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Positive solutions of elliptic systems with superlinear nonlinearities on the boundary

Shalmali Bandyopadhyay, Maya Chhetri, Briceyda Delgado, Nsoki Mavinga, Rosa Pardo

TL;DR

The paper analyzes a coupled elliptic system with nonlinear boundary conditions that are superlinear and subcritical at infinity, parameterized by $\lambda>0$. Using a rescaling technique, Leray–Schauder degree theory, and elliptic regularity, it proves the existence of a connected branch of positive weak solutions bifurcating from infinity as $\lambda\to 0^+$, and under near-zero hypotheses, a global branch emanating from the trivial solution with a unique infinity bifurcation point at $\lambda=0$. It further establishes a global connected component from zero via Crandall–Rabinowitz and Rabinowitz bifurcation theory, and, under monotonicity assumptions, multiplicity of positive solutions for a range of $\lambda$ values through degree theory and sub-/supersolution methods. The results extend scalar boundary-value problem techniques to strongly coupled elliptic systems with nonlinear boundary terms and provide a detailed bifurcation and multiplicity picture, including nonexistence for large $\lambda$. Overall, the work combines rescaling, fixed-point degree arguments, and variational-type methods to map the full bifurcation landscape of the system.

Abstract

We consider elliptic systems with superlinear and subcritical boundary conditions and a bifurcation parameter as a multiplicative factor. By combining the rescaling method with degree theory and elliptic regularity theory, we prove the existence of a connected branch of positive weak solutions that bifurcates from infinity as the parameter approaches zero. Furthermore, under additional conditions on the nonlinearities near zero, we obtain a global connected branch of positive solutions bifurcating from zero, which possesses a unique bifurcation point from infinity when the parameter is zero. Finally, we analyze the behavior of this branch and discuss the number of positive weak solutions with respect to the parameter using bifurcation theory, degree theory, and sub- and super-solution methods.

Positive solutions of elliptic systems with superlinear nonlinearities on the boundary

TL;DR

The paper analyzes a coupled elliptic system with nonlinear boundary conditions that are superlinear and subcritical at infinity, parameterized by . Using a rescaling technique, Leray–Schauder degree theory, and elliptic regularity, it proves the existence of a connected branch of positive weak solutions bifurcating from infinity as , and under near-zero hypotheses, a global branch emanating from the trivial solution with a unique infinity bifurcation point at . It further establishes a global connected component from zero via Crandall–Rabinowitz and Rabinowitz bifurcation theory, and, under monotonicity assumptions, multiplicity of positive solutions for a range of values through degree theory and sub-/supersolution methods. The results extend scalar boundary-value problem techniques to strongly coupled elliptic systems with nonlinear boundary terms and provide a detailed bifurcation and multiplicity picture, including nonexistence for large . Overall, the work combines rescaling, fixed-point degree arguments, and variational-type methods to map the full bifurcation landscape of the system.

Abstract

We consider elliptic systems with superlinear and subcritical boundary conditions and a bifurcation parameter as a multiplicative factor. By combining the rescaling method with degree theory and elliptic regularity theory, we prove the existence of a connected branch of positive weak solutions that bifurcates from infinity as the parameter approaches zero. Furthermore, under additional conditions on the nonlinearities near zero, we obtain a global connected branch of positive solutions bifurcating from zero, which possesses a unique bifurcation point from infinity when the parameter is zero. Finally, we analyze the behavior of this branch and discuss the number of positive weak solutions with respect to the parameter using bifurcation theory, degree theory, and sub- and super-solution methods.

Paper Structure

This paper contains 16 sections, 12 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Auxiliary Results
  3. Regularity of weak solutions and positivity
  4. Weak solution to classical solution
  5. Proof of Theorem \ref{['thm:local']}
  6. Rescaling
  7. A Compact Operator
  8. Solution to Rescaled Problem
  9. Global Bifurcation: Proof of Theorem \ref{['thm-2']}
  10. Proof of Theorem \ref{['thm-2']}
  11. Multiplicity Result: Proof of Theorem \ref{['thm-3']}
  12. Proof of Theorem \ref{['thm-3']}
  13. Step 1: Existence of a positive solution $(u_1 ^*,u_2^*)$ for each $\lambda\in \left(\mu_0, \overline{\lambda}\right)$ via degree theory
  14. Step 2: Existence of a second positive solution for each $\lambda\in \left(\mu_0, \overline{\lambda}\right)$ via sub and super solution theory
  15. Step 3: The existence of a solution for $\lambda=\overline{\lambda}$.
  16. ...and 1 more sections

Key Result

Theorem 1.1

Suppose that H_inf holds. Then, there exists $\widetilde{\lambda}>0$ such that for all $\lambda\in(0,\widetilde{\lambda}]$, pde has a positive weak solution $(\lambda,(u_1,u_2))$ such that Moreover, there exists a connected component of positive weak solutions of pde, namely $\mathscr{C}^+\subset \Sigma$, bifurcating from infinity at $\lambda=0$, such that the projection of $\mathscr{C}^+$ on the

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • Lemma 3.1
  • proof
  • ...and 12 more