Existence of positive solutions for a class of p-Laplacian superlinear semipositone problems

Abstract

We consider a quasilinear elliptic problem of the formwhere λ > 0 is a parameter, 1 < p < 2 and Ω is a strictly convex bounded domain in ℝN, N > p, with C2 boundary ∂Ω. The nonlinearity f : [0, ∞) → ℝ is a continuous function that is semipositone (f(0) < 0) and p-superlinear at infinity. Using degree theory, combined with a rescaling argument and uniform L∞a priori bound, we establish the existence of a positive solution for λ small. Moreover, we show that there exists a connected component of positive solutions bifurcating from infinity at λ = 0. We also extend our study to systems.

Existence of Positive Solution to Schrödinger-type Semipositone Problems with Mixed Nonlinear Boundary Conditions

“…For the case (f ∞ ), the scaling method combined with the degree theory [13] and the mountain pass lemma with C 1,α (Ω)-regularity [9,35] are frequently used to demonstrate the existence of a positive solution for semipositone Laplacian (or p-Laplacian) problems with Dirichlet boundary conditions. Here, more precise assumptions are given for this case.…”

A class of semipositone $p$-Laplacian problems with a critical growth reaction term

“…where Ω is a smooth bounded domain in R N , 1 < p < N, p < q ≤ p * , µ > 0 is a parameter, and p * = Np/(N − p) is the critical Sobolev exponent. The scaling u → µ 1/(q−1) u transforms the first equation in (1.1) into −∆ p u = µ (q−p)/(q−1) u q−1 − 1 , so in the subcritical case q < p * , it follows from the results in Castro et al [6] and Chhetri et al [7] that this problem has a weak positive solution for sufficiently small µ > 0 when p > 1 (see also Unsurangie [16], Allegretto et al [1], Ambrosetti et al [2], and Caldwell et al [5] for the case when p = 2). On the other hand, in the critical case q = p * , it follows from a standard argument involving the Pohozaev identity for the p-Laplacian (see Guedda and Véron [11,Theorem 1.1]) that problem (1.1) has no solution for any µ > 0 when Ω is star-shaped.…”

On a class of infinite semipositone problems for (p,q) Laplace operator

“…However topological tools such as degree theory, fixed point methods, bifurcation theory etc. are mainly found reliable in obtaining the positive solution to semipositone problems (see [12], [26], [30] and [17]). But all these methods depend strongly on the qualitative properties of the solutions, such as regularity, asymptotic behavior, a priori estimates, stability etc.…”