Maximum and Comparison Principles for Operators Involving thep-Laplacian

On positive solutions of the (p,A)-Laplacian with potential in Morrey space

“…The additional regularity assumptions on A and V for the case 1 < p < 2 in the Main Theorem seems to be technical, and might be nonessential. However, these assumptions guarantee the Lipschitz continuity of solutions of (1.4) (in fact they guarantee that solutions are C 1,α , see [26,Theorem 5.3]), a property which (as in [38,36]) is essential for the proof of the Main Theorem in this range of p. On the other hand, throughout the paper we do not use the boundary point lemma, which was an essential tool in [17,38,36].…”

“…[7,17,36], where stronger regularity assumptions on the coefficients and the boundary are assumed). In particular, we prove the existence of the principal eigenvalue, establish its main properties, and study the relationships between the positivity of principal eigenvalue, the weak and strong maximum principles, and the (unique) solvability of the Dirichlet problem.…”

Ground state alternative for p-Laplacian with potential term

“…We shall need also the following comparison principle of J. García-Melián, and J. Sabina de Lis [17,Theorem 5].…”

“…The authors wish to thank V. Liskevich and V. Moroz for valuable discussions, in particular for V. Moroz' comment on the convexity counterexamples in [9], [17] and [14]. Part of this research was done while K. …”

Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations

“…If Ω is a C 1,θ (0 < θ < 1) bounded domain and V ∈ L ∞ (Ω), it is well-known that the variational problem Ω (|∇φ| p + V |φ| p )dx Ω |φ| p dx admits a unique (up to a scalar multiplication) minimizer ϕ (see, e.g., [16,21,Lemma 3]). Moreover, ϕ is a C 1,µ (0 < µ < 1) positive solution of the quasilinear eigenvalue problem (1.5)…”