Weak solutions of quasilinear elliptic PDE's at resonance

Existence of at least three weak solutions for a quasilinear elliptic system

In this paper, applying two theorems of Ricceri and Bonanno, we will establish the existence of three weak solutions for a quasilinear elliptic system. Indeed, we will assign a differentiable nonlinear operator to a differential equation system such that the critical points of this operator are weak solutions of the system. In this paper, applying two theorems of R...

Positive Solutions of Quasilinear Elliptic Equations

(1.2) { −∆pu = λa(x)|u|p−2u, u ∈ D 0 (Ω), has the least eigenvalue λ1 > 0 with a positive eigenfunction e1 and λ1 is the only eigenvalue having this property (cf. Proposition 3.1). This gives us a possibility to study the existence of an unbounded branch of positive solutions bifurcating from (λ1, 0). When Ω is bounded, the result is well-known, we refer to the survey article of Amann [2] and t...

Krylov and Safonov Estimates for Degenerate Quasilinear Elliptic PDEs

We here establish an a priori Hölder estimate of Krylov and Safonov type for the viscosity solutions of a degenerate quasilinear elliptic PDE of non-divergence form. The diffusion matrix may degenerate when the norm of the gradient of the solution is small: the exhibited Hölder exponent and Hölder constant only depend on the growth of the source term and on the bounds of the spectrum of the dif...

Large Solutions for Quasilinear Elliptic Equations

EXISTENCE OF WEAK SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING THE p-LAPLACIAN

This paper shows the existence of nontrivial weak solutions for the quasilinear elliptic equation − ` ∆pu +∆p(u ) ́ + V (x)|u|p−2u = h(u) in RN . Here V is a positive continuous potential bounded away from zero and h(u) is a nonlinear term of subcritical type. Using minimax methods, we show the existence of a nontrivial solution in C loc (R N ) and then show that it decays to zero at infinity wh...