The topological sensitivity analysis method has been recognized as a promising, fast, and accurate approach for solving topology optimization and inverse problems. It is based on developing an asymptotic expansion of a design functional with respect to the creation of a small hole inside the computational domain. In this work, we extend this method to the narrow escape problem. The biological process is governed by a parabolic diffusion equation. We derive a sensitivity analysis for the parabolic problem solution with respect to the creation of a small absorbing boundary subset. We develop a rigorous mathematical framework that is valid in two- and three-dimensional space. It provides an asymptotic formula that describes the behavior of the perturbed solution with respect to the location and size of an arbitrary perturbed boundary subset. The Sobolev capacity notion has been employed to measure the smallness of the boundary subset and to describe the asymptotic behavior with respect to the perturbation size. The performed mathematical analysis is general and can be adapted for a large class of partial differential equations. The obtained asymptotic formula can serve as a useful tool to perform numerical algorithms for solving optimization and control problems.
In this paper, we study the existence of the solutions for the following \(p\)-Kirchhoff equation on bounded domain \[\begin{cases}-\Bigg(a+b\displaystyle\int_{\Omega}|\nabla u|^pdx\Bigg)\Delta_pu+\lambda|u|^{p-2}u=h(u), &x \in \Omega,\\ u=0, &x\in\partial\Omega,\end{cases}\] with prescribed mass \[\int_{\Omega}|u|^pdx=m^p,\] where \(2\leq p\lt 3\), \(a\gt 0\), \(b\gt 0\) are positive constants, \(h(u)\) is a general nonlinearity with Sobolev subcritical growth, \(\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator, \(\Omega\subset\mathbb{R}^3\) is a bounded domain, \(\lambda\in \mathbb{R}\) appears as a Lagrange multiplier. First, we prove that the equation admits a positive solution which is a local minimizer when \(h\) is \(L^p\)-subcritical or \(L^p\)-critical at infinity by Brezis-Nirenberg technique. Moreover, we get the multiplicity result by using the genus theory. Next, we prove that the equation has a local minimizer when \(h\) is \(L^p\)-supercritical. Besides, by using the Pohozaev analysis, we prove that in this case, if \(\Omega\) is a star-shaped domain with respect to the origin, the equation admits a second solution which is mountain pass type. To the best of our knowledge, this work seems to be the first contribution on the existence of normalized solutions for \(p\)-Kirchhoff equation on bounded domain.
This paper investigates the oscillatory behavior of solutions to a class of second-order nonlinear neutral delay differential equations with both positive and negative terms of the form \[\left(a(\theta) z^{\prime}(\theta)\right)^{\prime} - p(\theta)x(\theta) + q(\theta)x^{\alpha}(\sigma(\theta)) = 0, \quad \theta \geq \theta_{0},\] where \(z(\theta) = x(\theta) + b(\theta)x(\tau(\theta))\). To facilitate the analysis, the equation is transformed, via a positive solution of an auxiliary second-order ordinary differential equation, into a binomial form. By employing the comparison and integral averaging techniques together with the arithmetic-geometric mean inequality, we establish new sufficient conditions for the oscillation of all solutions. The results obtained extend and improve several existing criteria in the literature. Finally, illustrative examples are presented to demonstrate the effectiveness, novelty, and applicability of the proposed oscillation criteria.
This paper investigates a class of strongly indefinite Choquard equations formulated on the two-dimensional unit ball \(B\). Specifically, consider the following equation: \[\begin{cases}-\Delta u+V(x)u=(I_{\mu}\ast F(u))f(u), &x\in B, \\ u\in H^1_{0,\mathrm{rad}}(B), & \end{cases}\] where \(\mu\in(0,2)\) and \(I_{\mu}\) represents the classical Riesz potential. A fundamental hypothesis is that zero lies within a spectral gap of the Schrödinger operator \(-\Delta+V\). Furthermore, the continuous nonlinearity \(f(t)\) is characterized by a supercritical exponential growth governed by \(\exp[(\beta+|x|^{\alpha})t^{2}]\) with \(\beta, \alpha\gt 0\). Most existing works on this problem are limited to the subcritical or critical exponential growth cases. However, dealing with supercritical growth is much more difficult, especially when estimating the exact upper bound of the minimax levels. To overcome this, we use an approximation method and fine estimates to prove that this indefinite problem has a positive ground state solution. Our approach can also be applied to other elliptic problems with supercritical growth.
We explicate the difference between the function theory of one complex variable and the function theory of several complex variables. In particular, we use the inhomogeneous Cauchy-Riemann equations to explain why there is a Hartogs extension phenomenon in several complex variables but not in one complex variable.
We study the nonlinear elliptic system: \[\begin{cases} u \in W^{1,p}_0(\Omega): -\operatorname{div} (a(x)|\nabla u|^{p-2}\nabla u) + u = -\operatorname{div} (a(x) u|\nabla \psi|^{p-2}\nabla \psi) + f(x), \\ \psi \in W^{1,p}_0(\Omega): -\operatorname{div} (a(x)|\nabla \psi |^{p-2}\nabla \psi) = u^{\theta} \end{cases}\] in a bounded, open subset of \(\mathbb{R}^N\) for \(N \gt 2\) and \(2 \lt p \lt N\), where \(f\) satisfies: \[0 \leq f, \quad f\in L^{(p^*){'}}(\Omega), \quad p^*= \frac{Np}{N-p},\] \(a \in L^{\infty}(\Omega) \) is a given function such that there exist \(\alpha, \beta \in \mathbb{R}\) satisfying \[0\lt\alpha \leq a(x) \leq \beta, \quad x\in \Omega.\] We prove the existence of weak solutions in \([W^{1,p}_0(\Omega)]^2\) under the assumption \[0\lt \theta\lt 1- \frac{2}{p^*}.\]
We present a class of nonlinear mappings, properly containing the nonexpansive ones, enjoying the fixed point property in orthogonally convex Banach spaces.
We prove the existence of multiple positive solutions for elliptic systems with linear boundary conditions of Neumann type. We suppose that the nonlinearities grow quadratically with respect to gradient. A key step is to obtain a priori bound on the derivatives by using a Gronwall-type inequality. Our approach is topological and relies on the fixed point index.
We show how the Poincaré-Birkhoff theorem for Hamiltonian systems can be used to find multiple solutions of the antiperiodic problem. Applications are given to scalar second order differential equations whose nonlinearities provide a twist in the phase plane, among which those with a superlinear or sublinear behaviour at infinity.
In this paper, we study the existence of positive normalized solutions to the following \(p\)-Laplacian system: \[\begin{cases} -\Delta_p u+\lambda_1u^{p-1}=\mu_1u^{m_1-1}+\beta r_1u^{r_1-1}v^{r_2}&\text{in }\mathbb{R}^N,\\ -\Delta_p v+\lambda_2v^{p-1}=\mu_2v^{m_2-1}+\beta r_2u^{r_1}v^{r_2-1}&\text{in }\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^p=a, \quad \int_{\mathbb{R}^N}|v|^p=b,\end{cases}\] where \(1\lt p\lt N\), \(\mu_1,\mu_2,\beta,a,b\gt 0\) are prescribed, \(\lambda_1,\lambda_2 \in \mathbb{R}\) are known as the Lagrange multiplier, \(\Delta_p u= \mathrm{div} (|\nabla u|^{p-2} \nabla u)\) denotes the \(p\)-Laplacian operator. We prove the existence of positive solutions for the coupled purely mass super-critical case (i.e., \(\frac{p^2}{N}+p\lt m_1,m_2,r_1 + r_2\lt p^*\)) by a minimization argument based on a closed ball and the Pohozaev constraint.
The aim of this paper is to introduce a new comparison theorem (in both delayed and advanced cases) that allows us to investigate the properties of third-order differential equations with quasi-derivatives \[(r_1(t)(r_2(t)y'(t))')'-p(t)y(\tau(t))=0\] using the following simpler differential equations \[(r(t)(r(t)z'(t))')'-p(t)z(\tau(t))=0\] and \[y'''(t)-q(t)y(\sigma(t))=0.\] The obtained comparison principles allow for the immediate transcription of the oscillatory results known for the simpler equations into studied equation with quasi-derivatives. The progress achieved will be illustrated through several examples.
This work is devoted to the study of a class of Schrödinger-Poisson system with doubly critical growth on the first Heisenberg group. Utilizing the concentration-compactness principle associated with classical Sobolev space on the Heisenberg group and mountain pass theorem, we prove that the system admits multiple nontrivial solutions.
We investigate a relation between the Harnack inequalities and the (a priori) growth estimates for positive solutions of quasilinear elliptic equations with nonlinear terms involving the solution and its gradient in an arbitrary domain in \(\mathbb{R}^N\).
