Mountain Pass Method in Nonlinear Analysis

• Mountain Pass Method is a variational approach that identifies saddle-point critical points by exploiting a ‘mountain pass’ geometry in the energy landscape.
• It employs minimax principles, precise energy estimates, and trial functions to establish the existence of positive solutions even under critical nonlinearities and perturbations.
• The method is widely used in nonlinear elliptic PDEs, constrained optimization, and various physical models where standard minimization techniques fail.

The Mountain Pass Method is a fundamental variational approach for locating saddle-type critical points of nonlinear functionals, especially those exhibiting multiple nontrivial solutions due to the interaction between local minima and structural nonlinearities. The method is ubiquitously employed in nonlinear elliptic PDEs, critical point theory, constrained optimization, and numerous physical models where standard minimization arguments are inadequate. The essential principle is the identification of a “mountain pass” geometry in the energy landscape, enabling one to detect critical points at positive energy levels that are not global minima. Rigorous justification relies on the minimax principle, a geometric separation in the function space, and fine energy estimates ensuring local compactness where global compactness may fail. Recent research extends this theory to problems with critical exponents and delicate lower-order or logarithmic perturbations, such as the Brézis–Nirenberg problem with a logarithmic term (Zhang et al., 15 Apr 2025).

1. Energy Functional and Critical Point Equation

For the Brézis–Nirenberg problem with logarithmic perturbation, the functional is defined on $H_0^1(\Omega)$, $\Omega \subset \mathbb{R}^4$ bounded and smooth, as

$$J(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \frac{\lambda}{2} \int_{\Omega} |u^+|^2 \,dx - \frac{\mu}{4} \int_{\Omega} |u^+|^4 \,dx - \frac{\theta}{2} \int_{\Omega} (u^+)^2 \left( \log (u^+)^2 - 1 \right) dx,$$

where $u^+ := \max\{u,0\}$, $\lambda \in \mathbb{R}$, $\mu > 0$, and $\theta < 0$. The associated Euler–Lagrange equation is

$$-\Delta u = \lambda u + \mu |u|^2 u + \theta u \log u^2, \qquad u \in H_0^1(\Omega).$$

Solutions $u$ correspond to critical points of $J$. The logarithmic perturbation acts as a lower-order nonlinearity with negative sign, interacting subtly with the critical Sobolev exponent in dimension 4.

2. Mountain Pass Geometry and Path Construction

The necessary geometry for the Mountain Pass Theorem is established by:

• Existence of a local minimizer $\Omega \subset \mathbb{R}^4$0 such that $\Omega \subset \mathbb{R}^4$1, $\Omega \subset \mathbb{R}^4$2 and $\Omega \subset \mathbb{R}^4$3 for $\Omega \subset \mathbb{R}^4$4.
• For any $\Omega \subset \mathbb{R}^4$5, $\Omega \subset \mathbb{R}^4$6 as $\Omega \subset \mathbb{R}^4$7, i.e., paths leaving the local well descend to arbitrarily negative energy.

Hence, define the path set

$\Omega \subset \mathbb{R}^4$8

and the mountain pass level

$\Omega \subset \mathbb{R}^4$9

The geometry ensures $$J(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \frac{\lambda}{2} \int_{\Omega} |u^+|^2 \,dx - \frac{\mu}{4} \int_{\Omega} |u^+|^4 \,dx - \frac{\theta}{2} \int_{\Omega} (u^+)^2 \left( \log (u^+)^2 - 1 \right) dx,$$0 and $$J(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \frac{\lambda}{2} \int_{\Omega} |u^+|^2 \,dx - \frac{\mu}{4} \int_{\Omega} |u^+|^4 \,dx - \frac{\theta}{2} \int_{\Omega} (u^+)^2 \left( \log (u^+)^2 - 1 \right) dx,$$1.

