Related papers: Concentration Phenomena for $(p,N)$-Laplace Equati…
This article deals with the following fractional $(p,q)$-Choquard equation with exponential growth of the form: $$\varepsilon^{ps}(-\Delta)_{p}^{s}u+\varepsilon^{qs}(-\Delta)_q^su+…
We prove the existence of solutions for a class of quasilinear problems involving variable exponents and with nonlinearity having critical growth. The main tool used is the variational method, more precisely, Ekeland's Variational Principle…
In this work we consider the following class of fractional $p\&q$ Laplacian problems \begin{equation*} (-\Delta)_{p}^{s}u+ (-\Delta)_{q}^{s}u + V(\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*}…
In this paper, we investigate some existence results for double phase anisotropic variational problems involving critical growth. We first establish a Lions type concentration-compactness principle and its variant at infinity for the…
We analyze the existence and multiplicity of positive solutions to a nonlocal elliptic problem involving the spectral fractional Laplace operator endowed with homogeneous mixed Dirichlet-Neumann boundary conditions and weighted critical…
We prove the multiplicity and concentration of normalized solutions of critical biharmonic equations with combined nonlinearities in $\mathbb{R}^{N}$ \begin{equation*} \Delta^{2}u+V(\varepsilon x)u=\lambda u+\mu |u|^{q-2}u+|u|^{2^{**}-2}u…
In this paper, we are interested in studying the multiplicity, uniqueness, and nonexistence of solutions for a class of singular elliptic eigenvalue problem for the Dirichlet fractional $(p,q)$-Laplacian. The nonlinearity considered…
In this paper we study the multiplicity and concentration of positive solutions for the following $(p, q)$-Laplacian problem: \begin{equation*} \left\{ \begin{array}{ll} -\Delta_{p} u -\Delta_{q} u +V(\varepsilon x) \left(|u|^{p-2}u +…
The aim of this paper is to investigate the existence, multiplicity and concentration of positive solutions for the following nonlocal system of fractional Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}…
We study the concentration phenomenon for solutions of the fractional nonlinear Schr\"{o}dinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation…
We study asymptotic behavior of sub-solutions to non-uniformly elliptic equations with nonstandard growth. In particular, Harnack type inequalities are proved. Our approach gives new results for the cases with (p,q) nonlinearity and…
This paper is concerned with the following fractional $p$-Kirchhoff equation \begin{eqnarray*} \varepsilon ^{sp}M\left( {\varepsilon ^{sp - N}}\iint_{\mathbb{R}^{2N}}\frac{{{{\left| {u(x) - u(y)} \right|}^p}}}{{{{\left| {x - y} \right|}^{N…
In the paper we show that critical sequences associated with the mountain pass level for semilinear elliptic problems on $\R^N$ converge when the non-linearity is subcritical, superlinear and satisfies the penalty condition…
In this paper, we study the existence and concentration of positive solution for the following class of fractional elliptic equation $$ \epsilon^{2s} (-\Delta)^{s}{u}+V(z)u=f(u)\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where $\epsilon$ is…
We establish existence results for a class of mixed anisotropic and nonlocal $p$-Laplace equation with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To…
In our series of papers, we establish infinite concentration and oscillation estimates for supercritical semilinear elliptic equations in discs. Especially, we extend the previous result by the author (N. arXiv:2404.01634) to the general…
In this paper, we consider the existence and multiplicity of normalized solutions for the following $p$-Laplacian critical equation \begin{align*} \left\{\begin{array}{ll} -\Delta_{p}u=\lambda\lvert u\rvert^{p-2}u+\mu\lvert…
In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem $$\left\{\begin{array}{rcl} \mathcal{L}_{\varepsilon}u = f(u) \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u>0 \ \…
We establish the local boundedness of (sub-)solutions to nonlinear kinetic diffusion equations with $p$-growth, where the kinetic p-Laplace equation is a prototypical example. A key ingredient is the derivation of kinetic…
By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical…