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The main goal of this paper is the study of two kinds of nonlinear problems depending on parameters in unbounded domains. Using a nonstandard variational approach, we first prove the existence of bounded solutions for nonlinear eigenvalue…
In this paper we introduce a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. We prove certain properties of the corresponding Musielak-Orlicz Sobolev spaces (an equivalent…
We study the finite element approximation of problems involving the weighted $\Phi$-Laplacian, where $\Phi$ is an $N$-function and the weight belongs to the class $A_\Phi$. In particular, we consider a boundary value problem and an obstacle…
We consider a strongly nonlinear elliptic problem with the homogeneous Dirichlet boundary condition. The growth and the coercivity of the elliptic operator is assumed to be indicated by an inhomogeneous anisotropic $\mathcal{N}$-function.…
This paper deals with existence of positive solutions for a class of quasilinear elliptic systems involving the $\Phi$-Laplacian operator and convex-concave singular terms. Our approach is based on the generalized Galerkin Method along with…
The Musielak--Orlicz setting unifies the variable exponent, Orlicz, weighted Sobolev, and double-phase spaces. They inherit technical difficulties resulting from general growth and inhomogeneity. In this survey we present an overview of…
We prove H\"older continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This…
We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of…
This paper deals with the lack of compactness in nonlinear elliptic problems $(P)$. In particular, a domain $\Omega$ is provided where not converging Palais-Smale sequences exist at every energy level. Nevertheless, it is proved that…
We consider a class of nonlinear integro-differential operators and prove existence of two principal (half) eigenvalues in bounded smooth domains with exterior Dirichlet condition. We then establish simplicity of the principal…
The main goal is to establish necessary and sufficient conditions under which the fractional semilinear elliptic equation $\Delta^{\frac{\alpha}{2}} u=\rho(x)\,\varphi(u)$ admits nonnegative nontrivial bounded solutions in the whole space…
We consider a parabolic PDE with Dirichlet boundary condition and monotone operator $A$ with non-standard growth controlled by an $N$-function depending on time and spatial variable. We do not assume continuity in time for the $N$-function.…
We are concerned with solvability of nonlinear systems involving a discrete singular $\phi$-Laplacian operator of type \begin{equation*} u \mapsto \Delta\left[\phi(\Delta u(n-1))\right] \qquad (n\in \{1, \dots, T\}), \end{equation*}…
This paper is concerned with a nonlinear Steklov boundary-value problem involving weighted $p(.)$-Laplacian. Using the Ricceri's variational principle, we obtain the existence of at least three weak solutions in double weighted variable…
We study the question of the existence of infinitely many weak solutions for nonlocal equations of fractional Laplacian type with homogeneous Dirichlet boundary data, in presence of a superlinear term. Starting from the well-known…
A classical regularity result is that non-negative solutions to the Dirichlet problem $\Delta u =f$ in a bounded domain $\Omega$, where $f\in L^q(\Omega)$, $q>\frac{n}2$, satisfy $\|u\|_{L^\infty(\Omega)} \leq C\|f\|_{L^q(\Omega)}$. We…
We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic $N$-function, which is not necessarily of power type and need not satisfy the $\Delta_2$ nor the $\nabla _2$-condition. Fully anisotropic,…
In this paper we prove a Lions type result for a large class of Orlicz-Sobolev space that can be nonreflexive and use this result to show the existence of solution for a large class of quasilinear problem on a nonreflexive Orlicz-Sobolev…
We prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider \[\partial_t u-\mathrm{div} A(x,\nabla u)= f\in L^1(\Omega_T),\]…
The paper is devoted to the existence of positive solutions of nonlinear elliptic equations with $p$-Laplacian. We provide a general topological degree that detects solutions of the problem $$ \{{array}{l} A(u)=F(u) u\in M {array}. $$ where…