Aug 2026· Nonlinear Analysis· 0 citations· 13 references
Mathematics
Abstract
This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $01$ and $f\in L^{(p+1)/p}_g(\Omega)$. Under certain integrability condition on $g$, the existence of solution is proven for every $\lambda\in \mathbb{R}$. Moreover, the regularity of solution is also obtained.
This paper investigates the existence of normalized solutions to the following Schrödinger equation $$ \begin{cases} -\Delta u + [V(x) + \lambda] u = \mu |u|^{q-2}u + |u|^{2^*-2}u, & x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}u^2\mathrm{d}x = c, \end{cases} $$ where $N\ge 3,$ $2 < q < 2^*=\frac{2N}{N-2}$, $c > 0$ is a prescribed mass, and $\lambda\in\mathbb{R}$ is a Lagrange multiplier dependent on the solution $u$. Existing studies have confirmed the existence of local minimizers for the equation when $2 < q < 2+\frac{4}{N}$, but the existence of a second Mountain-Pass type solution remains unresolved. By improving variational methods and introducing a refined constraint set $A_{s_0}$, we prove the existence of a positive local minimizer $\tilde{u}_c$ for $c\in(0,c_0)$ under a mild potential assumption, with $\Phi(\tilde{u}_c) < 0$. Further, under additional assumptions on the radiality and decay of $V(x)$, leveraging the compactness of the radial Sobolev space, we construct a non-standard Mountain-Pass geometry and accurately estimate the minimax energy level, establishing the existence of a second radial Mountain-Pass type solution $u_c$ with $0 < \Phi(u_c) < m(c)+\frac{1}{N}\mathcal{S}^{\frac{N}{2}}$. Our results fill the gap in the multi-solution research of the equation for $2 < q < 2+\frac{4}{N}$, and the proposed variational framework provides a new approach for the study of constrained elliptic equations.
Xianhua Tang, Heng Yang· Advances in Differential Equ...· 0 citations
We investigate the Dirichlet problem for the variational integral $J[u] = \int_{\Omega} f(\nabla u) \, dx$ with density $f$ of linear growth satisfying appropriate ellipticity conditions. We show that the relaxed problem admits a unique solution $u$ in the space of functions of bounded variation, if the set $\Gamma_0$ of convex points $x \in \partial\Omega$ is sufficiently large. For example, the inequality $\mathcal{H}^{n-1}(\Gamma_0)>\frac{2}{3}\mathcal{H}^{n-1}(\partial\Omega)$ is sufficient. Moreover, the minimizer $u$ is smooth in the interior of $\Omega$ and attains the prescribed boundary data at least on $\Gamma_0$ in the classical sense.
In this work, we conduct a comprehensive study of problem \begin{equation*} \begin{cases} -\Delta_1 u + g(u)|Du| = h(u)f&\text{in }\Omega, u=0&\text{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^N$ is a bounded Lipschitz domain, $f\in L^1(\Omega)$ is a nonnegative datum, and $g,h$ are nonnegative continuous functions on $(0,\infty)$ that may be singular at the origin. Under the minimal assumptions that $g$ is integrable near zero and $h$ is bounded at infinity, we explore the existence of a global $BV(\Omega)$ solution. Furthermore, a comparison principle is proved under suitable monotonicity assumptions on $h$. This framework avoids any growth restrictions on $h$ near the origin, thus allowing for highly singular terms. To handle these nonlinearities, we introduce a novel approach that takes advantage of the rigid structure of the 1-Laplacian operator.
We study the uniform $L^\infty(\Omega)$ a priori boundedness of positive weak solutions to the fractional semilinear Dirichlet problem $(-\Delta)^s u = f(u)$ in a bounded, convex, $C^{1,1}$ domain $\Omega \subset \mathbb{R}^N$ with homogeneous exterior condition $u\equiv 0$ in $\mathbb{R}^N\setminus\Omega$. We consider slightly superlinear nonlinearities of the form $f(t) = t^q L(t)$, where $1 \le q \le \frac{N+2s}{N-2s}$ and $L$ is a slowly varying function. Although uniform estimates are well-established in the strictly subcritical regime $q<\frac{N+2s}{N-2s}$, the slightly subcritical case, $q = \frac{N+2s}{N-2s}$, is highly challenging due to the potential formation of bubbling profiles. In this work, we isolate a structural condition on the slowly varying perturbation, namely $$ \lim_{t \to \infty} \frac{t \, |L'(t)|}{L^{\frac{N}{2s}}(t)} = \infty, $$ which acts as an asymptotic barrier that prevents mass concentration. Under this assumption, we establish global uniform $L^\infty(\Omega)$ bounds for positive solutions, significantly expanding the class of known nonlinearities for which such estimates hold.
Juan-Carlos Felipe-Navarro, Rosa Pardo· 0 citations
In this paper we consider nonlinear biharmonic equations with $p$-Laplacian ($p\ge2$) of the form $$ \left\{ \begin{array}{l} \Delta^2 u - \Delta_p u + V (x) u = f (x, u) \text{,} u \in H^2 (\mathbb{R}^N) \text{,} \end{array} \right. $$ where the potential $V(x)$ may be indefinite. Using local linking and Morse theory, nontrivial solutions are obtained. In case the nonlinearity $f(x,\cdot)$ is odd, we obtain a sequence of large energy solutions. In the second part of the paper, for bounded positive potential, we get multiple solutions for the case that $$f(x,u)=\lambda g (x) | u |^{q - 2} u + | u |^{m - 2} u$$ with exponent $m$ critical or subcritical.
In this paper, we study isolated singularities of the following semilinear elliptic equation $-\Delta u+\frac12 x\cdot \nabla u+\frac{1}{q-1}u-u^q=0$ in $\Omega \setminus \{0\}$, where $n\ge 3$, $\Omega \subset \mathbb{R}^n$ is a domain, $0 \in \Omega$ and $q>1$. This equation arises in the study of blow-up profiles of semilinear heat equations. For $\frac{n}{n-2}<q<\frac{n+2}{n-2}$, we establish a complete classification of isolated singularities for nonnegative solutions and characterize the precise asymptotic behavior of singular solutions. Our results improve those of Guedda and Kirane (Trans. Amer. Math. Soc., 1995: 3595-3603), where analogous results were obtained only for radially symmetric positive solutions. In addition, we also derive the asymptotic behavior of solutions in the Serrin critical case $q=\frac{n}{n-2}$ and the supercritical case $q>\frac{n+2}{n-2}$.
Meiqing Xu, Hui Yang· 0 citations
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