[Paper Review] On Strongly Nonlinear Eigenvalue Problems in the Framework of Nonreflexive Orlicz-Sobolev Spaces

Edcarlos D. Silva, J.V. Gonçalves+1 more|arXiv (Cornell University)|Oct 9, 2016

Nonlinear Partial Differential Equations12 references3 citations

TL;DR

This paper establishes the existence and multiplicity of non-negative weak solutions for a strongly nonlinear eigenvalue problem driven by the $Φ$-Laplacian operator in nonreflexive Orlicz-Sobolev spaces, without requiring the $Δ_2$ condition. The key contribution is proving that $m-1$ solutions exist for large parameter values $\lambda$, even when the underlying space lacks reflexivity, using variational and topological methods in carefully constructed subspaces of nonreflexive Orlicz-Sobolev spaces.

ABSTRACT

It is established existence and multiplicity of solutions for strongly nonlinear problems driven by the $Φ$-Laplacian operator on bounded domains. Our main results are stated without the so called $Δ_{2}$ condition at infinity which means that the underlying Orlicz-Sobolev spaces are not reflexive.

Motivation & Objective

• To establish existence and multiplicity of weak solutions for a strongly nonlinear eigenvalue problem involving the $Φ$-Laplacian operator in nonreflexive Orlicz-Sobolev spaces.
• To remove the standard $Δ_2$ condition at infinity, which is known to imply reflexivity, thus extending results to nonreflexive settings.
• To analyze the problem under general growth conditions on the nonlinearity $f(u)$, including oscillatory behavior between positive and negative values in specified intervals.
• To provide a framework for handling energy functionals that are not well-defined or $C^1$ in the full nonreflexive Orlicz-Sobolev space.

Proposed method

• Utilizes the $Φ$-Laplacian operator defined via a $C^1$ function $\phi$ satisfying specific growth and monotonicity conditions.
• Works in the Orlicz-Sobolev space $W_0^{1}L_{\Phi}(\Omega)$, where $\Phi(t) = \int_0^t s\phi(s)\,ds$, without assuming the $\Delta_2$ condition.
• Employs variational and topological methods, including lower and upper solutions, to prove existence of multiple solutions.
• Constructs appropriate subspaces of $W_0^{1}L_{\Phi}(\Omega)$ where the energy functional is well-defined and admits critical points.
• Applies generalized Green's formula and trace theory in nonreflexive Orlicz spaces to characterize the zero trace condition for functions in $W_0^{1}L_{\Phi}(\Omega)$.
• Uses the extension of $f$ as an odd function to obtain $2(m-1)$ solutions, symmetric in sign.

Experimental results

Research questions

• RQ1 Can the existence of multiple non-negative weak solutions be guaranteed for the $Φ$-Laplacian problem in nonreflexive Orlicz-Sobolev spaces without the $\Delta_2$ condition?
• RQ2 What conditions on the nonlinearity $f(u)$ ensure the existence of $m-1$ distinct solutions with prescribed $L^\infty$-norm bounds?
• RQ3 How can variational methods be adapted when the energy functional is not $C^1$ or even well-defined on the entire nonreflexive Orlicz-Sobolev space?
• RQ4 What is the role of the conjugate function $\tilde{\Phi}$ and trace theory in characterizing the zero trace space $W_0^{1}L_{\Phi}(\Omega)$ in nonreflexive settings?

Key findings

• For each $\lambda > \overline{\lambda}$, there exist at least $m-1$ non-negative weak solutions $u_1, \dots, u_{m-1}$ in $W_0^{1}L_{\Phi}(\Omega) \cap L^\infty(\Omega)$ with $\|u_k\|_\infty \in (a_k, a_{k+1}]$.
• The existence of $m-1$ solutions is guaranteed under the conditions $(f_1)$--$(f_3)$, even when $W_0^{1}L_{\Phi}(\Omega)$ is nonreflexive due to the absence of the $\Delta_2$ condition.
• The converse holds: if a non-negative solution $u$ satisfies $\|u\|_\infty \in (a_k, a_{k+1}]$, then $\int_{a_k}^{a_{k+1}} f(s)\,ds > 0$ must hold.
• The results extend previous work by Hess and Loc–Schmitt from the $p$-Laplacian to the more general $Φ$-Laplacian in nonreflexive spaces.
• Examples of nonreflexive $Φ$-functions covered include $\Phi(t) = e^t - t + 1$, $\Phi(t) = (1+t^2)^\gamma - 1$ for $\gamma > 1/2$, and $\Phi(t) = t^p \log(1+t)$ for $p \geq 1$.
• By extending $f$ as an odd function, at least $2(m-1)$ weak solutions exist: $m-1$ positive and $m-1$ negative, each with $L^\infty$-norm in the intervals $(a_k, a_{k+1}]$.