Weak solutions of degenerated quasilinear elliptic equations of higher order

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...

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...

Large Solutions for Quasilinear Elliptic Equations

Classical Solutions for Some Higher Order Semilinear Elliptic Equations under Weak Growth Conditions

We want to explain the crucial difference between the second order (m = 1) and higher order (m > 1) case. If m = 1 a very satisfactory result is known. Indeed, the sign condition (2) alone is sufficient to ensure classical solvability of the Dirichlet problem (1). There are two very strong devices in the theory of second order elliptic equations, which make the proof of the needed a-priori maxi...

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...