3. Upper Estimate for the Mountain Pass Level

Local compactness (Palais–Smale condition) is obstructed by the critical exponent; it holds only below a specific threshold. Precise upper bounds are engineered via trial functions based on the Aubin–Talenti bubble,

$$J(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \frac{\lambda}{2} \int_{\Omega} |u^+|^2 \,dx - \frac{\mu}{4} \int_{\Omega} |u^+|^4 \,dx - \frac{\theta}{2} \int_{\Omega} (u^+)^2 \left( \log (u^+)^2 - 1 \right) dx,$$2

where $$J(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \frac{\lambda}{2} \int_{\Omega} |u^+|^2 \,dx - \frac{\mu}{4} \int_{\Omega} |u^+|^4 \,dx - \frac{\theta}{2} \int_{\Omega} (u^+)^2 \left( \log (u^+)^2 - 1 \right) dx,$$3, $$J(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \frac{\lambda}{2} \int_{\Omega} |u^+|^2 \,dx - \frac{\mu}{4} \int_{\Omega} |u^+|^4 \,dx - \frac{\theta}{2} \int_{\Omega} (u^+)^2 \left( \log (u^+)^2 - 1 \right) dx,$$4 near $$J(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \frac{\lambda}{2} \int_{\Omega} |u^+|^2 \,dx - \frac{\mu}{4} \int_{\Omega} |u^+|^4 \,dx - \frac{\theta}{2} \int_{\Omega} (u^+)^2 \left( \log (u^+)^2 - 1 \right) dx,$$5. One shows for small $$J(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \frac{\lambda}{2} \int_{\Omega} |u^+|^2 \,dx - \frac{\mu}{4} \int_{\Omega} |u^+|^4 \,dx - \frac{\theta}{2} \int_{\Omega} (u^+)^2 \left( \log (u^+)^2 - 1 \right) dx,$$6,

$$J(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \frac{\lambda}{2} \int_{\Omega} |u^+|^2 \,dx - \frac{\mu}{4} \int_{\Omega} |u^+|^4 \,dx - \frac{\theta}{2} \int_{\Omega} (u^+)^2 \left( \log (u^+)^2 - 1 \right) dx,$$7

where $$J(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \frac{\lambda}{2} \int_{\Omega} |u^+|^2 \,dx - \frac{\mu}{4} \int_{\Omega} |u^+|^4 \,dx - \frac{\theta}{2} \int_{\Omega} (u^+)^2 \left( \log (u^+)^2 - 1 \right) dx,$$8 is the least energy of negative critical points and $$J(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \,dx - \frac{\lambda}{2} \int_{\Omega} |u^+|^2 \,dx - \frac{\mu}{4} \int_{\Omega} |u^+|^4 \,dx - \frac{\theta}{2} \int_{\Omega} (u^+)^2 \left( \log (u^+)^2 - 1 \right) dx,$$9 is the best Sobolev constant in dimension 4: $u^+ := \max\{u,0\}$0 Delicate expansions handle the logarithmic and critical terms, using extremal properties of $u^+ := \max\{u,0\}$1 to identify maximizers.

4. Palais–Smale Condition and Profile Splitting

The Palais–Smale condition is established below the critical threshold: $u^+ := \max\{u,0\}$2 Given a $u^+ := \max\{u,0\}$3 sequence $u^+ := \max\{u,0\}$4 with $u^+ := \max\{u,0\}$5 and $u^+ := \max\{u,0\}$6,

• Use energy inequalities and the structure of $u^+ := \max\{u,0\}$7 to show boundedness in $u^+ := \max\{u,0\}$8.
• Apply Brezis–Lieb splitting: for $u^+ := \max\{u,0\}$9, $\lambda \in \mathbb{R}$0, yielding

$\lambda \in \mathbb{R}$1

and via Sobolev, $\lambda \in \mathbb{R}$2. If $\lambda \in \mathbb{R}$3, $\lambda \in \mathbb{R}$4 contradicts minimality, so $\lambda \in \mathbb{R}$5, and $\lambda \in \mathbb{R}$6 strongly in $\lambda \in \mathbb{R}$7.

5. Existence and Positivity of the Mountain Pass Solution

By Willem's Mountain Pass Theorem, there exists $\lambda \in \mathbb{R}$8 with

$\lambda \in \mathbb{R}$9

Positivity follows from maximum principles and comparison:

• Testing $\mu > 0$0 on $\mu > 0$1 ensures $\mu > 0$2.
• Moser iteration yields $\mu > 0$3.
• Elliptic regularity improves $\mu > 0$4 to $\mu > 0$5.
• The strong maximum principle confirms $\mu > 0$6 in $\mu > 0$7.

Thus, a new positive mountain pass solution of the logarithmic Brézis–Nirenberg problem is obtained at strictly positive energy.

6. Extensions to Higher and Lower Dimensions

The mountain pass approach and upper-bound analysis generalize to $\mu > 0$8 and $\mu > 0$9:

• Replace the critical exponent $\theta < 0$0 by $\theta < 0$1.
• Adjust the critical Sobolev constant as $\theta < 0$2 and the gap as $\theta < 0$3.
• Careful choice of parameter $\theta < 0$4 controls the logarithmic term in the concentration estimates.
• The compactness argument (Palais–Smale condition) and minimax construction proceed as in $\theta < 0$5.
• For $\theta < 0$6, the same bubble truncation arguments are no longer effective, indicating the necessity for new techniques in higher dimensions.

7. Summary Table: Energy Levels and Compactness Gaps

Dimension $\theta < 0$7	Critical exponent $\theta < 0$8	Sobolev constant factor	Palais–Smale gap threshold
$\theta < 0$9	$$-\Delta u = \lambda u + \mu |u|^2 u + \theta u \log u^2, \qquad u \in H_0^1(\Omega).$$0	$$-\Delta u = \lambda u + \mu |u|^2 u + \theta u \log u^2, \qquad u \in H_0^1(\Omega).$$1	$$-\Delta u = \lambda u + \mu |u|^2 u + \theta u \log u^2, \qquad u \in H_0^1(\Omega).$$2
$$-\Delta u = \lambda u + \mu |u|^2 u + \theta u \log u^2, \qquad u \in H_0^1(\Omega).$$3	$$-\Delta u = \lambda u + \mu |u|^2 u + \theta u \log u^2, \qquad u \in H_0^1(\Omega).$$4	$$-\Delta u = \lambda u + \mu |u|^2 u + \theta u \log u^2, \qquad u \in H_0^1(\Omega).$$5	$$-\Delta u = \lambda u + \mu |u|^2 u + \theta u \log u^2, \qquad u \in H_0^1(\Omega).$$6
$$-\Delta u = \lambda u + \mu |u|^2 u + \theta u \log u^2, \qquad u \in H_0^1(\Omega).$$7	$$-\Delta u = \lambda u + \mu |u|^2 u + \theta u \log u^2, \qquad u \in H_0^1(\Omega).$$8	$$-\Delta u = \lambda u + \mu |u|^2 u + \theta u \log u^2, \qquad u \in H_0^1(\Omega).$$9	$u$0

The Palais–Smale condition holds below the specified gap threshold in each dimension, permitting the mountain pass construction of positive non-minimizing solutions (Zhang et al., 15 Apr 2025).

References and Historical Context

The mountain pass approach to critical exponent semilinear elliptic equations originates from the seminal work of Brézis–Nirenberg (1983), generalizing to more delicate structures such as those with logarithmic and lower-order perturbations. This method provides existence, energy, and qualitative properties of solutions where topological and variational complexity precludes direct minimization. Recent results explicitly resolve conjectures regarding solutions with positive energy in dimension four and extend these findings to other critical dimensions through precise quantitative energy estimates and compactness arguments (Zhang et al., 15 Apr 2025).