1][2][3][4], Cao [8], do Ó et al [34,35], de Figueiredo et al [16,17], Y.X. Li et al [25][26][27], Ruf [38], Lu and Yang [30,31], etc.…”