Motivation & Objective

To address the existence of nonnegative solutions for fractional $p$-Laplacian problems on unbounded domains where compactness is lost due to the unbounded domain and weight function.
To establish existence results under conditions that allow the weight $\varphi(x)$ to be positive on a bounded set and nonpositive outside, overcoming loss of compactness.
To extend existence theory to the nonlocal, nonlinear fractional $p$-Laplacian operator with critical or subcritical nonlinearities $f(u)$, including $f(u) = u^q$.
To provide a rigorous variational framework for problems with weight functions that are not strictly positive, using a concentration-compactness approach.

Proposed method

Formulates the problem as a distributional weak solution in $D^{s,p}(\mathbb{R}^N)$, requiring test functions in $C_c^\infty(\mathbb{R}^N)$.
Employs a sequence of approximating problems on expanding balls $B(0,R_n)$, with $u_n$ solving the problem in $B(0,R_n)$ and vanishing outside.
Uses the mountain pass theorem and bounded Palais-Smale sequences to construct a weak limit $u$ in $D^{s,p}(\mathbb{R}^N)$.
Applies the compact embedding $L^{p^*_s}(K) \hookrightarrow L^r(K)$ for bounded $K$ and $r < p^*_s$ to pass to a.e. convergence of $u_n \to u$.
Uses the duality pairing and dominated convergence to pass to the limit in the weak formulation, ensuring convergence of the right-hand side involving $\varphi(x)f(u_n)\psi$.
Employs the condition $(f_3)$ to control the difference $f(s)s - pF(s)$ via $Cs^m$ with $m < p$, enabling uniform integrability and compactness.

RQ1Under what conditions on the weight $\varphi(x)$ and nonlinearity $f(u)$ does a nonnegative solution exist for the fractional $p$-Laplacian problem on $\mathbb{R}^N$?
RQ2How can one overcome the loss of compactness in the fractional $p$-Laplacian setting when the domain is unbounded and $\varphi$ changes sign?
RQ3Can the existence of a nontrivial solution be guaranteed when $f(u)$ grows subcritically but not superlinearly, especially for $q < p-1$?
RQ4Is it possible to extend existence results to the fractional $p$-Laplacian with weight functions that are positive on a bounded set and nonpositive elsewhere?

The problem admits a nontrivial nonnegative distributional solution $u \in L^{Np/(N-sp)}(\mathbb{R}^N) \setminus \{0\}$ with $u \geq 0$ and $\|u\|_{D^{s,p}} < \infty$.
The solution satisfies the weak formulation: $\int_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y)) (\psi(x)-\psi(y))}{|x-y|^{N+sp}} \, dx\,dy = \int_{\mathbb{R}^N} \varphi(x)f(u)\psi \, dx$ for all $\psi \in C_c^\infty(\mathbb{R}^N)$.
The solution is nontrivial ($u \not\equiv 0$) due to the positivity of the limit of the mountain pass level $c_n \to c > 0$, contradicting $u \equiv 0$.
The result holds under the weight condition $(W)$: $\sup_{\mathbb{R}^N \setminus \Omega} \varphi \leq 0 < \inf_\omega \varphi$ for bounded $\omega \subset \Omega$.
The existence result is new even for the linear case $p=2$, establishing existence of a nonnegative solution for $1 < q < 2^*_s - 1$ in $D^{s,2}(\mathbb{R}^N)$.
The method applies when $f$ satisfies $f(s) \sim s^q$ with $p-1 < q < p^*_s - 1$ or $0 \leq q < p-1$, under appropriate growth and convexity conditions.