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On $(p,N)$-Laplace multivalued equations with critical exponential nonlinearity in $\mathbb{R}^N$

Ankit, Abhishek Sarkar

TL;DR

The paper addresses the existence of nonnegative solutions to multivalued $(p,N)$-Laplace equations with discontinuous nonlinearities exhibiting exponential critical growth in $\mathbb{R}^N$. It develops a non-smooth variational framework in the weighted space $\mathbf{X}=W_V^{1,p}(\mathbb{R}^N)\cap W_V^{1,N}(\mathbb{R}^N)$, leveraging Orlicz spaces and Trudinger–Moser type inequalities to handle the exponential nonlinearity. By combining Ekeland’s variational principle and a non-smooth Mountain Pass theorem, the authors prove the existence of two nonnegative solutions for small $\varepsilon$, with one solution having negative energy and the other positive energy, and in a coupled setting, two nonnegative solutions exist. The results extend multiplicity theory to non-differentiable, non-Hilbert settings with exponential critical growth, providing a framework applicable to free-boundary and reaction-diffusion-type models in unbounded domains.

Abstract

In this paper, we study the existence of nonnegative solutions for a class of multivalued $(p,N)$-Laplace problems having discontinuous nonlinearity with critical exponential growth in $\mathbb{R}^N$. To demonstrate the existence results, we utilized variational methods for non-differentiable functions.

On $(p,N)$-Laplace multivalued equations with critical exponential nonlinearity in $\mathbb{R}^N$

TL;DR

The paper addresses the existence of nonnegative solutions to multivalued -Laplace equations with discontinuous nonlinearities exhibiting exponential critical growth in . It develops a non-smooth variational framework in the weighted space , leveraging Orlicz spaces and Trudinger–Moser type inequalities to handle the exponential nonlinearity. By combining Ekeland’s variational principle and a non-smooth Mountain Pass theorem, the authors prove the existence of two nonnegative solutions for small , with one solution having negative energy and the other positive energy, and in a coupled setting, two nonnegative solutions exist. The results extend multiplicity theory to non-differentiable, non-Hilbert settings with exponential critical growth, providing a framework applicable to free-boundary and reaction-diffusion-type models in unbounded domains.

Abstract

In this paper, we study the existence of nonnegative solutions for a class of multivalued -Laplace problems having discontinuous nonlinearity with critical exponential growth in . To demonstrate the existence results, we utilized variational methods for non-differentiable functions.

Paper Structure

This paper contains 9 sections, 24 theorems, 231 equations.

Table of Contents

  1. Introduction and Main Results
  2. Preliminaries
  3. Non smooth critical point theory
  4. Some version of Trudinger-Moser type inequality and related results
  5. Orlicz spaces
  6. Embedding Results
  7. Properties of functional Lg
  8. Existence of first solution
  9. Existence of the second solution

Key Result

Theorem 1.2

Assume V1-V2, f1-f2, f4,f6 and g1 hold. Then, there exist $\hat{\epsilon}>0$ and $t_*>0$ such that, the problem (main problem) possesses a nonnegative solution $\ u_\epsilon\in \mathbf{X}$ with $I_{\epsilon}(u_\epsilon)=c_{\epsilon}<-\delta^*<0$, for all $\epsilon \in (0,\hat{\epsilon})$ and $t_0\in

Theorems & Definitions (55)

  • Example 1.1: Example of an $f$ satisfying \ref{['f1']}--\ref{['f6']}
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Generalized Directional Derivative
  • Definition 2.2: Generalized Gradient
  • Remark 2.3
  • Definition 2.4: Critical Point
  • Definition 2.4: Critical Point
  • Theorem 2.5: Mountain Pass Theorem for Non-differentiable Functions radulescu1993mountain
  • ...and 45 more