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Loop type subcontinua of positive solutions for indefinite concave-convex problems

Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu

Abstract

We establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a non-regular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn's topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved in [15], and extend previous results established in the powerlike case.

Loop type subcontinua of positive solutions for indefinite concave-convex problems

Abstract

We establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a non-regular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn's topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved in [15], and extend previous results established in the powerlike case.

Paper Structure

This paper contains 7 sections, 15 theorems, 119 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and Examples
  3. Regularization schemes and transversality conditions
  4. A priori bounds
  5. Proofs of Theorems \ref{['resu01']} and \ref{['resu02']}
  6. Proof of assertion (i) in Theorems \ref{['resu01']} and \ref{['resu02']}
  7. Proof of assertion (ii) in Theorems \ref{['resu01']} and \ref{['resu02']}

Key Result

Theorem 1.1

Under $\mathcal{B}u=u$, we assume H_1g, H_2f, ab:posi, H_3, H_2', and $(H_\psi)$ with $\psi = \pm a$. In addition, suppose either Then, the following two assertions hold:

Figures (3)

  • Figure 1: The loop type subcontinua $\mathcal{C}_{0}$ and $\mathcal{C}_{\ast}$.
  • Figure 2: The bounded component $\mathcal{C}_{\varepsilon}$ for $(P_{\mathcal{B},\varepsilon})$ with $\mathcal{B}u=\frac{\partial u}{\partial \mathbf{n}}$. (i) Case $\int_\Omega a < 0$. (ii) Case $\int_\Omega a > 0$.
  • Figure 3: The situations of $\Sigma_{\varepsilon}^\pm$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Remark 3.1
  • Remark 3.2
  • ...and 14 more