This paper focuses on the following planar Schrödinger-Poisson system with critical exponential growth and nonlocal interaction \[\begin{cases}-\Delta u+\lambda u+\mu(\log|\cdot|*u^2)u = \gamma \left( I_\alpha * |u|^q \right) |u|^{q-2} u+\left(e^{u^2}-1-u^2\right)u, & x\in \mathbb{R}^2, \\ \displaystyle \int_{\mathbb{R}^2}u^2\mathrm{d}x=c,\end{cases}\] where \(c\gt 0\), \(\mu,\gamma\gt 0\), \(\lambda \in \mathbb{R}\) appears as a Lagrange multiplier, \(\alpha \in (0,2)\), \(1+\frac{\alpha}{2} \leq q \lt +\infty\), \( I_\alpha:\mathbb{R}^2\to\mathbb{R}\) denotes the Riesz potential and \(1+\frac{\alpha}{2}\) is the lower critical exponent with respect to the Hardy-Littlewood-Sobolev inequality. Through delicate energy estimates, under explicit conditions on \(c\), we prove the existence of two normalized solutions: one is a local minimizer and the other is of mountain-pass type. The presence of the logarithmic kernel and the competition between the two nonlocal terms necessitates the development of new tools to address the loss of compactness caused by the critical exponential growth, for which the variational techniques developed for the local problem are no longer applicable. Our work not only generalizes the special case \(\gamma=0\), but also provides an analytical approach that is applicable to more \(L^2\)-constrained problems with competing nonlocal terms modelling long-range attraction in particle physics.
In this paper, we establish a priori estimates and existence of positive solutions for elliptic problems under integral Neumann boundary conditions.
We provide an alternative approach, based on the Leray-Schauder fixed point index in cones, to a fixed point theorem for operator systems due to Precup. Our focus is on the case of operators whose components are entirely of compressive type. The abstract technique is applied to a system of second-order differential equations providing a coexistence positive solution by means of an eigenvalue type criterion.
A subset \(J\) of vertices is said to be a \((1,k)\)-dominating set if every vertex \(v\) not belonging to the set \(J\) has a neighbour in \(J\) and there exists also another vertex in \(J\) within the distance at most \(k\) from \(v\). In this paper, we study the problem of the existence of independent \((1,k)\)-dominating sets for \(k\in\{1,2\}\) in the tensor product and in the strong product of two graphs. We give complete characterisations of these graph products, which have independent \((1,1)\)-dominating sets or independent \((1,2)\)-dominating sets, with respect to the properties of their factors.
We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schrödinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. As a continuation of our previous study on the existence and uniqueness of solutions, we prove here some strong stabilization properties. In particular, we show the finite time extinction of solutions induced by the nonlinear saturation mechanism, which, sometimes, can be understood as a bang-bang control. The analysis relies on refined energy methods. Our results provide a rigorous justification of nonlinear dissipation as an effective stabilization mechanism for this class of complex equations where the maximum principle fails.
In this paper, we study the normalized solutions of the following critical growth Choquard equation with mixed local and nonlocal operators: \[\begin{split}-\Delta u +(-\Delta)^s u &= \lambda u +\mu |u|^{p-2}u +(I_{\alpha}*|u|^{2^*_{\alpha}})|u|^{2^*_{\alpha}-2}u \quad\text{in}\quad \mathbb{R}^N,\\ \| u\|_2 &= \tau,\end{split}\] where \(N\geq 3\), \(\tau\gt 0\), \(I_{\alpha}\) is the Riesz potential of order \(\alpha\in (0,N)\), \(2^*_{\alpha}=\frac{N+\alpha}{N-2}\) is the critical exponent corresponding to the Hardy-Littlewood-Sobolev inequality, \((-\Delta)^s\) is the nonlocal fractional Laplacian operator with \(s\in (0,1)\), \(\mu\gt 0\) is a parameter and \(\lambda\) appears as Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of the mass-subcritical perturbation \(\mu |u|^{p-2}u\) with \(2\gt p\gt 2+\frac{4s}{N}\) under some assumptions on \(\tau\).
We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schrödinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. Under suitable structural conditions on the complex coefficients, we establish the existence and uniqueness of global solutions. The analysis relies on the delicate proofs that the maximal monotone operator theory can be adapted to this framework, even for unbounded domains.
